中山大学：高级金融理论：高级金融理论讲义（中山大学岭南学院 李仲飞）

I a0a1a2 a3a4a5a6a7a8a6a9 c?September 6, 2005 c?a0a1a2 1 September 6, 2005 September 6, 2005 2 c?a0a1a2 a10 a11 a12a13a14 a15a16a17a18a19a20a21a22a23a24 5 §1.1 a25a26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 §1.2 a27a28a29a30a31a32a33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 §1.3 a34a35a36a37a38a39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 §1.4 a40a41a42a43a44a45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 §1.5 a46a47a48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 §1.6 a36a37a49a50a51a52a31a53a54a55a56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 §1.7 a53a57a58a59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 §1.8 a60a61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 a12a62a14 a15a16a63a64a65a66a67 13 §2.1 a25a26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 §2.2 a53a68a29a69a31a70a30a30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 §2.3 a71a72a31a73a74a29a68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 §2.4 a53a35a75a76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 §2.5 a73a74a58a59a29a68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 a12a77a14 a78a79a80a81a19a20 19 §3.1 a25a26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 §3.2 a82a83a84a85a38a39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 §3.3 a86a87a88a36a37a38a39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 §3.4 a82a83a84a85a38a39 a3 a74a89a56a68a90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 §3.5 a86a87a88a36a37a38a39 a3 a74a89a56a68a90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 §3.6 a82a83a84a85a38a39 a3 a74a58a59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 §3.7 Pareto-a91a92a58a59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 a12a93a14 a94a95 27 §4.1 a25a26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 §4.2 a96a97 a6a98a99 a29a47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 §4.3 a100a68a101a102a74a103a53a30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 a12a104a14 a105a106a95a107a108a109a110a21a67a111a112a108a113a95a114 33 §5.1 a25a26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 §5.2 a89a56a68a90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 §5.3 a115a116a117a118a119a31a120a121a122a123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 §5.4 a116a117 a3 a30a124a125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 a126 a127 a126 a127 §5.5 a116a117 a3 a30a124a125a128a74a35a129a118a119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 §5.6 a116a117 a3 a30a100a68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 §5.7 a68a130a74a131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 §5.8 a29a68a132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 §5.9 a60a61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 a12a133a14 a17a18a22a134a64a65a135 45 §6.1 a25a26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 §6.2 a136a72a31a120a121a136a72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 §6.3 a120a121a136a72a31a137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 §6.4 a136a72a31a137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 a12a138a14 a139a140a141a142a22a17a18a113a95 51 §7.1 a25a26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 §7.2 a35a129a143a144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 §7.3 a116a117a145a146 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 §7.4 a55a56a147a148a149a31a148a149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 §7.5 a98a150a151a152a74a36a37a29a68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 §7.6 a153a45a154a155a128a74a36a37a29a68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 §7.7 a156a35a157a158a159a46a125a74a160a82a30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 §7.8 a60a61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 a12a161a14 a162a106a163a164a113a95a65 CAPM 57 §8.1 a165a85a166a167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 §8.2 a168a153a169ta89a56a74a53a35a36a37a38a39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 §8.3 a55a56a170a171a29a68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 §8.4 a172a173a86a174a143a144a74a55a56CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 §8.5 a34a35a38a39a118a119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 §8.6 a175a84a85a38a39 a3 a74a55a56CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 September 6, 2005 4 c?a0a1a2 a176a177a178 a179a180a181a182a183a184a185a186a187a188 §1.1 a189a190 a168a191a35a36a37a38a39a75a76 a3 a192 a36a37a193a168a35a194a195a196a53a174a197a198a199a75a76a91a200a50 a150 a36a37a74a116a117a31a118a119a45a74 a201a202a203a204 a116a117a74a58a59a205a206 a3 a74a36a37a74a207a144a74a208a209a197 a168a191a35a75a76 a3 a192 a210 a173a74a27a28a29a30a211a53a174a212a213a74a197 a214 a121a215a74a211 a192 a216a217 a27a28a29a30a211a218a219a220a212a213a74a197 a221a222 a27a28a29a30a74a212a213 a192 a46a47a48a53a174a223a53a174a224a195a196a36a37a197 a99a225a226a227a228a229a225 a74a34a35a75a76a230a231a27a28a29a30 a74a218a232a212a213 a192 a203a204a233a234a235 a173 a201 a36a37a68a90 a226a236a237 a74a238a32a33a153a230a231a36a37a74a239a195a196a197 §1.2 a240a241a242a243a244a245a246 a73a247a168a191a35a75a76 a3a248a249 a192 a168a34a35a75a76 a3 a192 a27a28a29a30a144a250a83a251 (a252 a249a99 a44a45) ?a253a254a255a197a0a53a1 a250a83a211a2 a210 a173a153a169t = 0,1,...,T a74a3a4a5a6a74a53a1a7a8a197a168a153a1690 a192 a46a47a48a27a166a9a10a53a1a250a83a11 a12 a215a121a197a13a211 a221a222 a153a45a74a14a15 a192a16a17 a234a235a18 a253 a18 a34a74 a201a150 a250a83a74a32a33a197a168a153a169 T a192a16a17 a11a166a9a19 a215a74a250a83a197 a73a88a224 a192 a46a47a48a168a153a169ta74a32a33a251a144a250a83a251?a74a53a1a155a20Ft a253a7a8a197a53a1a155a20 Ft a21 a211?a74 a53a1a123a251a22 a192 a198a23a123a251a24a24a27a25a195a26a27a28a29a30a31a1 a249a99 a44a45 a192a32a33a34 a254 a192 a0a1a250a83 ω ∈ ?a35 a150 a26 a36 a35 a150F t a74a53a1a37a38a197a39a212a40a211a41a168a153a169 ta46a47a48a166a9a19a215a250a83a35 a150F t a74a10a1a37a38a192a13a27a166a9 a198a1a37a38 a3 a74a10a1a250a83a211a19a215a250a83 a192a42a43 a166a9a27a35 a150 a198a1a37a38a74 a248 a23a250a83a11a27a27a215a121a197 a44a45 1.1 a46a47a48a49a50a51a52a53a54a55a56a57a58a52 a192a59 a50a51a53a54a55a60a51a57a58a52a61a62a197 ? a168a153a1690a192a46a47a48a63a173a250a83a74a64a65a32a33 a192a66 a153a169 -0a155a20a67a68a69a155a20F0 = {?,?}. ? a168a153a169T a192a46a47a48a70a173a84a71a32a33 a192 a122 a43 a153a169 -T a155a20a67a72a155a20 FT = {{ω} : ω ∈ ?}. ? a168a153a1691,...,T ?1a192a46a47a48a70a173a74a32a33a73a74 a150a75 a24a76 a3 a45a77 ? Ft+1 a78 Ft a79a80 (a27a53a29a81a90 a79a80 )a192a252a53a1a250a83a210a168a74a153a169 -(t+1)a155a20a74 a248 a1a37a38a211a82 a210 a168 a74a153a169-ta155a20a74 a248 a1a37a38a74a123a251a197a83a68a224 a192 a247a84a24a1a250a83a35 a150 t a153a169a155a20a74a24a1a27a85a37a38 a192 a86 a82 a17 a27a154a27a35 a150ta203a227 a64a65a153a169a155a87a74a85a53a37a38a77 ? a88a89a46a47a48a90a91a27a27a92a61 a16a17a93 a3a166a9a74a32a33 a192 a252 a16a17 a201a150 a250a83a74a32a33a211a175a94a74a197 ? a155a20a74T + 1a37a49{F0,F1,...,FT}a95a67a32a33a96a123 a192 a61a67F. ? a168a173a97a1a250a83a74a98a99 a192 a32a33a96a123a74a100a53a1a101a102a211a103a104a105(a106a1071.1) – a155a20Ft a74a0a1a37a38a95a67a53a1a153a169-ta89a56 a192 a203a204 a89a56a108a74a53a1a109 a110 a192 a61a207ξt; – a89a56ξ0 = F0 a95a67a111a109 a110; – a89a56ξt a74a112a53a211a89a56ξτ ? ξt, τ > t. a89a56ξt a74a113a53a211a89a56ξt+1 ? ξt. – a89a56ξt a74a114a115a211a89a56ξτ ? ξt, τ < t. a89a56ξt a74a116a115a211a103a53a74 a192 a67a89a56ξt?1 ? ξt,a61a207ξ?t . ? a168 a210 a173a74a11a253a153a169 t = 1,...,T a74 a210 a173a89a56a117a43a74a251a50a61a67 Ξ. a118a61 k = sharp(Ξ) a67 Ξ a3a74a89a56 a74a119a73a197 a150 a211a120a121a122a123a124a74a89a56a119a67 k + 1. 5 1.2a125a126a127a128a129a130a131 a132a133 a134 a135a136a137a138a139a140a141a142a143a144 a145 a145 a146 a147 a148 a148 a149 a150 a150 a151 a152 a153 ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9 ω10 ω11 ω12 ω13 ω14 ω15 ω16 ω17 ω18 ω19 ω20 ω21 ω22 a45 t a153a1690 a153a1691 a153a1692 a153a1693 a1541.1: a155a156a157 September 6, 2005 6 c?a158a159a160 a132a133 a134 a135a136a137a138a139a140a141a142a143a144 1.2 a125a126a127a128a129a130a131 a161 1.2.1 a162a163a57a164a61a62 a165 a56a166a167a168a169a52a170a171a172a173a197 a174 a167a172a173a175a176 a177a178 (G) a192 a175a176 a177a179 (B) a197 a180 a167a168a169 a181a182a1831 a184a185a172a173 a192a186 a180 a167a168a169 a181a182a1832 a184a185a172a173 a192 a187a188a189a190? a191a166a167a172a173a52a192a193 a194a195a196a197a198a199 a41 ξgg ξgb ξbg ξbb ξg ξb ξ0 a1541.2: a155a156a157 ? = {GG,GB,BG,BB} a61a62 a200 a113a201 F0 = {?, ?} F1 = {{GG,GB},{BG,BB}} F2 = {{GG},{GB},{BG},{BB}} a181a182a183 0 a192 a53a54a55a202a176 a203a204a205a206 a192 a181a182a183 1 a205a206 a168a1691 a52a170a171a172a173 a192 a181a182a183 2 a205a206 2 a167a168a169 a52a170a171a172a173a197 a182a183 1 a52a103a104a51 ξg = {GG,GB}, ξb = {BG,BB} a182a183 2 a52a103a104a51 ξgg = {GG},ξgb = {GB},ξbg = {BG},ξbb = {BB} a50a51a207a208a103a104 a190 a201 Ξ = {ξg,ξb,ξgg,ξgb,ξbg,ξbb} ? a46a47a48a201a150a250a83a74a32a33a209a73a210a224a211a212a168a210a173a3a4a213a73a3a192a247a214a215a108a36a37a68a90a108a216a72a108a36a37a49a50 a217 a173 a108a151a152a218 a20 a192 a83a83a197a172a219a253a254 a192 a247a84a27a27 a98a150 a153a169ta46a47a48a154 a234a235 a74a32a33a220a155a221a23a250a83 a192 a248a222a223a224 a153a169 t a198a23a250a83a128a74 a151a152a218 a20a225a36a37a68a90a211a63a173a226a227a74a197a254a255a198a23a213a73a74a53a1a148a228 a211 a192a229 a82 a17a230a231 a43a250a83a251?a232a74a102a119 a192 a118a88a89a82 a17 a201a150 a155a87Ft a211 a194a233 a52. ? a29a227a41a247a84ta153a74 a151a152 a144a102a119ct : ? →Ra230a231a74 a34a192 a248a222c t a201a150F t a74a154a234a30a211a235 ct a168a0a1 a89a56ξt ∈Ft a232a211a210a119 a192 a252 ω,ωprime ∈ ξt ? ct(ω) = ct(ωprime) a236 a17a229 a198a1a237a85a74a130a61a207ct(ξt),a238a61a67c(ξt). a128a85a197 ? a144ct a230a231a2 a210 a173ξt ∈Ft, a239c(ξt)a49a43a74a240a73 a192a241 a119a83 a150F t a3 a74a89a56a119sharp(Ft). c?a0a1a2 7 September 6, 2005 1.3a135a136a137a138a139a140 a132a133 a134 a135a136a137a138a139a140a141a142a143a144 – a198 a249 a192 a236 a17 a144a85a53a242a243 ct a230a231 a207a67Ft a154a234a102a119 a203a204 a207a67a240a73a74 a151a152a218 a20a197 – a244a245a224 a192 a144 c a230a231Ft a154a234a102a119 ct a74 T + 1 a37a49{c0,c1,...,cT}, a85a153 a230a231 k + 1 a241 a240a73 (c(ξ) : ξ ∈ Ξuniontext{ξ0}) ? a29a227a41a247a84T + 1a37a49c a3a74a0a53a1a102a119ct a211Ft a154a234a74a192a248a222a95ca246a247 a150 a32a33a96a123F. §1.3 a248a249a250a251a252a253 ? a217a254a255J a1a0a1 a192 a0a1a2a3a4a120a5a6a1a7a8a9a7a10a11a7a10a12a13 ? a14a15a0a1a16 a17a255a18 a15a19a20a21a22a23a24a25a20a26a13 ? a24a25a27a28a0a1a29a2a30a2a11a31a2a23a32a33a21a22a13a34a8a9a35a24a25a36a27a37a38a39a8a40a23a41a42a25a43a13a34a6 a1a35a24a25a36a27a44a9a21a22a45a46a47a10a21a22a13 ? a48xj(ξt)a49a50a0a1j a255a51a52ξt a23a24a25 a48x(ξt)a49a50J a15a0a1 a255a51a52ξ t a23a24a25a53 a54 a35a55x(ξt) = (x1(ξt),...,xJ(ξt)) a48xjt a49a50a0a1j a255a56a2ta19a20 a51a52ξ t a23a24a25xj(ξt)a57a58a23a53 a54a59 xjt = (xj(ξt) : ξt ∈Ft) a48xt a49a50 a56 a2J a15a0a1 a255a56 a2ta19a20 a51a52 a23a24a25a57a58a23a53 a54 xt = (x1t,...,xJt) ? a255a19a200a60a2a24a25a35 ? a2a61a62a63a15a0a1a64 a255a65 a15a19a20a2a66a67a24a25a13a3a68 a59a255 a19a20ta47a10a23a35 a69a70a711 a23a67a44a6a1a35 a255 a14a15a19a20ta51a52a23a24a25a72a731a35a74 a255a75a17 a19a20a24a25 a710 a13 ? a0a1 a255a76a77a78a79 a19a20T a80a23 a56 a2a19a20a81a82a13 ? a48pj(ξt)a49a50a0a1j a255a51a52ξt a23a83a84a35t = 0,...,T a48p(ξt)a49a50J a15a0a1 a255a51a52ξ t a23a83a84a53 a54a59 p(ξt) = (p1(ξt),...,pJ(ξt)) ? a85a73a86a87a88a23a89a90a35 a91a92 a2a19a20T a23a83a84p(ξT),a93a94a81a82 a255 a19a20T a95a96a97a98 a99a100 a83a84a101 a710. a48pjt a49a50a0a1j a255a56a2ta19a20 a51a52ξ t a23a83a84pj(ξt)a57a58a23a53 a54 pjt = (pj(ξt) : ξt ∈Ft) a48pt a49a50 a56 a2J a15a0a1 a255a56 a2ta19a20 a51a52 a23a83a84a57a58a23a53 a54 pt = (p1t,...,pJt) ? a48hj(ξt)a49a50a0a1j a255a51a52ξt a23a29a2 a54 a98 a48h(ξt)a49a50J a15a0a1 a255a51a52ξ t a23a0a1a102a103a35 a71 a53 a54 h(ξt) = (h1(ξt),...,hJ(ξt)) September 6, 2005 8 c?a104a105a106 a107a108a109 a110a111a112a113a114a115a116a117a118a119 1.4 a120 a121a122a123a124a125 – a14a15a0a1a23a29a2 a54 a61a45a27a126a23a35a67a35a127a128a23 (a76a66a2a129a130a131a132)a35 – a133a134a35a85a73a86a87a88a23a89a90a35a19a20 T a102a103h(ξt)a101 a71 a67a13 a48ht a49a50 a255a56 a2a19a20ta51a52ξt a23a0a1a102a103h(ξt)a57a58a23a53 a54 ht = (h(ξt) : ξt ∈Ft) T + 1a135a102h = {h0,...,hT}a136 a71 a0a1a102a103a137a138a13 ? a63a15a0a1a102a103a137a138ha255a51a52ξt a23a21a22(a139a140a141a85)a35 a142a71z(h,p)(ξ t),a27a143a102a103 a255a144a145a146ξ? a23 a147 a24a25a23a21a22a148a149 a255 a143 a145a146 a23a83 a70 ( a127a30a150 a255a145a146ξ t a81a82a151a23a83 a70 a148a149a81a82a152a23a83 a70) a35a55 z(h,p)(ξt) ≡ [p(ξt) + x(ξt)]h(ξ?t )?p(ξt)h(ξt),t = 1,...,T a99a153a154 a15a53 a54a155a156a157a158a71 a34a159a37 a54a155a156a160a155a161 a13 ? a48zt(h,p)a49a50 a255a56 a2a19a20-ta51a52ξt a23a21a22z(h,p)(ξt)a57a58a23a53 a54 zt(h,p) = (z(h,p)(ξt) : ξt ∈Ft) ? a0a1a102a103a137a138ha255a19a200a23a83 a70a71p(ξ 0)h(ξ0) ? a162 1.3.1 – a163a164a165a166a167a168a169a170ha59a171a172a173t ≥ 1,a171a174a175ξt a176a177 1a178a165a166j,a171ξt a179a180a181a182 a183a184a185a186a187a188 a35a189 hj(ξt) = 1, hj(ξ) = 0, ?ξ negationslash= ξt; hi(ξ) = 0, ?ξ,i negationslash= j z(h,p)(ξt) = ?pj(ξt) z(h,p)(ξt+1) = pj(ξt+1) + xj(ξt+1), ?ξt+1 ? ξt z(h,p)(ξ) = 0, ?ξ negationslash∈ Ξ\({ξt}∪{ξt+1 : ξt+1 ? ξt} – a190a191a192a29a2a137a138: a171a174a175a193 a179a180a181a182 a174a175a194a195 a181 a178a165a166 j, hj(ξt) = 1, ?ξt ∈Ft,t = 0,1,...,T ?1 hj(ξT) = 0, ?ξT ∈FT hi(ξ) = 0, ?ξ,i negationslash= j z(h,p)(ξt) = xj(ξt), ?ξt ∈Ft,t = 0,1,...,T z(h,p)(ξ0) = ?pj(ξ0) ? a19a20ta23a24a25xjt,a83a84pjt,a102a103ht,a21a22zt(h,p)a196a27Ft a61a197a198a199a13 §1.4 a200a201a202a203a204a205 ? a206a207a208a209a210a211a88a23a81a82a61a212a213a23a21a22a57a58a23a214a103a136 a71a215a216a217a218a219a220 a35a221a222 a71 M(p) = {(z1,...,zT) ∈Rk : zt = zt(h,p) for some h,t = 1,...,T} c?a104a105a106 9 September 6, 2005 1.5a223a224a225 a107a108a109 a110a111a112a113a114a115a116a117a118a119 ? a2261.3.1 a227a208a209a102a103a137a138a23a21a22a228a229a230a73a231a232a97a58a130a233a13 a234a235 a35a24a25 (xj1,...,xjT) ∈M(p),j = 1,...,J, ?p. ? a154a10a236a237a238a239a10a236a237a23a63a15a240a241a242 a235a59 a243a154 a10a236a237a227a35a231a232a97a58a130a233a27a80a97a23a35a244a245a246a73a28a221a23a208a209a21a22a98 a243 a239a10a236a237a227a35a231a232a97a58a130a233a27a247a97a23a35a245a246a73a208a209a83a84a98 ? a208a209a210a211a136 a71 a27a248a249a250 a251 a179 (a245a83a84p)a35a68a252a32a33a253a254a19a20(a19a201a47T)a23a255a0a1a2a61a3 a71 a4 a15a208a209a102a103a137a138a23a21a5a254a212a213a35a55 M(p) = Rk. a208a209a210a211a136 a71 a27a6a250 a251 a179 a35a68a252M(p)a27Rk a23a7a8a130a233a13 §1.5 a9a10a11 a2431.2a145 a227a35a12a221a222 a77 a255a0a23a13 a54c(ξ t),ct,c. a14a15 1.2 a16a17a18 a195a19a20a171a21a22a23a24 c = (c 0,c1,...,cT) a25a26a168 Rk+1 a27a179a28a29a30a31 a35a32a33a34a35a36 a179a37a38a39a179 a13 a206a40a41a42a101a255a0a27a126a23a35a43a19a35a44 a157a45 a23a46a48a198a199 a71 ui : Rk+1+ →R. a44 a157a45i a23a47a48 a71wi = (wi 0,...,wiT) ∈R k+1 + . a49a59 ? a46a48a198a199ua243a19a20ta27a50a51a23a35a68a252a34?(c0,c1,...,cT),a244a241cprimet ≥ ct a35a36a52 u(c0,...,cprimet,...,cT) ≥ u(c0,...,ct,...,cT). ? ua27a50a51a23a35a68a252ua243a14a15a19a20a196a27a50a51a23a13 ? ua243a19a20ta27a53a84a50a51a23a35a68a252a34?(c0,c1,...,cT),a244a241cprimet > ct a35a36a52 u(c0,...,cprimet,...,cT) > u(c0,...,ct,...,cT). ? ua27a53a84a50a51a23a35a68a252ua243a14a15a19a20a196a27a53a84a50a51a23a13 §1.6 a54a55a56a57a58a59a60a61a62a63a64 ? a63a15a46a48a198a199 a71u a23a44 a157a45 a23a255a0-a208a209a102a103a65a66a67a68 a71 max u(c) (1.6.1) s.t. c(ξ0) = w(ξ0)?p(ξ0)h(ξ0) (1.6.2) c(ξt) = w(ξt) + z(h,p)(ξt), ?ξt,t = 1,...,T (1.6.3) c ≥ 0 (a68a252a52a43a131a132a23a69) September 6, 2005 10 c?a104a105a106 a107a108a109 a110a111a112a113a114a115a116a117a118a119 1.6a112a113a70a71a72a73a74a108a75a76a77 ? a78a79a80a81 (1.6.2) a238 (1.6.3) a82a159 a71≤ a237a95a72a83a35a84a16a73a46a48a198a199a85a42a101 a71 a50a51a23a35a86a61a87a58 a72a83a13 ? a78a79a80a81(1.6.2)a238(1.6.3)a61a37 a235 a87a58 c0 = w0 ?p0h0 (1.6.4) ct = wt + zt(h,p), t = 1,...,T (1.6.5) ? a67a68(1.6.1)a23Lagrangea198a199 a71 L(c,h,λ) = u(c) + λ(ξ0)[w(ξ0)?p(ξ0)h(ξ0)?c(ξ0)] + Tsummationdisplay t=1 summationdisplay ξt∈Ft λ(ξt)[w(ξt) + z(h,p)(ξt)?c(ξt)] a75 a227λ(ξt)a71a155a159a73a78a79a80a81(1.6.3)a23Lagrangea156a8a13 ? a67a68(1.6.1)a23a63a88a89a241a90 a52a71 ?L ?c(ξt) = 0, ?ξt,t = 0,...,T ?L ?h(ξt) = 0, ?ξt,t = 0,...,T a55 ?u ?c(ξt) ?λ(ξt) = 0, ?ξt,t = 0,...,T (1.6.6) λ(ξt)p(ξt) = summationdisplay ξt+1?ξt [p(ξt+1) + x(ξt+1)]λ(ξt+1) (1.6.7) ? a68a252ua27a91a92a23a35a93a88a94a90 a52a95 a78a79a80a81a41a27a96a37a23a13 ? a68a252a42a101?u/?c(ξt) > 0,a93a90 a52(1.6.7) a97a58 p(ξt) = summationdisplay ξt+1?ξt [p(ξt+1) + x(ξt+1)] ?u/?c(ξt+1)?u/?c(ξ t) a253 a75 a87a58a37 a54 a23a98a83 a71 pj(ξt) = summationdisplay ξt+1?ξt [pj(ξt+1) + x(ξt+1)] ?u/?c(ξt+1)?u/?c(ξ t) , j = 1,...,J a88a83a49a99a35a208a209 j a243a145a146ξt a23a83a84a72a73a208a209 j a243ξt a23a8 a145a146 ξ t+1 a23 a147 a24a25a23a21a22 a156 a45 a243a145 a146ξ t+1 a23a255a0 a95a243a145a146ξ t a23a255a0a100a233a23a101a102a103a44a104a152a105a73 ξt a23 a56 a52a8 a145a146a106 a238a13 ? a243a32a33a19a20a23a208a209a83a84 a95a107 a63a19a20a23a208a209a21a22a100a233a23 a99 a15a105a73 a243 a239a108a236a237a238 a154 a108a236a237a227a27a63 a134a23a13 c?a104a105a106 11 September 6, 2005 1.7 a108a109a118a119 a107a108a109 a110a111a112a113a114a115a116a117a118a119 §1.7 a61a110a111a112 ? a239a108a208a209a210a211a227a23a113a114a16a63a15a208a209a83a84a53 a54 p. a63a115a208a209a102a103a137a138 {hi}a238a63a115a255a0a1a2{ci} a102a58a35a116a117 1. a208a209a102a103a118a137hi a238a255a0a1a2 ci a27a44 a157a45i a23a245a83a84pa23a65a66a67a68a23 a158a119 2. a210a211a85a120a35a55 summationdisplay i hi = 0 (1.7.1) summationdisplay i ci = summationdisplay i wi (1.7.2) ? a206a207a253a18a44a157a45a23a78a79a113a114a80a81a155a161a35a61a121 (1.7.1)?(1.7.2) – a68a252a60a52a122a123a23a208a209(a55a59 z(h,p) = 0 ? h = 0),a93(1.7.2)?(1.7.1) – a68a252a124 a243 a122a123a23a208a209a35a93a34a159a73a116a117(1.7.2)a23a255a0a1a2a102a23 a56 a52a208a209a102a103a137a138a102a227 a125a126 a52a63a15a116a117(1.7.1)a35a55a210a211a85a120a13 ? (1.7.1) a49a50a208a209a27a67a127a159a23a13a208a209a29a52 a54 a159a128 a71a129 a81a82 a54 a13a68a252a14a15a208a209a27a126a127a159a23a35a101a14 a15a44 a157a45i a23a130a131a208a209a102a103 a71?hi 0,a84 a71a132a133a134a135 a42a101a60a52a130a131a255a0a47a48a35a93a43a19a35 a136a137 a208a209a102 a103a137a138 ˉhi a23a210a211a138a120a90 a52 a159 a71 summationdisplay i ˉhi(ξt) =summationdisplay i ?hi0, ?ξt, a68a252a253hi a157a158a71a129a81a82a59 hi = ˉhi0 ??hi0,a93a88a83 a95(1.7.1) a63a139a13 §1.8 a140a141 ? a142a143 a158 a118a95a144a221a145a23 a51a52a146 a236a237a27a95a147a48a23a35a228 – a19a233a27a148a149a23a13 – a150a151a214a27a152a153a23a13 ? a243a148a149a19a233a101a154 a107 a35 a107 a19a20 t a44 a157a45 a23a155a44a48 a51a52 a23 σ- a44a199 (σ- a156) a254a157a94a35a74a95a27a48a37a2 a254a157a94a13 September 6, 2005 12 c?a104a105a106 a158a159a160 a161a162a163a164a165a166a167 §2.1 a168a169 a95a243a65 a108a236a237a63a134a35 a243 a239a108a236a237a227a35a253a254a21a22 a95a17a92 a23a228a151a83a84a100a233a23a105a170a52 a154 a15 a234a235 a240a241 a23a145a171 a59a172 a145a145 a95 a126a145a13 §2.2 a61a173a174a175a60a176a177a177 ? a239a108a210a211a227a23a63a83a221a178 a59 a32a33 a154 a15a179a52 a155 a133a21a22a23a208a209a102a103a137a138a35a52 a155 a133a23a19a20 0a83a84a35a55 z(h,p) = z(hprime,p) ? p0h0 = p0hprime0 (2.2.1) ? a90a52(2.2.1)a72a83a73 z(h,p) = 0 ? p0h0 = 0 a99 a27a90 a71 a35a180a181a221a222a35z(h,p)a27ha23 a172 a145a198a199a13 ? a68a252a44 a157a45 a23a46a48a198a199 a243 a19a20 0a53a84a50a51a35a93a63a83a221a178 a243 a113a114a19a58a182a13 (a183a208a99) ? a63a83a221a178a58a182a23a184a63a15a96a37a90 a52 a27 a59 ( a185a208a99) 1. a124a243a179a52a126a23a66a67a21a22a23a208a209a102a103a137a138a35 2. a46a48a198a199 a243a186 a15a21a22a95 a71 a67a23a32a33a19a20a27a53a84a50a51a23a13 ? a42a101a63a83a221a178a58a182a13 ? a21a22a221a83a187a198a27a188a189q : M(p) →R, q(z) = p0h0, ?z ∈M(p), a75a227ha190a213z = z(h,p) ? a191a229a97a58a21a22z a23a208a209a102a103a137a138ha61a62a95a192a63a13a84a63a83a221 a157a193 a208a35 a56 a52 a99a100 a137a138a23a19a200a83 a84 p0h0 a27 a155 a133a23a13 a99 a134a21a22a221a83a187a198a27a52a221a222a23a13 ? a21a22a221a83a187a198q a39a14a15a21a22a28a38 a77 a97a58 a99 a15a21a22a23a208a209a102a103a137a138a23a19a20 -0a83a84a13 ? q a27M(p)a88a23 a172 a145a187a198 a59 q(z + zprime) = q(z)+ q(zprime), ?z,zprime ∈M(p) a208a99 a59 ∵ z,zprime ∈M(p), ∴ ?h,hprime a190a213 z = z(h,p), zprime = z(hprime,p). a16a73z(h,p)a27 a172 a145a23a35 z + zprime = z(h,p)+ z(hprime,p) = z(h + hprime,p). a90a43 q(z + zprime) = p0(h + hprime)0 = p0h0 + p0hprime0 = q(z) + q(zprime). 13 2.3a194a195 a74a196a117a197a198 a107a199a109 a110a111 a194a195 a74a196a200 ? a180a181a2261.3.1,a14a15a208a209a23a24a25a16a63a15a190a191a192a29a52a208a209a102a103a137a138a97a58a13a90a43xj ∈M(p), ?p . a74 a99 a15a190a191a192a29a52a208a209a102a103a137a138a23a19a20 -0a83a84a27pj0,a73a27 q(xj) = pj0. a221a83a23a138a252 a95a201 a102a63a139a13 §2.3 a202a203a60a204a205a174a173 ? a239a108a210a211a227a23a63a15a206a207a208a27 a99 a134a63a15a208a209a102a103a137a138h,a17a179a52a126a23a21a22z(h,p)a238a53a84a128a23a19 a200a83a84p0h0. ? a63a15a207a208a27 a99 a134a63a15a208a209a102a103a137a138a35 a17 a241a209a27a63a15a210a211a25a35a241a209a179a52a126a23a66a67a21a22a238 a71 a67 a23a19a20-0a83a84a35a55 p0h0 ≤ 0, z(h,p) ≥ 0 a212 a75 a227 a125a126 a52a63a15a53a84a95a72a83a98 ? a210a211a25 ? notdblarrowleft a211a25 ? a162 2.3.1 a163a164a2131.2.1. a214a215 a195 a181 a165a166 a171a174a175 ξ gg a216 ξgb a179a217 a208a218 1a35 a171a186 a32a219a218 0a13 a220 a165a166 a171 a172a1730 a221a222a34 a195a223a224 a179 a35 a225a171a172a173 1 a226a227 a218a228a223a224 a179 a13 a229a230a220 a165a166 a179a231a232 a218 p(ξ0) = 0, p(ξg) = ?1, p(ξb) = 0, a233a234a35 a171 ξ g a176a177 a220 a165a166a235 a171a236 a182a237 a183a184a238 a218a239 a194a195a240a241a242a243a244a245a246a247a248 h a34 a242a249 a207a208 a225a250 a34 a242a249 a206a207a208 a240a251 a218 ha252a253a254 -0 a231a232 p0h0 = 0 ha255 ξg a252a0a1 z(h,p)(ξg) = ?p(ξg) = 1 ha255 ξb a252a0a1 z(h,p)(ξb) = 0 ha255 ξgg a252a0a1 z(h,p)(ξgg) = x(ξgg) = 1 ha255 ξgb a252a0a1 z(h,p)(ξgb) = x(ξgb) = 1 ha255 ξbg a252a0a1 z(h,p)(ξbg) = 0 ha255 ξbb a252a0a1 z(h,p)(ξbb) = 0 a2z(h,p) = (1,0,1,1,0,0) > 0. ? a3a4a5a210a211a6? [a7a8a9a10a11a12a13a14a15h, z(h,p) ≥ 0 ? p0h0 ≥ 0] a10a16 a17 ?) a18z(h,p) ≥ 0. a19p0h0 < 0a240a20ha21a22a23a6a24a25a26a27a28p0h0 ≥ 0. ?)a19a4a5a22a23a6h,a20p0h0 < 0,z(h,p) ≥ 0,a29a30a31a32a33a34a18a35a36a24 ? a3a4a5a23a6? [?h, z(h,p) > 0 ? p0h0 > 0] a10a16 a17 ?) a18z(h,p) > 0. a19p0h0 < 0a240a20ha21a22a23a6 a240a37a38 a21a23a6a39a19 p0h0 = 0a240a20 ha21 a40 a23a6a24a25a26a41a28p0h0 > 0. ?)a19a4a5a23a6h,a20 p0h0 ≤ 0, z(h,p) ≥ 0, a42a43a44 a45a46 a28 a40a47a48a49 a3a50a51a24 a25a26 a240 a19 z(h,p) > 0, a20a52a53a51a28 p0h0 ≤ 0 a240a29a30a31a32a33a34a18a35a36a39a19 z(h,p) = 0, a20a52a40a54 a55a56 a28p0h0 = 0a240a29a30 a53 a51a35a36a24 September 6, 2005 14 c?a57a58a59 a60a61a62 a63a64a65a66a67a68a69 2.4 a70 a64a71a72 ? a55a73a17a74a75a55a54a76a77q a78a21a79a80a81 a240a82a83 q(z) ≥ 0, ?z ≥ 0, z ∈M(p); a78a21a79 a48a49 a80a81 a240a82a83 q(z) > 0, ?z > 0, z ∈M(p). ? q a21a48a49a80a81?a3a4a5a23a6 ?notdblarrowleft a3a4a5a22a23a6?q a21a80a81a24 ? a84a85 2.3.2 a0a1a86a87a88a89a90a91a92a93a252?a250a94a255a95a96a97 a10a16 a17 a3a4a5a23a6? [z(h,p) > 0 ? q(z(h,p)) = p0h0 > 0] ? q a79 a48a49 a80a81a97 ? a84a85 2.3.3 a0a1a86a87a88a89a90a93a252?a250a94a255a98a95a96a97 a10a16 a17 a3a4a5a22a23a6? [z(h,p) ≥ 0 ? q(z(h,p)) = p0h0 ≥ 0] ? q a79a80a81a24 ? q a21 a48a49 a80a81 ? notdblarrowleft q a79a80a81a24 a99 2.3.4 a100 2.3.1 a101a252a102a103 a243a244 a252a0a1a86a87a88a89a104 q(z) = 0, ?z ∈M(p) a105 a90 a251 a104a253a2540a252 a243a244 a87a92a1040 a97 a105a249 a239a88a89a90a93a252a106 a250 a90a91a92a93a252 a240a107a108a250a94 a255a109 a110 a98a95 a96a97 §2.4 a111a112a113a114 ? a115a116a117a118 a52 a115 a47a40 a116a117a118a12a119a97 ? a120a116a117a118a44a81(a22)a23a6a121a122a123a124a125a126a115a116a117a118a81a8a127a128a129a130a131a132a97 ? a5a133a134t < T a81a131a132ξt a135 a81 a40 a116a22a23a6a79 a40a47 a10a11a12a13 h(ξt)a240a136a137a28a80a81 a40 a116 a74a75 [p(ξt+1) + x(ξt+1)]h(ξt) ≥ 0, ?ξt+1 ? ξt a138 a48a49a139 a81 a54a49 p(ξt)h(ξt) < 0 ? a5a133a134 t < T a81a131a132 ξt a135 a81 a40 a116a23a6a79 a40a47 a10a11a12a13 h(ξt) a240a136a140a141a79 a40a47a40 a116a22a23a6 a240a140 a141a137 a28a80a81a128a142 a40 a116 a74a75 a138 a142 a54a49 a97a50 a54a143 a240 p(ξt)h(ξt) ≤ 0 [p(ξt+1) + x(ξt+1)]h(ξt) ≥ 0, ?ξt+1 ? ξt a42a43a44 a45a46 a28 a40a47a48a49 a3a50a51a119a144a97 ? a5a145 a47 a128a129a130a131a132a3a4a5 a40 a116a23a6?a3a4a5a115a116a23a6 a10a16 a17 ?) a146a147a148a149a150a151a81a115a116a152a23a6a81a31 a140a153 a132 a240a154a155 a10a16 a82a156a157a158 a119a144 z(h,p) > 0 ? p0h0 > 0 c?a57a58a59 15 September 6, 2005 2.4 a70 a64a71a72 a60a61a62 a63a64a65a66a67a68a69 a159z(h,p) > 0a240a20 x(ξT)h(ξ?T ) ≥ 0, ?ξT (2.4.1) [p(ξT?1)+ x(ξT?1)]h(ξ?T?1) ≥ p(ξT?1)h(ξT?1), ?ξT?1 (2.4.2) ... [p(ξ1)+ x(ξ1)]h(ξ0) ≥ p(ξ1)h(ξ1), ?ξ1 ? ξ0 (2.4.3) a42a43a44 a45a46 a28 a40a47a48a49 a3a50a51a119a144a97 a52 a126a5a145 a47 ξT?1 a3a4a5 a40 a116a23a6 a240(2.4.1) a160a161 p(ξT?1)h(ξT?1) ≥ 0, ?ξT?1 (2.4.4) a37a38a52(2.4.2) a28 [p(ξT?1) + x(ξT?1)]h(ξ?T?1) ≥ 0, ?ξT?1 a162 [p(ξT?1) + x(ξT?1)]h(ξT?2) ≥ 0, ?ξT?2, ?ξT?1 ? ξT?2 (2.4.5) a52 a126a5a145 a47ξT?2 a3a4a5 a40 a116a23a6 a240a53 a51a160a161 p(ξT?2)h(ξT?2) ≥ 0, ?ξT?2 (2.4.6) a163 a26a164a165 a240 a28 p(ξ0)h(ξ0) ≥ 0 a52 a126(2.4.1)～(2.4.3) a44 a45a46 a28 a40a47a48a49 a3a50a51 a240a53 a51 a40 a55 a21 a48a49 a3a50a51 a240a162p(ξ 0)h(ξ0) > 0,a166 a162p 0h0 > 0. ?)a167a10a168a24a19a169a170a3a171a240a20a4a5a172a131a132 ξt, (t < T)a173a174a5ξt a4a5a40a116a23a6h(ξt),a162 p(ξt)h(ξt) ≤ 0 [p(ξt+1) + x(ξt+1)]h(ξt) ≥ 0, ?ξt+1 ? ξt a43a44 a45a46 a28 a40a47a48a49 a3a50a51a119a144a24 a175a176 a40a47 a10a11a12a13a14a15ha17a5ξt a177 a28a12a13h(ξt),a38a5a43a178 a131a132 a177 a28a142a12a13a24 a20a179t > 0 a133 a240 a28 p(ξ0)h(ξ0) = 0 z(h,p)(ξt+1) = [p(ξt+1) + x(ξt+1)]h(ξt) ≥ 0, ?ξt+1 ? ξt z(h,p)(ξt) = 0?p(ξt)h(ξt) ≥ 0 z(h,p)(ξ) = 0, ?ξ ∈ Ξ\({ξt}∪{ξt+1 : ξt+1 ? ξt}) a42a43a44 a45a46 a28 a40a47a48a49 a3a50a51a24a29a30a3a4a5a115a116a23a6a35a36a24 a179 t = 0 a133 a240 a28 p(ξ0)h(ξ0) ≤ 0 z(h,p)(ξ1) = [p(ξ1) + x(ξ1)]h(ξ0) ≥ 0, ?ξ1 ? ξ0 z(h,p)(ξ) = 0, ?ξ ∈ Ξuniondisplay ξ1?ξ0 {ξ0,ξ1} a42a43a44 a45a46 a28 a40a47a48a49 a3a50a51a24a29a180a181a30a3a4a5a115a116a23a6a35a36a24 September 6, 2005 16 c?a57a58a59 a60a61a62 a63a64a65a66a67a68a69 2.5a68a182a183a184a185a186 ? a5a145 a47 a128a129a130a131a132a3a4a5 a40 a116a22a23a6 ? notdblarrowleft a3a4a5a115a116a22a23a6a97 a10a16 a17 a30a148a164a187a97 a167a188 a17a189a190 a1882.3.1,a5ξg a4a5 a40 a116a22a23a6 a240a191 a3a4a5a115a116a22a23a6 (a25a21 a74a75 a76a77 a79a80a81)a97 §2.5 a192a193a194a195a196a197 a198a199a200a201 a84a202a203a204 a17 a205 a124a126a206a207a10a11 a54a49 a81 a74a75 a55 a54a76a77 a97 a84a85 2.5.1 a208a209a210a211a212a213a214a215a89a216a90a91a92a217a218a213a219a220a221a222a223a87a92a224a225 a94 a221a95a96a97a226a227a219a222a223a0a1 a86a87a88a89a90a91a92a93a213a97 a10a16 a17 a167a10a168a97a34a19a5a206a207 a54a49 a135 a4a5a23a6a219 a20 a4a5a10a11a12a13a14a15 ha173a174 p0h0 ≤ 0, z(h,p) ≥ 0, (2.5.1) a43a44 a45a46 a28 a40a47a48a49 a3a50a51a97 a159hi a138ci a79a228a229a230ia81a206a207a10a11a12a13a14a15 a138a231a232a233a234 a219 a20a136a235a236a237 a238a239a240a241 ci(ξ0) = wi(ξ0)?p(ξ0)hi(ξ0) ci(ξt) = wi(ξt)+ z(hi,p)(ξt), ?ξt, t = 1,...,T. a37a38 a28 ci(ξ0)?p(ξ0)h(ξ0) = w(ξ0)?p(ξ0)(hi + h)(ξ0) ci(ξt)+ z(h,p)(ξt) = wi(ξt)+ z(hi + h,p)(ξt), ?ξt, t = 1,...,T. a25a26hi + ha138ci + (?p0h0,z(h,p))a242 a236a237a238a239a240a241 a97a243 a52 a126a244a125 a77a245 ui a48a49a246a247 a219 a38a52(2.5.1), ci + (?p0h0,z(h,p)) > ci a248a231a232a233a234 ci +(?p 0h0,z(h,p))a48a49a249a126ci a219a174a250a35a36a97a25a26a5a206a207a54a49a135 a3a4a5a23a6a219 a37a38a52 a55 a2292.3.2,a206a207 a74a75 a55 a54a76a77 a21 a48a49 a80a97 a84a85 2.5.2 a208a209a210a211a212a213a214a215a89a216a90a217a218a213a219a251a221a252a253 0 a91a92a217a218a219a220a221a222a223a87a92a224a225 a94 a221a98a95 a96a219a226a227a222a223a0a1a86a87a88a89a90a93a213a97 a10a16 a17 a164a187a123a10a97 a156 a149a254a255(a29a0a34a18a28a133a79a1a2a81) a17 a115a116a117a118a44a81 a231a232a3a4a5 a5a6a7 a138 a129a169a133a134a97 a29a0a8a9 a156 a219 a55 a2292.5.1a3a10a124a125a219a25a21a244a125 a77a245 a5a44 a11 a133a134a3a79 a48a49a246a247 a81a97 a191 a123a173a125 a82a156 a12 a0a97 a84a85 2.5.3 a208a209a210a211a212a213a214a215a89a216a90a217a218a213a251a221a252a253T a91a92a217a218a219a227a251 a94 a221a13a14a15a16a17a18a96a221a19 a20 a252a253a21a90a93a213a227a221a252a253 T a90a93a213a22a23 a24 a219a220a221a222a223a15a16a87a92a225 a94 a221a95a96a219a226a227a222a223a0a1a86a87a88 a89a90a91a92a93a213a97 a10a16 a17 a159 a10a11j a173a174xjt ≥ 0, ?t ≥ 1,a42xjT > 0. a20a206a207 a54a49pjt a27a25a5a145 a47 a133a134a81a145 a47 a131a132a26a79 a48a49 a80a81a219 a27a20 a228a229a230a123a28a5 a54a49 a21 a139 a81 (≤ 0)a131a132a29a30a10a11j a219 a31a32a136 a177 a28a33a133a134 T a219a29a181a123 a48 a49a247a34 a228a229a230a5a133a134T a81 a231a232 a97 a37a38 a206a207a133a81 a231a232a35 a3a79a36 a249 a81a219a25a21a244a125 a77a245 a5T a48a49a246a247a97 c?a57a58a59 17 September 6, 2005 2.5a68a182a183a184a185a186 a60a61a62 a63a64a65a66a67a68a69 a159hi a138ci a79a228a229a230ia81a206a207a10a11a12a13a14a15 a138a231a232a233a234 a97a34a19a4a5a23a6a219 a20 a4a5a10a11a12a13a14a15 ha173a174 p0h0 ≤ 0, z(h,p) ≥ 0, a43a44 a45a46 a28 a40a47a48a49 a3a50a51a97 Case 1. a82a83 zT(h,p) > 0, a20a30a55a229 2.5.1a81a10a16 a37a38 a40 a181a219a174a33 a40a47 a30 (h i,ci) a81a36 a249 a33 a205 a35a36a81a169 a83 a97 Case 2. a82a83 zT(h,p) = 0, a191 p0h0 < 0, a20a28a119a39?p0h0 a29a30 α a40a10a11 j (αpj0 = ?p0h0), a31a32a136 (a41 h) a177 a28a33a133a134 T, a123 a48a49a247a34 a228a229a230a5a133a134 T a81 a231a232 a97 a137a42a43a44 a219a7a14a15 ?h = h + ˉh, a43a44 ˉh(ξt) = (0,...,α,...,0), ?ξt ∈ Ft,?t < T, α a21a45j a47a32a46a219 hi + ?ha138ci + (?p0?h0,z(?h,p))a236 a237a238a239a240a241 a219a42a26 a231a232a233a234 a48a49a249 a126ci,a174a250a35a36a24(a47a9 a17 a5ξ0 a81 a231a232 a3 a12 a219a5a44 a11 a133a134a81 a231a232 a3a48) Case 3. a82a83zT(h,p) = 0a219p0h0 = 0,a191a7a172ξt(0 < t < T), z(h,p)(ξt) > 0a219 a20 a30Case 2a164a187a219a5ξt a29a30a10a11j (a119a39a21z(h,p)(ξt)),a31a32a136(a41h) a177 a28a33T a219 a32 a48a49a247a34 a228a229a230a81a244a125a219a180a181a174 a33a35a36a97(a47a9 a17 a5ξt a41a43a49a50a81 a231a232 a3 a12 a219a5a43a51a228a81 a231a232 a3a48a24) September 6, 2005 18 c?a57a58a59 a52a53a54 a55a56a57a58a59a60 §3.1 a61a62 ? a10a11a63a64a79a65a66 a37a67 a81 (a163a54a49 p) ? Rk = M(p) := {z(h,p) : ?h} ?a8a127 a32a43 a133a134a81 a231a232a233 a234 a206a123a68a21a172 a47 a10a11a12a13a14a15a81 a74a75 a43a69 a174a97 ? a10a11a63a64a79a128 a37a67 a81? M(p) subsetornotdbleqlRk ? a5a120a116a117a118a44a219a10a11a63a64a81a37a67a33a140a70a4a5a45a46a30a71a72a73a66a40a181a115a81a10a11 ? a5a115a116a117a118a44a219a28a74a75a5 a32a43 a133a134a76a77a10a11a219a29a173a174a63a64a79a65a66 a37a67 a81a78a27 a155 a81a10a11 a245 a41a131 a132 a245 a46a79 a115a97 ? a39a80a81a250a65a66 a37a67 a63a64a81 a40a47 a134a82a219 a31 a10a16a206a207a81 a231a232 a32a83a79 Paretoa36 a249 a81a97 §3.2 a84a85a86a87a88a89 ? a90a91a10a11a63a64a92a8a127 a54a49 a206a79a65a66 a37a67 a81a10a11a81a188a93a79 Arrow a15a16a97 ? a131a132ξt a81Arrowa10a11a219a43a94a6a5a133a134ta81a131a132ξt a211,a5a78a28a43 a136 a131a132a210. a29 a47 a10a11a81 a74a75 a95 a46a79Rk a44a81 a40a47 a120a96 a95 a46 a17 (0,...,0,1,0,...,0) defines e(ξt). ? a97a28k a47Arrowa10a11a219a145a47a7a124a126Ξ a44a81a40a47a131a132(a98a99)a97 ? a82a83a78a28 k a47Arrowa10a11a26a100a76a77a219a101 a141Rk a44a8a127 a231a232a233a234 a123a125 a40a47 a29a30 a31 a177 a28a10a11a12a13 a14a15a102 a5 a97 ? a28a103Arrowa10a11a219 a162 a173a76a77 a3a104 a126a133a134 0,a63a64a242a79a65a66 a37a67 a81a97 ? a32a43a133a134a81a76a77a74a75a105a105 a143 a48 a46 a103a65a66 a37a67 a63a64a78 a155a140 a81a10a11 a245 a219 ? a165a106a120a116a117a118a44a81 a37a67 a63a64a81a134a82a219a123a174a33a65a66 a37a67 a63a64a81 a40a47a107 a120a134a82a97 ? a5a133a134t < T a131a132ξt a81 a40 a116 a74a75a108a109 a79 a40a47J ×k(ξt) a108a109 a219a43a45j a110a111a112a79 pj(ξt+1)+ xj(ξt+1), ?ξt+1 ? ξt a29a113k(ξt) = sharp{ξt+1 : ξt+1 ? ξt}a21ξt a81a93a98a99 a245 a97 ? a84a85 3.2.1 a114a115a90a116a117a118 a119 a213?a221a19 a20 a23 a120a121a122a123 ξ t a213a13 a110a124a125a126a127 a213a128a104 k(ξt). a10a16 a17 a63a64a79a65a66 a37a67 a81?a5a145 a47 a128a129a130a131a132ξt a28a41a7a8a9a81a5ξt a81 a40 a116 a74a75 a219a4a5 a40a47 a10a11a12a13 a5 a119a101a129 a74a75 ( a130a131 a189 a31 a10a16). ?a5a145 a47 ξt a81 a40 a116 a74a75a108a109 a81a132a21 k(ξt)(a147 a108 a109 a229a170). ? a133a134 a17 a63a64a79a65a66 a37a67 a81a78 a140a70 a81a36 a46 a10a11 a245= a37 a131a132a135a81a136a98a99a250 a4 a81a36a105a32 a74 a245 max 0≤t≤T?1 k(ξt) ? a72a38a219a137a10a11 a245a138 a27a139a79a31a32a81a219a10a11 a54a49 a123a173a174a10a11a81 a40 a116 a74a75 a5a172a129a131a132a79a140a141a81a219 a37 a38 a63a64a123a28a79a128 a37a67 a81a219 a162 a173a4a5a27 a140 a245a142 a81a10a11a97 ? a99 3.2.2 a221a1001.2.1 a101a219 a143 a19 a20 a23 a120a121a144a145a146a147a148a149a20a150a124 a219 a151a152 a114a115a90a116a117a118 a119 a213a153a154a155 a123 a90 a17a156a157 a94 a221 a149a20 a15a16a97 a105a20 a155 a123a158 a225a159 a150 a97 a122a160a161 a219a162a163 a94 a221 a149a20 a15a16a219a18a96 a150a164 a104 x1(ξg) = x1(ξb) = 0, x1(ξgg) = x1(ξbb) = 1, x1(ξgb) = x1(ξbg) = 0, x2(ξg) = x2(ξb) = 0, x2(ξgg) = x2(ξbb) = 0, x2(ξgb) = x1(ξbg) = 1, 19 3.3a61a165a166a167a168a169a170 a60a171a62 a172a173a174a175a169a170 a176 x1 = (0,0,1,0,0,1), x2 = (0,0,0,1,1,0). a221a252a253 1 a213 a149a20a122a123 ξ g a22ξb a213a13 a110a124a125a126a127 a104 parenleftBigg 1 0 0 1 parenrightBigg , parenleftBigg 0 1 1 0 parenrightBigg a177a178 a213a128a222a104 2. a179a227a219a208a209a19a103a15a16a213a87a92a221a252a253 1 a213 a149a20a122a123 ξ g a22 ξb a222a104 1/2, a180a181a221 a252a253 0 a213a13 a110a124a125a126a127 a104 parenleftBigg 1 2 1 2 1 2 1 2 parenrightBigg a17a128a104 1. a151a152a114a115a90a23a118 a119 a213a97(a182a183 a17 a184a185a186a187 a221a252a253 0a188a189a15a16a190a191a192a221 a149a20 a252a253 -1a122 a123 a213a225a193a13 a110a124a125 a97) §3.3 a194a195a196a197a198a88a89 ? a199a200a51a131a132a135 – a137a28a201a202a115a47(a28 a104 a47) a133a134 – a145 a47 a128a129a130a133a134a131a132a28a203 a47 a93a98a99 a17 up a138down. ? a199a200a51a131a132a135a36 a107 a120a81a188a93a204 a205a206 1.2 a44a81a250a97 ? a99 3.3.1 – a163 a148a149a20 a221a19 a20 a252a253a188a189a213a15a16 a17 a ( a207a208a209a210), b (a221a252a253 T a211 a110 a213a212a213a214a16). – a214a16a221a252a253 T a213a18a96a104 1, a221a252a253 t(t < T)a213a87a92a104 pb(ξt) = ˉr?(T?t), ?ξt – a209a210a221a252a253 0 a213a87a92a104 pa0 = 1, a108a215a19a13 a110 a101a17a87a92a154a181 a161a216 a104 u a217a219a154a181 a218a219 a211 d(d < u)a217a219a220a221a222 a122a123 up a223a90 down a146a213a97 a151a152 a219a209a210a221a252a253 -t a122a123 ξt a213a87a92a104 pa(ξt) = ut?ldl a17a101l a104a224a211 ξt a104a225 down a146a213a213a226a216 (0 ≤ l ≤ t). – a209a210a213a18a96a227a221 a120a228 a252a253 T a23 a24a158 a104 xa(ξt) = uT?ldl, ?ξT a17a101l a104a224a211 ξT a104a225 down a146a213a213a226a216a97 – a221a19a13 a110 a219a214a16a213a229 a230 a104 ˉr, a209a210a213a229 a230 a104 ua231 d. – a221a19 a20 a23 a120a121a122a123 a219a13 a110a232a233a126a127 a104 parenleftBigg ˉr ˉr u d parenrightBigg a17a234a104 2 (a151u > d). – a151a152a219 a105a20a235a236a237 a15a16a114a115 (a238 a148a149a20 a15a16a22 2T a20a252a253 -T a122a123) a239a240a117a118 a119 a213a241 September 6, 2005 20 c?a242a243a244 a245a171a246 a172a173a174a175 a169a170 3.4 a172a173a174a175 a169a170a247a248 a249a250a251a252 §3.4 a84a85a86a87a88a89a253 a254a255a0a1a2 ? a3a4a204 a205a5a6a7a8a9a10a11a12a13 q a14M(p)a15a16a17a18a19a20 a12a13a21 ? a22a23a24a25a14a26a27a28a29a16 a7a30M(p) = Rk a7a31a32q a14Rk a15a16a19a20 a12a13a21 ? a33a34 ξ a35a36a37 a38a7a39a40e(ξ), a14a41a42a43ξ a16Arrowa44a45a16a46a47a48 e(ξ) = (0,...,0,1,0,...,0) ∈Rk, a49a501a51a42a43ξ. a52a53e(ξ) (ξ ∈ Ξ) a54a55Rk a16a17a18a56 a21 ? a10a57q(ξ) = q(e(ξ)),a58a59q(ξ)a40ξ a16a33a34a60a61 a21 ? a62a18z ∈Rk a63a64a65a40 z = summationdisplay ξ∈Ξ z(ξ)e(ξ) a31a32 q(z) = q ? ?summationdisplay ξ∈Ξ z(ξ)e(ξ) ? ?=summationdisplay ξ∈Ξ z(ξ)q(e(ξ)) = summationdisplay ξ∈Ξ z(ξ)q(ξ) (3.4.1) ? a15a66a14a67a68a69a11a70a64a65a16a8a9a10a11a12a13a241 a22a23a67a71a17a72a73q a64a65a8a9a10a11a12a13a74k-a75a68a69 a11a70a76a77(q(ξ) : ξ ∈ Ξ),a30 q(z) = qz (3.4.2) ? a68a69 a11a70a78 a14(a79 a70) a80a16?a8a9a10a11a12a13a14(a79 a70) a80a16?a81a82a83(a84a47)a85a84a47 a21 ? a86a87 a7a88a89 a68a69 a11a70 a58a90a91a92a4a14a93(a79 a70) a80 a7 a14a94a95a24a25a14a93a82a83(a84a47)a85a84a47a16a17a18a96 a97a21 ? a41a42a43a44a45 a11a70p a16a68a69 a11a70 a63a98 a86a87a99 a88a89 a48 1. a100a101 a8a9a40e(ξ),?ξ a16a44a45a54a102a103a104 a21 2. a68a69 a11a70q(ξ) a105a14 a8a9a40e(ξ) a16a44a45a54a102a54a102a103a104a16a106a107 -0a11a70a7a86a14a108 a40a7 a83(3.4.1) a50 a109z a40e(ξ) a105a110 q(e(ξ)) = summationdisplay ζ∈Ξ e(ξ)(ζ)q(ζ) = q(ξ) ? a80a22a83a111a112a44a45a24a25a50a113a87a7a114a68a69a11a70a115a116a40a19a20a96a117a54a16a118a14a119a96a120a16a21 a121a122 3.4.1 a33a34a60a61a123a124 q(ξt)pj(ξt) = summationdisplay ξt+1?ξt q(ξt+1)[pj(ξt+1) + xj(ξt+1)], ?ξt, t ≥ 0, ?j (3.4.3) a125a126q(ξ 0) a127a128a129 1 a21 c?a242a243a244 21 September 6, 2005 3.4a130a131 a132a133a134a135 a247a248 a249a250a251a252 a245a136a246 a130a131 a132a133a134a135 a44a137a48 a138a139 a86a87a17a18a44a45a54a102a103a104 ?h: a83a106a107t ≥ 1a16a68a69ξt a140a141 a17a142a44a45j,a143a144a83a106a107t+1 a16a62a17 a63a145 a16ξt a16a146a68a69ξt+1 ? ξt a147a148 a7 a3a4 a53 z(?h,p)(ξt) = ?pj(ξt) z(?h,p)(ξt+1)) = pj(ξt+1) + xj(ξt+1), ?ξt+1 ? ξt z(?h,p)(ζ) = 0, ?ζ ∈ Ξuniondisplay ξt+1?ξt {ξt,ξt+1} a149 a43 ?h0 = 0a7a53 q(z(?h,p)) = p0?h0 = 0 a31a32a149(3.4.2) a110 0 = ?q(ξt)pj(ξt) + summationdisplay ξt+1?ξt q(ξt+1)[pj(ξt+1) + xj(ξt+1)]. a86a105a14a150(3.4.3)a51t ≥ 1a55a151 a21 a51 t = 0 a16a96a117 (3.4.3)a63a152a153a154a155a48a83a106a107 0 a140a141 a17a142a44a45 j, a143a144a83a106a107 -1 a156a68a69 a147 a148 a21a157 a44a45a54a102a103a104a83a106a107 -1a68a69ξ1 a16 a8a9a40p j(ξ1) + xj(ξ1),a83a49a92a68a69a16 a8a9a40 0, a83a106 a1070a16 a11a70a40 a86pj(ξ0). a43a14 a149(3.4.1)a53 pj(ξ0) = summationdisplay ξ1?ξ0 q(ξ1)[pj(ξ1) + xj(ξ1)], a158a159t = 0 a16(3.4.3). ? a160 a10 a44a45 a11a70p, a118a96a117a54(3.4.3)a63a161 a148 a68a69 a11a70q. a162a163 a40 a48 1. a161a118a106a107 1 a16a68a69 a11a70a21 a83 (3.4.3) a50 a164 t = 0 a7 a110a165 a53 J a18a96a117a166 k(ξ0)a18a167 a5a168 a16a96 a117a54 a7 a49a169 a168a170a171 a105a14a83 ξ0 a16a17a112 a8a9a170a171a21a172 a24a25a14a26a27a28a29a16a106 a7a157a170a171 a16a173a174a43 k(ξ0),a108 a32a157 a96a117a54 a53a175 a17a16a118q(ξ1),ξ1 ? ξ0. 2. a51a62a17ξ1,a161a118a49a146a68a69a16a106a107-2a68a69 a11a70a21 a83(3.4.3) a50 a164t = 1, a58a176 a10ξ 1,a110a96a117a54 summationdisplay ξ2?ξ1 q(ξ2)[pj(ξ2) + xj(ξ2)] = q(ξ1)pj(ξ1), j = 1,...,J a157 a96a117a54 a53J a18a96a117a166k(ξ1)a18a167 a5a168a7 a49a169 a168a170a171 a105a14a83 ξ1 a16a17a112 a8a9a170a171a21a172 a24a25 a14a26a27a28a29a16a106 a7a157a170a171 a16a173a174a43 k(ξ1) a7 a108 a32a157 a96a117a54 a53a175 a17a16a118q(ξ2),ξ2 ? ξ2. 3. a177a178a15 a116a179 a117a241 ? a83a68a69 a11a70 a81 a400 a16a180a181 a7 a3a4 a63a182a183 a41a51a68a69 a11a70 q(ξt+1) q(ξt) a99a177a184a96a117a54(3.4.3),a185 a161 a148 a41a51a68a69 a11a70a7 a143a144 a31 a41a51a68a69 a11a70 a161 a148 a68a69 a11a70a21 a24a25a16a26a27 a28a29a20a186a44a187a118a16 a175 a17a20 a21 ? a188a189a190a23 a114 a83a1918a192 a154a193 a101a194a28a29a24a25 a21 September 6, 2005 22 c?a242a243a244 a245a136a246 a130a131 a132a133a134a135 3.5 a195a196a197 a198a199 a134a135 a247a248 a249a250a251a252 §3.5 a200a201a202a203a204a205a206a253 a254a255a0a1a2 a47a67a96a117(3.4.3)a7a2073.3.1 a50a16a208a209a66a44a45a24a25a50a16a68a69 a11a70 a63a210a211a161 a148 a21 a51a86a212a18a44a45 a7 a83 a62a18a68a69ξt a3a4 a53 a212a18a96a117a48 q(ξt) = uq(ξut+1) + dq(ξdt+1) q(ξt) = ˉrq(ξut+1) + ˉrq(ξdt+1) a49a50ξut+1 a74ξdt+1 a14ξt a16a212a18a146a189a213 a21 a118a214a110a41a51a68a69 a11a70 q(ξut+1) q(ξt) = vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle 1 d 1 ˉr vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle u d ˉr ˉr vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = ˉr?dˉr(u?d) q(ξdt+1) q(ξt) = vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle u 1 ˉr 1 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle u d ˉr ˉr vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = u? ˉrˉr(u?d) a86a215a41a51a68a69 a11a70 a51a62a18ξt a78a14a17a87a16a21a108a158a7a22a23a101a106a107-ta68a69ξt a40a216down a148a217 a187la218 a7a30 ξt a16a68a69 a11a70a40 q(ξt) = parenleftbigg u? ˉr ˉr(u?d) parenrightbigglparenleftbigg ˉr?d ˉr(u?d) parenrightbiggt?l a219 a143 a7q(ξ t)a14a79 a70 a80a16? u > ˉr > d ?a81a82a83a84a47 q(ξt)a14a80a16? u ≥ ˉr ≥ d ?a81a82a83a85a84a47 §3.6 a220a221a222a223a205a206a224 a225a226a227 ? a228a112a44a45a24a25a50 a229a230a231 a16a232a233 -a44a45a54a102a234a235a236a237 a40 max c,h u(c) (3.6.1) s.t. c0 = w0 ?p0h0 (3.6.2) ct = wt + zt(h,p), t = 1,...,T (3.6.3) a149 a43 p0h0 = q(z(h,p)) = q(c1+ ?w1+) a49a50 c1+ = (c1,...,cT), w1+ = (w1,...,wT) a238 a95 a64a65 a31 a106a1071a239a16a232a233 a88a240a74a241a242a7a243a89a244a245 (3.6.2) a63 a184 a40 c0 = w0 ?q(c1+ ?w1+) c?a246a247a248 23 September 6, 2005 3.7 Pareto-a249a250a251a252 a253 a136a254 a130a131 a132a133a134a135 a32(3.6.3) a63 a184 a40 c1+ ?w1+ ∈M(p) a108 a158a7a255a0 a236a237(3.6.1)～(3.6.3)a63a177a184 a40 maxc u(c) (3.6.4) s.t. c0 = w0 ?q(c1+ ?w1+) (3.6.5) c1+ ?w1+ ∈M(p) (3.6.6) a22a23a24a25a14a26a27a28a29a16 a7a113a1M(p) = Rk a7a31a32a244a245(3.6.6) a2a143a55a151 a21a3 a17a162 a7a149 a43q(·)a14 a19a20a16 a7a244a245(3.6.5) a63a4 a184a55 c0 + qc1+ = w0 + qw1+ (3.6.7) a49a50q a14a41a42a43a44a45 a11a70p a16a68a69 a11a70a76a77a21 ? a15a5a16 a238a6 a150a137 a7 a22a23a232a233 a88a240 c a7a8(3.6.2)a74(3.6.3)a7a30ca7a8(3.6.7). a9a214 a7 a22a23 ca7 a8 a244a245(3.6.7),a30c a10a43z(h,p) = c1+ ?w1+ a7a8 a243a89a244a245(3.6.2)a74(3.6.3)a21 ? a108 a158a7a255a0 a236a237(3.6.1)～(3.6.3)a174 a11 a43 maxc u(c) (3.6.8) s.t. c0 + qc1+ = w0 + qw1+ (3.6.9) a86a14 a229a230a231 a5a51a28a29a16a11 a53a12a13 a24a25a106a16a232a233a234a235a236a237 a7 a49a14a213a118a16a17a15a16a69 a40 q(ξ) = ?u ?c(ξ) ?u ?c(ξ0) , ?ξ ∈ Ξ (3.6.10) a15a66a150a137 a7a17a18a19a229a20 a174a43a68a69 a11a70a21 a108 a158a7a52a53a229a230a231 a16 a17a18a19a229a20a21a22 a41a174 a21 ? a23 a11a70q(ξ)a229a230a231 a63a24 a140 a17a111a25a83a68a69ξ a16a232a233 a7a32 a17a111a25a106a1070a16a232a233a16 a11a70a401 a21 ? a232a233-a44a45a54a102a234a235a236a237(3.6.1)-(3.6.3)a26a232a233a234a235a236a237(3.6.8)-(3.6.9)a16a174 a11 a20a27a28a3a4a48 a232a233 a238a29 {ci} a74 a11 a53a12a13a11a70 q a14a11 a53a12a13 a24a25a50a16a17a18a30a31 ? a71a87a16 {ci} a74a44a45a11a70 p a14a26a27a28a29a24a25 (a83 p a32) a50a16a17a18a30a31 a7 a49a50a30a31a44a45 a11a70 p a74 a11 a53a12a13a11a70 q a33a34a66 (3.4.3)a41a35a169 a7a159q a14a41a42a43pa16a68a69 a11a70a21 §3.7 Pareto- a36a37 a226a227 ? a26a111a112a38a39a17a87 a7 a17a18a232a233 a238a29 a59 a40 a14Paretoa40a41a42 a7 a22a23a81 a63a145 a177a43 a238a29a44 a16 a241a242a45 a110a46 a229a230a231 a16a47a67a79 a70 a48a49a32 a81a79 a70a50a51 a49a52 a229a230a231 a16a47a67 a21 a165a53a99a150 a7a238a29{ci} a14Pareto-a54 a255 a16 a7 a22a23a81a82 a83a49a92 a238a29{cprimei}a45 a110 Isummationdisplay i=1 cprimei = Isummationdisplay i=1 wi, (3.7.1) ui(cprimei) ≥ ui(ci), ?i, (3.7.2) September 6, 2005 24 c?a246a247a248 a253 a136a254 a130a131 a132a133a134a135 3.7 Pareto- a249a250a251a252 a55 a66(3.7.2)a56 a51 a51a46a18ia40a79 a70 a81a174a66a57 a66(3.7.1)a64a65a238a29{cprimei}a14 a63a58 a16a59 a159 a83a60a61a106a107a156 a229a230a231 a16a232a233a214 a74 a174a43 a241a242 a214 a74a62 a66 (3.7.2)a64a65a62a18 a229a230a231i a63a64a65cprimei,a54a144a17a18a16a69 a64a65 a56 a51 a17a18 a229a230a231i a79 a70 a64a65cprimei. ? a191a17a66a47a67a230a48a12a13a24a25a50a16a30a31a238a29a14 Paretoa54a255a16a21 a22a23a44a45a24a25a14a26a27a28a29a16a59 a113a1a149§3.6 a68a59a62a18a30a31a232a233 a238a29a69 a14a28a29a11 a53a12a13 a24a25 a16a17a18a30a31 a238a29a21 a182a183 a191a17a66a47a67 a230 a59a86a18a30a31a232a233 a238a29 a14 Paretoa54 a255 a16 a21 a121a122 3.7.1 a70a71a72a73a74a75a76a77a78a60a61a79a80a81a82a83 a84 a42a59 a85a86a87a88a89 a42a90a91a92a93a80a94a61a95a96a42a59 a97a98a99a100 a77a78a101a102 a103a104 a80 Pareto a40a41a42 a21 a44a48a105{c i} a14a17a18Paretoa54 a255a238a29 a59 a30ci a14a236a237 maxc ui(c) s.t. c0 + qc1+ ≤ wi0 + qwi1+ (3.7.3) a16a54 a255 a118 a21 a22a23{ci}a81a14Paretoa54 a255 a16a59 a30 a82a83 a63a58 a238a29{ˉci}a45 a110 ui(ˉci) ≥ ui(ci), i = 1,...,I, (3.7.4) ui0(ˉci0) > ui0(ci0), a51a46 i0 ∈{1,...,I}. (3.7.5) a149 a43a156ui a14a79 a70a106a107 a16a59 a149(3.7.4)a53 ˉci0 + qˉci1+ ≥ wi0 + qwi1+, i = 1,...,I, (3.7.6) a93 a30 a63 a100a101(3.7.3)a16 a63a58 a118a59 a108ˉci a16a109 a110a111 a119a112a59 a31a32a108ci a16a109 a110a111 a119a112a59a86a26ci a14(3.7.3) a16a54 a255 a118a113a114 a21a149(3.7.5)a53 ˉci0 + qˉci01+ > wi00 + qwi01+, (3.7.7) a93 a30 a59 a115a1ˉci 0 a40(3.7.3) a16 a63a58 a118a59 a115a1 a63 a100a101 a108ˉci 0 a16a109 a110a111 a119a112a16(3.7.3)a16 a63a58 a118a59 a31a32 (3.7.5)a26ci0 a40(3.7.3)a16a54 a255 a118a113a114a57 a114(3.7.6) a10a43 a52a53a229a230a231 a41a116a59a58a117a118a101 (3.7.7),a110a101 Isummationdisplay i=1 ˉci0 + q Isummationdisplay i=1 ˉci1+ > ˉw0 + q ˉw1+, a49a50ˉw =summationtextIi=1 wia40a241a242a44a74a57a15a66a26a232a233 a238a29{ˉci} a16 a63a58 a20(a159summationtextIi=1 ˉci = ˉw0,a31a32summationtextIi=1 ˉci0+ qsummationtextIi=1 ˉci1+ = ˉw0 + q ˉw1+)a41a113a114a57 c?a246a247a248 25 September 6, 2005 3.7 Pareto-a249a250a251a252 a253 a136a254 a130a131 a132a133a134a135 September 6, 2005 26 c?a246a247a248 a119a120a121 a122a123 §4.1 a124a125 a126a127 a67a128 a12a13 q a129a129a67 a57 a83a130a131a132a55a133a134 M(p) a15 a21 a83a135a18a11 a53a115 a161a136 ( a167a67 a136a137) a133a134Rk a15a67 a57a138 a128a139a140a141 a53 a67a16 a21 ? a121a142 4.1.1 a143a60a144a92a80a145 a100a146a147 a144a92 Q : Rk →R, a148 a80a149a150a128a60a144a92 q a151a152 a153a154a155a156a157M(p) a158a159a128a160a161 a156a157Rk a42a145 a100a162a163 a59a164 Q(z) = q(z), ?z ∈M(p). ? a86a87 a63 a67 a138 a128a139a140a51a60a118 a126a127a3 a58 a67a128a59a49a67a128a190a165a26a166a167a67 a126a127 a67a128a139a140a168a41a71a169 ? a170a171a172a173a174a168a141a79 a70 a80(a80)a168 a138 a128a139a140a59a108 a40 a86a175a176a177a118a178a179a81a82a83a84a47(a85a84a47)a169a83a32a180 a192a92a171a181 a114 a67a43a228a112a38a39a50a168a68a69a128 a70a74a182a183 a50a176a184 a20 a169 §4.2 a185a186a187a188a189a190a191 ? a192a193a194a56a188a67 a230 a150a168a141a79 a70 a80 (a80)a168 a138 a128a139a140a168a82a83a176a169 a121a122 4.2.1 ( a195a196a197 a198a199 a128 a88) a72a73a60a61a200a201a202 a203a204? a205a76a94a61a206a42a143a60a144a92a169 a121a122 4.2.2 ( a195a196a197 a198a199 a128 a88 a59a207a208a209) a72a73a60a61a200a201a202 a210a203a204? a205a76a206a42a143a60a144a92a169 a44a137a48 ?)a211a82a83a79 a70 a80(a80)a168 a138 a128a139a140a59 a30 a82a83a79 a70 a80 (a80)a168 a126a127 a67a128a139a140(a138a128a139a140a83 M(p)a212a168a213a214a105a141a180a175 a126a127 a67a128a139a140) a59 a31a32a149 a67 a2302.3.2(2.3.3), a81a82a83a84a47(a85a84a47)a169 ?) a105a44a45a128 a70a215a216 a187a84a47 (a85a84a47) a59 a30a149 a67 a230 2.3.2(2.3.3) a59 a126a127 a67a128a139a140 q : M(p) → R a141a79 a70 a80(a80)a168a169a32a5a170a171 a114q a217a218 a40Rk →R a168a79 a70 a80(a80)a168a219a176a140 a168 a169 a220a179 a211a221a218a217a218 a222 a62a218a223a217a2181a75a99a28a55a169a191a180a162a59 a109 a180a175a167a67 a136a137 ?z ∈Rk \M(p) a59a58 a114q a217a218a101 a149M(p) a74 ?z a224a55a168a146a133a134a59a86a175a146a133a134a168a75 a168 a174a43M(p)a168a75 a168 a1161 a169 a115a225 a217 a86a175a217a218a59 a226 a160 ?z a227 a67a180a175a139a140 a111pi. a40 a186a228 a157 a217a218a141a79 a70 a80(a80)a168a59a234a235a168pia21a22a7a8a48M a50a112a43 a222 a81a174a43 ?z a168a60a61 a126a127 a165 a53 a79 a70 a112(a112)a43pi a168a128 a70 a59M a50 a229 a43 a222 a81a174a43 ?z a168a60a61 a126a127 a165 a53 a79 a70a229(a229) a43 pi a168a128 a70 a169a86a215a213a214a67 a57 a187a180a175a230a231 pi a168a232a134a169a233a175a217a218a141 a149 a165a234 a126a127 {x 1,...,xJ}a235a128 a236{p 1,...,pJ}a168J a175a237a238a235a239a234 a126a127 ?z a235a128 a236 pi a168a237a238a240a241a168a242a243a168 a126a127 a67a128a139a140a169a244a208 a245 a59 a246 a180a175a247a248a233J +1a237a238a168a130a131a132a241a133a134a249a168a250a67a251 a137 a59 a252a253 a244a180 a245 a249a217a218a254a168 a126a127 a67a128 a139a140a217a218a255a0a167a1J + 1a175a237a238a235a233a175a2a168a250a67a251 a137 a224a241a168a3a133a134a169 a4a5k?d (d = M a168a6 a7)a245 a254a8a170a171a9a10a11a180a175a255a135a175a12a13a133a134Rk a168a217a218a169a0 a14 a233a175a15a16a168a17 a245 a141a18a19a168a8a170a171 a20a21a22 a244a180 a245 a169 ? a23z ∈Rk,a21z a168a24a25a26a27a212a28(a29a30a29a31z a32a237a238a240a33a34a35a32a36a37a24 a236) qu(z) = min h {p0h0 : z(h,p) ≥ z} 27 4.2a38a39a40 a41a42a43a44 a45a46a47 a48a49 a235a50a28(a29a30a51z a29a31a32a237a238a240a33a34a35a32a36a52a24 a236) ql(z) = max h {p0h0 : z(h,p) ≤ z} a233a53a8a54a55 {h : z(h,p) ≥ z} = ?, a56a57a26 qu(z) = ∞; a54a55 {h : z(h,p) ≤ z} = ?, a56a57a26 ql(z) = ?∞a58a59a54a60a61M = span{(1,0)}, z = (1,1),a56{h : z(h,p) ≥ z} = ?. a62a63 4.2.3 a64a65a66a67a68a69a70a71a72 a73a74a75 a8a76 qu(z) = ql(z) = q(z), ?z ∈M(p). a237a60a77a78a79a50a28a32a26a27a8a234 qu(z) ≤ q(z) ≤ ql(z), ?z ∈M(p). a80 a61a81a248z ∈M(p)a82a11qu(z) < q(z)a58a56a81a248a237a238a240a33a34a35 hprime a82a11 z(hprime,p) ≥ z, p0hprime0 < q(z). a83h a84a85a86a237a238a240a33a34a35a82a11 z(h,p) = z, p0h0 = q(z) a14 a84 z(hprime ?h,p) ≥ 0, p0(hprime ?h)0 < 0 a87a88hprime ?h a84a85a86a89a90a91a8a92a93 a14a80 a23a58 a87a88 qu(z) = q(z), ?z ∈M(p) a94a95a96 a237ql(z) = q(z), ?z ∈M(p)a58 a50 a62a63 4.2.4 a64a65a66a67a68a69a70a71a72 a73a74a75 a8a76 qu(z) ≥ ql(z), ?z ∈Rk a237a60 a80 a61a81a248z ∈Rk a82a11qu(z) < ql(z)a58 a97 a77a78a79a50a28a32a26a27a81a248a237a238a240a33a34a35hprime a235hprimeprime a82 a11 z(hprime,p) ≤ z ≤ z(hprimeprime,p), p0hprime0 > p0hprimeprime0 a14 a84 z(hprimeprime ?hprime,p) ≥ 0, p0(hprimeprime ?hprime)0 < 0 a98hprimeprime ?hprime a84a85a86a89a90a91a8a92a93 a14a80 a23a58 a50 a62a63 4.2.5 a64a65a66a67a68a69a70a71a72 a74a75 a8a76 qu(z) > ql(z), ?z ∈Rk \M(p). September 6, 2005 28 c?a99a100a101 a45a46a47 a48a49 4.2 a38a39a40 a41a42a43a44 a237a60 a97 a77a102a1034.2.4,a104a105a237a106 qu(z) negationslash= ql(z), ?z ∈Rk \M(p) a80 a61a81a248z ∈Rk \M(p)a82a11qu(z) = ql(z)a8a56a81a248a237a238a240a33a34a35hprime a235hprimeprime a82a11 z(hprime,p) ≤ z ≤ z(hprimeprime,p), p0hprime0 = p0hprimeprime0 a0 a14z ∈Rk \M(p), z a247 a96a107 a0 a108 a86a237a238a240a33a34a35a109a241a8 a98z(hprime,p) ≤ z a235z ≤ z(hprimeprime,p) a249a32a110 a111a112 a247 a96a107 a241a113a8 a87a88 a8 z(hprimeprime ?hprime,p) > 0, p0(hprimeprime ?hprime)0 = 0 a98hprimeprime ?hprime a84a85a86a90a91a8a92a93 a14a80 a23a58 a50 ? a78a114a115 a4a21a22a116a117a118 a250a26a251a119a32a25a32a78a79a50a28a8a50a114a120a121a54a122a91a123a78a79a50a28a124a125a126a29a30a26a24 a127a128 a58 ? a129a26a85a86a250a26a251a119 ?z negationslash∈M(p)a58a26a27 N = {z + λ?z : z ∈M(p), λ ∈R} a56N a84Rk a32a3a130a131a8N a132a133M(p)a235 ?z a58N a32a6 a7 a110 a14M(p) a32a6 a7+1 a8N a84a29a30a134 {x1,...,xJ}a235 ?z a32J + 1a86a237a238a32a135a136a109a241a130a131a58 ? a54a55a247a81a248a89a90a91a8a110a24a137a8a54a55a29a30a26a24 a127a128 a84a138a32a8a56a0a102a1033a139qu(?z) ≥ ql(?z). a140a141a142 a143a96a246pi a144a145 ql(?z) ≤ pi ≤ qu(?z) a253q a125a126a134N a78a32a146a147a148a149Q : N →R, Q(z + λ?z) = q(z)+ λpi ? a50a114a237a106Qa84a142a143a150a151a32q a32a138a32a125a126a58 a62a63 4.2.6 a152a153 4 a64a65 q : M(p) →Ra154a155a156a8a76 Q : N →Ra157a154a155a156a58 a237a60 a83y ∈N, y ≥ 0. a56?z ∈M(p),λ ∈R,a82a11 y = z + λ?z 1. a23λ > 0a8a56a0y ≥ 0a8a158 ?z ≥?zλ. a97 a77ql a32a26a27a8ql a84a159a160a32a58 a253q l a161 a123a255a78a162a163a164a11 ql(?z) ≥ ql(?zλ). a0 a14?z λ ∈M(p), a97 a77a102a1034.2.3, ql(?zλ) = q(?zλ) = ?1λq(z). c?a99a100a101 29 September 6, 2005 4.2a38a39a40 a41a42a43a44 a45a46a47 a48a49 a87a88 a8 pi ≥ ql(?z) ≥ ql(?zλ) = ?1λq(z), a98 Q(y) = q(z)+ λpi ≥ 0. 2. a23λ < 0a8a56a0y ≥ 0a8a158 ?z ≤?zλ. a97 a77qu a32a26a27a8qu a84a159a160a32a58 a253q u a161 a123a255a78a162a163a164a11 qu(?z) ≤ qu(?zλ). a0 a14?z λ ∈M(p), a97 a77a102a1034.2.3, qu(?zλ) = q(?zλ) = ?1λq(z). a87a88 a8 pi ≤ qu(hatwidez) ≤ qu(?zλ) = ?1λq(z), a98 Q(y) = q(z)+ λpi ≥ 0. 3. a23λ = 0a8a56y = z ≥ 0, Q(y) = q(z) ≥ 0 (a87a134q a84a138a32). a165 a78a142 a143 a115a237a11Qa84a138a32a58 a50 ? a54a55a247a81a248a90a91a8a110a24a137a8a54a55a29a30a26a24 a127a128 a84a166 a236 a138a32a8a56a142 a143a246 pi a144a145 ql(?z) < pi < qu(?z) a167 a102a1034.2.6a32a237a106 a168a169 a18a19a8a142 a143a96 a11 a62a63 4.2.7 a64a65 q : M(p) →Ra154a170a69a155a156a8a76 Q : N →Ra157a154a170a69a155a156a58 ? Qa171a172a78a84a24 a236 a134{p1,...,pJ}a235pi a32J + 1a86a237a238a32a135a136a109a241a130a131N a78a32a29a30a26a24 a127a128 a58 a87a88 a8Qa248N a78a84a166 a236 a138(a138)a32?a29a30a134{x1,...,xJ}a235 ?z a32J +1a86a237a238a32a242a243a84a173a90 a91(a89a90a91)a32a58 ? a50a114a174a85a86a59a3a124a175a106a78a79a50a28a58 a176 4.2.8 a177a178 1.2.1 a179a8a180a181a182a183a66a67a60a184a185 1 a186a187a156a188a189a190a67 (a66a67 1) a8a184a185 2 a186a187a156a188 a189a190a67 (a66a67 2). a191a192a8a193a187a190a67a156a194 a75 a154 x1(ξg) = x1(ξb) = 1, x1(ξ) = 0, ?ξ ∈F2 a195 a187a190a67a156a194 a75 a154 x2(ξg) = x2(ξb) = 0, x2(ξ) = 1, ?ξ ∈F2 a180a193a187a190a67a177a184a185 0 a156a68a69a196 p1(ξ0) = 0.9 September 6, 2005 30 c?a197a198a199 a45a46a47 a48a49 4.2 a38a39a40 a41a42a43a44 a195 a187a190a67a156a68a69a196 p2(ξ0) = 0.75, p2(ξg) = 0.9, p2(ξb) = 0.8 a200a201 a154a202a203 a204 a156a8 a205 a196a206a2073.2.1 a156a208a209a210a177a184a1851a156a182a183a211a210a212a213a214a215(a182a183a193a187a216a217a218a219 a212a196 parenleftBigg 0 0 1 1 parenrightBigg a8 a220 a208a1961a8a213a221a222a223a224 a225 a183a2262 )a58 a227 a206a228a229a230a231Rk a1546a232a156(a233k = 6), a234a235a236a237 a214a230a231M(p) a154 4 a232a156a58a211a238 a239 a8a193a183 a227 a206a228a229 z = [z(ξg),z(ξb),z(ξgg),z(ξgb),z(ξbg),z(ξbb)] a240a241a242 a193a183a66a67a243a244a245a246 a237 a214a8a247 a248a249 a247 z(ξgg) = z(ξgb), z(ξbg) = z(ξbb). a250a251 a193a183 a227 a206a228a229 ?z: ?z1 = (0,0), ?z2 = (2,1,1,0) a252a253 ?z negationslash∈M(p) a8 ?z a156a68a254a156 a239a255 q u(z) = minh{p0h0 : z(h,p) ≥ z}a0a1a2a3a64a4a5a6a7a8a153 a9a10 a60 min h p1(ξ0)h1(ξ0)+ p2(ξ0)h2(ξ0) s.t. h2(ξg) ≥ 2 h2(ξg) ≥ 1 h2(ξb) ≥ 1 h2(ξb) ≥ 0 h1(ξ0) + 0.9[h2(ξ0)?h2(ξg)] ≥ 0 h1(ξ0) + 0.8[h2(ξ0)?h2(ξb)] ≥ 0 a191a154a193a183a146a147a11a12a8a153a8 a220 a3a154a60 h2(ξg) = 2, h2(ξb) = 1, h2(ξ0) = 10, h1(ξ0) = ?7.2 a5a6a13 a14 a254a196 1.02a8a233qu(?z) = 1.02. a191a183a66a67a243a244a245a246a156a216a217a177a184a185 1 a196 (0,0), a177a184a185 2 a196 (2,2,1,1). a15a154 a235a236a237 a214a230a231 M(p) a179 a16 a1 ?z a156a5a17 a227 a206a228a229a58 a242 a222a66a67a68a69a70a71a72 a74a75 a8a191a183a66a67a243a244a245a246a156a184a185 -0a68 a69 1.02a18a206a154a5a17a156a58 a177a191a183a178a223a179a8a5a6a66a67a243a244a245a246 a240a241 a191a192a19a206a60a20a21a177 a235a236a237 a214a230a231a179 a22 a186 a16 a1 ?z a156 a5a17 a227 a206a228a229a8 a253a23a24a25a237 a214a191a183 a227 a206a228a229a156a66a67a243a244a245a246a58a26a191a27a2a3a28a29a193a30a213a31a32a8 a205 a196a0a33 a235a236a237 a214a230a231a179a191a183a5a17a34a213a35a177a58 a205a36 a193a30a37a2a3a146a147a11a12a8a153a58 c?a197a198a199 31 September 6, 2005 4.3a48a49a38a39a40a41a42a43 a45a46a47 a48a49 a44 a207a8?z a156a68a254a156a4 a255q l(z) = maxh{p0h0 : z(h,p) ≤ z}a0a1a2a3a64a4a5a6a7a8a153 a9a10 a60 max h 0.9h1(ξ0)+ 0.75h2(ξ0) s.t. h2(ξg) ≤ 2 h2(ξg) ≤ 1 h2(ξb) ≤ 1 h2(ξb) ≤ 0 h1(ξ0) + 0.9[h2(ξ0)?h2(ξg)] ≤ 0 h1(ξ0) + 0.8[h2(ξ0)?h2(ξb)] ≤ 0 a220 a5a6a3a45a46a47 a44 a8a71a72h2(ξ0) = 9a48a49a5a6a13 a14 a254a1960.27, a233ql(?z) = 0.27. a191a183a66a67a243a244a245 a246 a236a237 a156a216a217 (0,0,1,1,0,0)a154 a235a236a237 a214a230a231a179a17a222a221a222 ?z a156a5a50a216a217a58 ? a18a51 a14a112a52 a29a30a26a24 a127a128 a32a85a86a166a53a138 (a138) a32a54a24 a127a128a55a96a56 a0 a57a58a59 a32a60a61 0 a62 a253 a124a60 a61a63a64a65a131a32a66a172a67 a57a68a21a22 a58a54a55 a57a58a59 a32 a112a52 a63a64a84a69 a70a71a72a73 a123 a128a7 a84a166a53a159a160(a159a160) a32a8 a74a75 a66a172a67 a57a68 a121 a76( ?u ?c(ξ)/ ?u ?c(ξ0),ξ ∈ Ξ)a26a27a85a86a166a53a138(a138)a32a54a24 a127a128summationtext ξ∈Ξ z(ξ) bracketleftBig ?u ?c(ξ)/ ?u ?c(ξ0) bracketrightBig . a77a78a79a80a81 a60a82a83a84a84a85a32a82a85a56a86a84a85a82 §4.3 a87a88a89a90a91a92a93a94 ? a29a30a26a24 a127a128a95 a54a24 a127a128 a32a125a126a85a96a124a175a86a84a84a85a32a58 a77a78a79a97a80a81 a60a82a98 a117 a122a86a99a135a136a109 a100 a130a131a32a101a26a102a119a82a81a99a85a86a103a104a105a32a25a82a106a85a86a25a26a27 a116 a29a30a26a24 a127a128 a32a85a86a166a53a138 (a138) a32a125a126a58 a77a78a79 a84a107a108 a80a81 a32a60a82a135a136a109 a100 a130a131M(p)a110 a14 a101a26a102a119a130a131Rk a82a29a30a26a24 a127a128 a62a54a24 a127a128 a84a19a85a86a82 a14 a84a142 a143 a158 ? a109a110 4.3.1 a180a66a67a68a69a70a71a72 a74a75 a82a66a67 a200a201 a154a111a112a203 a204 a156a247 a248a249 a247a35a177a113a193a156a170a69a155a156a114 a68a148a149a58 September 6, 2005 32 c?a99a100a101 a115a116a117 a118a119a120a121a122a123a124a125a126a127a128a122a129a120a130 §5.1 a131a132 ? a169a133a21a22a54a24a127a128a32a163a86a134a135a136a137a32a138a139—a85a86a123a140a141a24a53a82 a142 a85a86a123a143a144a145 a146a147a68 a82 a55a148 a149 a29a30a150a151 a127a128 a123a150a151a152a32a138a139a58 ? a166a53a138(a138)a32a140a141a151a53a32a81a99 a146a153 a139a154a86a81a99a90a91(a89a90a91)a58a140a141a151a53a32a84a85 a146a153 a139a154 a78a79 a84a107a108 a80a81 a58 ? a140a141a151a53 a161 a134a155 a146a156a157a158 a32a159 a96a56a160a149 a58a85a161a140a141a151a53a115a139a82 a117 a122a29a30a32a151a53 a112a96a162a95 a82a173 a141a105a163a164a109 a100a165 a86a29a30a32a166a167 a158 a33a34a35a58 ? a143a144a145 a146a147a68 a84a123a168a169 a87a170a171a172a173a174a116 a32a140a141a151a53a58a150a151a152a32a81a99 a146 a84Riesz a138a139a150 a58 a32a175 a176 a58 §5.2 a177a178a88a179 a54a55a166a167 a78a79 a84a107a108 a80a81 a32a82 a74a75 a29a30a150a151 a127a128 q a150a180a99a181a86a182a150a102a119a130a131Rk a78a82 a183a71 a140a141 a151a53q(ξ)a150a180a134Arrowa166a167 e(ξ)a32a151a53q(e(ξ))(a184a185a186 a133). a54a55 a78a79 a84 a97a80a81 a32a82 a74a75 a135a136a109 a100 a130a131M(p)a84a182a150a102a119a130a131Rk a32a187 a170 a130a131a82 a108a188Arrow a166a167a86 a107 a123a29a30a150a151 a127a128 a150a151a58 a189a190a191a168a169 a150 a58 4.2.1(4.2.2) a192a193a142 a143 a60a54a55a166a167a151a53a194a195 a116 a90a91 (a89a90a91) a82 a74a75 a29a30a150a151 a127a128a96a56 a125a126a134a150a180a99a181a86a182a150a102a119a130a131a78a32a166a53a138(a138)a32a54a151 a127a128 a82a140a141a140a141a151a53 a96a56 a123a54a151 a127a128 a124a150a180a58 a83Q a134a54a151 a127a128 a82 a183a83 q(ξ) = Q(e(ξ)), ?ξ ∈ Ξ, (5.2.1) a72 a145e(ξ) a134R k a145a32a140a141 -ξ a196a197a121 a76 a82 a98 a136a51a198 ξ a32 Arrowa166a167a32a199a91a58a200 q(ξ) a84a140a141 ξ a99a54 a151 a127a128Q a50a32a211a210a68a69a58a54a55Qa84a166a53a138(a138)a32a82a56a106a86a140a141a151a53a201a84a166a53a138(a138)a32a58 a202 a198a106a86a182a150a102a119z ∈Rk a96a56a203a100 z = summationdisplay ξ∈Ξ z(ξ)e(ξ), a142 a143 a158 Q(z) = summationdisplay ξ∈Ξ z(ξ)q(ξ) = qz, a72 a145q = (q(ξ),ξ ∈ Ξ)a134a140a141a151a53a121 a76a204a205 a162 Q(z) = qz (5.2.2) a84a54a151 a127a128 a123a140a141a151a53a32a138a139 a204 a98a29a30 z ∈M(p),a158 q(z) = qz (5.2.3) 33 5.2a206a207 a49a208 a45a209a47 a206a207 a49a208 a79a210a211a212 a43a213a214 a79 a43a49a215 a165a216 a82a217a218a219a151a53a220a221a222a140a141a151a53a223a224a225a82a226a86a227a228a150a229 a100a74a230 a217a218a219a166a167 a158a231a232a233a234 a235a236a77a78a79a237 a107a108 a80a81 a219 a74a216 ( a1853.4a174)a82a86 a80a81a78a79 a145a219a140a141a151a53a238a239a240a241a242a155 a146a156a157a158 (3.4.3)a219 a235 a159a225 a95a204 a242a243a166a244 a165a245a70 a82a246a247 a245a230 a166a167 a158a231a232a233 ?h a248a99a249a250t ≥ 1a219a140a141ξt a251a252 a245 a253 a166a167j a183a99a106a245a230a170a140a141 ξt+1 ? ξt a254 a83a255 a149 a82a0 z(?h,p)(ξt) = ?pj(ξt), z(?h,p)(ξt+1) = pj(ξt+1) + xj(ξt+1), ?ξt+1 ? ξt z(?h,p)(ζ) = 0, ?a1a83ζ a202 a198 ?h(ξ0) = 0, a2a3a4q(z(?h,p)) = p(ξ0)?h(ξ0) = 0. a254 q(z) = qz a5a222a198a217a218z(?h,p),a2a3a4 q(ξt)pj(ξt) = summationdisplay ξt+1?ξt q(ξt+1)[pj(ξt+1) + xj(ξt+1)], j = 1,...,J, ?t ≥ 1, ?ξt ∈Ft. (5.2.4) a6a7 a220a166a8a9a10a198t = 0a11a12a13a14a1a145q(ξ0) = 1. a156a157a158(5.2.4) a15a16a17a18a19a20a21a145a219 a156a157a158 (3.4.3)a237 a136a22a219 a204a23a24a245 ξ t a14 a156a157a158(5.2.4) a4J a230a156a157a25k(ξ t) a230a26a27a28q(ξ t+1)/q(ξt). a29a30 a14a2a3a31 a32 a166a244a243a136a5a198a33a151a34a35a219a140a141a151a36 a237a156a157a158 (5.2.4) a219a159 a234 a226a37a14 a38 a36 a235 (a235) a219a219a33a151a34a35a150a180a243 a38 a36 a235(a235) a219a159 a234a236a39 a20a21 a237a40 a18a19a219a14a0a41 a23a42a230 a33a151a34a35(a184a150a434.3.1) a14a44a226 a156a157a158(5.2.4) a4 a42a230 a159 a234 a45a46 5.2.1 a47a48a49a50a51a52a53a54a55a56?a57a58a59 (5.2.4) a47a48a49a50a51a52a60a61 a62a63 a49a50a51a52a60 q a64a65a66 a67 a63 a49a50a51a52a53a54a55a56 Q(z) = qz. a68 a244a248 ?)a8a69a219a70a71a31 a72a73a68 a244 a234 ?)a74qa237a75a76a77(5.2.4)a219 a245a230a38 a36 a235 a219a78a14a0a79Q(z) = qza80a81a219a34a35Qa237a82a83a219 a25a38 a36 a235 a219 a234 a84a85a86a87a68 a244Q a23M(p) a8a219a88a89a90 a237a91 a218a80a92a34a35 q a234a93 z ∈ M(p) a14a0a41 a23a68a94a77a231a232a233 h September 6, 2005 34 c?a95a96a97 a98a99a100 a101a102a103a104a25a105a106a107a108a109a110a25a111a103a112 5.2a101a102a103a104 a113 a225z = z(h,p) a234a114a115a116 a222 a75a76a77(5.2.4) a14a4 q(z) = p0h0 = summationdisplay j pj(ξ0)hj(ξ0) = summationdisplay j hj(ξ0) summationdisplay ξ1?ξ0 [pj(ξ1) + xj(ξ1)]q(ξ1) = summationdisplay ξ1?ξ0 q(ξ1) summationdisplay j [pj(ξ1) + xj(ξ1)]hj(ξ0) = summationdisplay ξ1?ξ0 q(ξ1) ? ?z(h,p)(ξ1) +summationdisplay j pj(ξ1)hj(ξ1) ? ? = q1z1 + summationdisplay ξ1?ξ0 summationdisplay j q(ξ1)pj(ξ1)hj(ξ1) = q1z1 + summationdisplay ξ1?ξ0 summationdisplay j ? ?summationdisplay ξ2?ξ1 q(ξ2)[pj(ξ2) + xj(ξ2)] ? ?hj(ξ1) = q1z1 + summationdisplay ξ2∈F2 q(ξ2) summationdisplay j [pj(ξ2) + xj(ξ2)]hj(ξ?2 ) = q1z1 + summationdisplay ξ2∈F2 q(ξ2) ? ?z(h,p)(ξ2) +summationdisplay j pj(ξ2)hj(ξ2) ? ? = q1z1 + q2z2 + summationdisplay ξ2∈F2 summationdisplay j q(ξ2)pj(ξ2)hj(ξ2) = ... = q1z1 + q2z2 +···+ qTzT + summationdisplay ξT∈FT summationdisplay j q(ξT)pj(ξT)hj(ξT) = qz = Q(z) a117Q(z) = q(z), ?z ∈M(p). a118 a30Qa237 a33a92a34a35 a234 a6a7a119 a14 a45a46 5.2.2 a47a48a51a52a53a54a55a56?a57a58a59 (5.2.4) a47a48a51a52a60a61 a62a63 a51a52a60 q a64a65a66 a67 a63 a51a52a53a54 a55a56 Q(z) = qz. a80a435.2.1a1205.2.2a121a244 a75a76a77(5.2.4) a122a123a243a124a125a92a36a219a18a126a250a127 a234a128a129 a14a124a125a92a36a220a130a92 a119 a80a81 a242 a75a76a77(5.2.4) a219a78 a234a131a132a133a134a135 a80a43a220a136a137a242a248 a68a94 a92a36a138a139a243a140 a116( a141a140 a116)?a75a76a77(5.2.4) a41 a23a38 a36 a235(a235) a219a78 a234 a236a39a68a94 a92a36 a237a142a143 a92a36a14 a144a145a146a147 a242a148 a149a150a24a230a151 a43a152 a150a153a154a155a151a156a157a158 a80a81a243 a245a230 (a245a159 a223a121a160a22 a150) a124a125a92a36 a157a158( a1614.2a162). a163 5.2.3 a164a165 4.2.8 a14a57a58a59 (5.2.4) a166a167a168a169a170a248 q(ξgg) + q(ξgb) = 0.9q(ξg) q(ξbg) + q(ξbb) = 0.8q(ξb) q(ξg)+ q(ξb) = 0.9 0.9q(ξg) + 0.8q(ξgb) = 0.75 c?a171a172a173 35 September 6, 2005 5.3a174 a105a106a175a176a177a178a179a180a181 a98a99a100 a101a102a103a104a25a105a106a107a108a109a110a25a111a103a112 a182a183 a57a58a184 a67a185a186 a64a66 a187a188 1 a189a190 q(ξg) = 0.3, q(ξb) = 0.6 a191a187a188 2 a189a190a54a50a192a193a194a195 a63 a57a58 q(ξgg) + q(ξgb) = 0.27 q(ξbg) + q(ξbb) = 0.48 a52a196a197a51 (a198a49a50a51a52) a52a60a61a49a50a51a52a189a190a54a50a52a47a48a199a200a201 a202 a47a48a203a204a14 a205 a189a190a54a50a52a206a184 a67 a199a200a201 a207a208 a193a206a209 a210 a52 a234 §5.3 a211a212a213a214a215a216a217a218a219a220 ? a68a94j a23a124a125ξt+1 (a221a222)a150a67a223a224a225a237a226a227a23ξt+1 (a221a222)a150a245a228(a229a230)a91a231a139a232 a227a23ξ t+1 a150a233 a162 a149ξ t = ξ?t+1 a150 a92a36a14 rj(ξt+1) = pj(ξt+1)+ xj(ξt+1)p j(ξt) (5.3.1) ? a234rj,t+1 a235a236 a68a94j a23 a249a250t + 1 a150a245a228a237a238 a14 a117 rj,t+1 = (rj(ξt+1),ξt+1 ∈Ft+1) ? a23a249a250 t + 1 a150a245a239a245a228a237a238a240a242a241 a242a243a244a52a14a245 a39a227a23a246a247a248a239a249 a4a250a22 a233 a162 a149a150 a249a250 -t+ 1a124a125a251a250a22 a150a252a253 ? a234 ˉr(ξt+1) a235a236 a23ξ t+1 a221a222 a150a245a228a254a255a0a237a238a253 ? ˉr(ξt+1)a160a1a2a3a124a125ξt+1,a86a4ξt+1 ? ξt. a5a1a2a3a124a125ξt. a118 a30 a14 a23 a249a250t+ 1a221a222 a150a245a228a254a255a0a237a238 ˉr t+1 a6 a242a7a17 a150 a35 a28 a241Ft a8a9 a150 a61 ? a249a4a23a249a250t+1a150a245a228a254a255a0a237a238a150a68a94a150a10a11a248a23a249a250ta12a13 a150a245a228a254a255a0a14a94a15 a249a2500 a12a13 a150a25 a249a250t+ 1a16 a228a150a17 a222 a14a94a253 ? a2a3a18a18a19a74a248 a23a24a245 a249a250 a150a24a245 a124a125a41 a23a249 a4 a245a228a254a255a0a237a238a150a68a94 ( a20 a68a94a77a21)a253 ? a245 a39a23a24a245 a249a250 a150a24a245 a124a125a41 a23a68a94( a20 a68a94a77a21)a249 a4 a38 a36a22 a150a25a254a255a0a150a245a228a237a238 a14a0a2 a3 a8 a232a80a81 a23 a124a125ξt a150a23a24a25a26 a242a27a28 a254a255a0a237a238a150a29a28 a248 ρ(ξt) = tproductdisplay τ=1 [ˉr(ξτ)]?1, t = 1,...,T (5.3.2) a1a30ξt a242ξτ a150a233a31a14 a117ξ τ ? ξt a253 ? ρ(ξt)a10 a249 a4a250a22 a233 a162 a149a150a144a32 a249a250-ta124a125ξt a241a250a22 a150 a14 a117ρ t a241Ft?1-a8a9 a150a253 ? a93ρ(ξ0) = 1. a79(5.3.2)a4 ρ(ξt) = ˉr(ξt+1)ρ(ξt+1) (5.3.3) September 6, 2005 36 c?a95a96a97 a98a99a100 a101a102a103a104a25a105a106a107a108a109a110a25a111a103a112 5.4a105a106a107a108a109a110 §5.4 a212a213a33 a34a35a36 ? a23a249a250T a150a124a125ξT a150a243a244a37a199a38a39a80a81a242a227a150a124a125a92a36a15a17a222a118a11a150a40a156a248 pi?(ξT) = q(ξT)ρ(ξ T) (5.4.1) a23 a249a250t < T a150 a124a125ξt a150a255a0 a30 a83a41a156 a80a81a242 pi?(ξt) = summationdisplay ξT?ξt pi?(ξT) (5.4.2) ? a255a0a30 a83a41a156 a241 a38 a36a22(a22)a150?a124a125a92a36a241 a38 a36a22(a22)a150a253 ? a246a247a124a125ξt a150a255a0a30 a83a41a156a42a43 pi?(ξt) = q(ξt)ρ(ξ t) (5.4.3) a68 a244a248 a44a45 a14 a46a47 a80a81(5.4.1)a14a8a9(5.4.3)a10a249a250T a124a125a12a13 a253 a1a48a14 a254 (5.4.1)a151a49 (5.4.2) a50 pi?(ξt) = summationdisplay ξT?ξt q(ξT) ρ(ξT) (5.4.4) a254 a124a125a92a36 a42a43a150a75a76a77(5.2.4) a5a234a3 a23ξ t a150a254a255a0a68a94 a14 a50 q(ξt) = summationdisplay ξt+1?ξt ˉr(ξt+1)q(ξt+1) (5.4.5) a79(5.3.3)a4 ˉr(ξt+1) = ρ(ξt)ρ(ξt+1) a14 a151a49(5.4.5)a50 q(ξt) = summationdisplay ξt+1?ξt ρ(ξt) ρ(ξt+1)q(ξt+1) a116 a234 a30a75a76a51a52 a13a53a54a14 a50 a16 q(ξt) = summationdisplay ξt+1?ξt summationdisplay ξt+2?ξt+1 ρ(ξt) ρ(ξt+1) ρ(ξt+1) ρ(ξt+2)q(ξt+2) = summationdisplay ξt+2?ξt ρ(ξt) ρ(ξt+2)q(ξt+2) ... = summationdisplay ξT?ξt ρ(ξt) ρ(ξT)q(ξT) (5.4.6) a234(5.4.4)a120(5.4.6)a117a50(5.4.3)a253 ? a10a249a250-0a124a125ξ0 a14(5.4.3)a12a242 pi?(ξ0) = q(ξ0)ρ(ξ 0) = 11 = 1 c?a95a96a97 37 September 6, 2005 5.5a105a106a107a108a109a110a55a56a57a58a175a176 a98a99a100 a101a102a103a104a25a105a106a107a108a109a110a25a111a103a112 ? a47a80a81(5.4.2)a14a4 1 = pi?(ξ0) = summationdisplay ξT?ξ0 pi?(ξT) a118 a30 a14pi? a59a221a241a245a239a41a156 a9a60 a253 ? a61a9(5.4.3) a235 a244 a255a0 a30 a83a41a156 a241a136a62a63a162a243 a150 a124a125a92a36 a253 ? a131a132a133a150a134a135a80a43 a8 a136a137a242a248 a38 a36a22(a22)a150a255a0a30 a83a41a156a150 a41 a23a83 a130a92a3 a68a94 a92a36a138a139a243a140 a116( a141a140 a116)a253 ? a255a0a30 a83a41a156 a241a64 a245a150? a20a21a241a16a17a18a19 a150a253 ? a245 a39a255a0 a30 a83a41a156 a241 a38 a36a22 a150 a14a0 a8 a232a80a81a65a125 a41a156 pi?(ξt+1 | ξt) = pi ?(ξt+1) pi?(ξt) , ?ξt+1 ? ξt (5.4.7) a79(5.4.7)a25(5.4.3)a66(5.3.3)a14a4 pi?(ξt+1 | ξt) = q(ξt+1)q(ξ t) ˉr(ξt+1) (5.4.8) a254 a30a151a49(5.2.4)a50 pj(ξt) = [ˉr(ξt+1)]?1 summationdisplay ξt+1?ξt pi?(ξt+1 | ξt)[pj(ξt+1)+xj(ξt+1)], ?t = 1,...,T, ?ξt, ?j. (5.4.9) a8a9a122a123a243 a255a0 a30 a83a41a156a150 a18a126a250a127 a253a227 a3 a8 a232a234a67a68a69a65a125 a255a0 a30 a83a41a156 a14a44a70 a153a154a255a0 a30 a83a41a156 a8 a116 a234a61a9(5.4.7)a53a54 a119a71a50 a248 pi?(ξt+1) = pi?(ξt+1 | ξt)·pi?(ξt), pi(ξ0) = 1. §5.5 a212a213a33 a34a35a36a72a73a74a75 a214a215 ? a76a77a78a79 a255a0 a30 a83a41a156 a14a7a17a80S a8a81 a12a82 a239a41a156a83a84a253S a8a85a4 a150 a8a9 a35 a28 a14a245a86a87ta146 a147 a68a88a14 a68a94a77a21a89a90 a14 a68a94 a92a36a14a91 a116 a14a130a130a14 a8a81 a12a92a93a94 a158a253 ? a92a93a94 a158a150a228a95a252 a14 a40 a245a121 a68a94j a23 a86a87ta150a82 a228a237a238r jt a96 a3 a255a0 a30 a83a41a156pi? a150a228a95a252a97 a98E?[r jt],a99a100a235a236 a228a95 a241 a96 a3pi? a251 a150 a61 a85 a69a14 a101a102 a234E[rjt] a235a236a96 a3“a103 a104a41a156”(a114a105a151 a43 a152a10a7a17 a150a106a107a108a109)a150a228a95a253 ? a234E?[rj,t+1 | ξt] a235a236 a23a41a156pi? a85r j,t+1 a96 a3a124a125ξt a150a65a125 a228a95 a14 a117 E?[rj,t+1 | ξt] = summationdisplay ξt+1?ξt pi?(ξt+1 | ξt)rj(ξt+1) (5.5.1) a234 E?t[rj,t+1] a235a236 rj,t+1 a96 a3 Ft a150a65a125 a228a95 a14 a227 a241a82 a239 F t a8a9a92a93a94 a158 a14 a23 a124a125 ξt a251 a252 E?[rj,t+1 | ξt]a253 ? a116a234a65a125 a228a95a150a110 a100a14 a8 a102 (5.4.9) a111a12 pjt = (ˉrt+1)?1E?t[pj,t+1 + xj,t+1] (5.5.2) September 6, 2005 38 c?a95a96a97 a98a99a100 a101a102a103a104a25a105a106a107a108a109a110a25a111a103a112 5.5a105a106a107a108a109a110a55a56a57a58a175a176 a128 a90a241a121a14 a68a94 j a150 a86a87 -t a92a36a130a3 a227a150 a82 a228a91a231a150 a65a125 a228a95a23 a82 a228a254a255a0a237a238a85a150a17 a222 a252 a14a112a30 a228a95 a241 a96 a3 a255a0 a30 a83a41a156 a251 a150a253 ? (5.5.2)a113 a8 a232 a46a47a237a238 a111a12 ˉrt+1 = E?t[rj,t+1] (5.5.3) a117 a14 a24 a82 a68a94a150 a82 a228a237a238a150 a65a125 a228a95 a130a3a82 a228a254a255a0a237a238 a14a112a30 a228a95 a241 a96 a3 a255a0 a30 a83a41a156 a251 a150a253 ? a163 5.5.1 a48a165 4.2.8 a37a14 a67a223 a242a243a244 a224a225a114 ˉr1(ξ0) = 1p 1(ξ0) = 10.9 = 1.11 ˉr2(ξg) = 1p 2(ξg) = 10.9 = 1.11 ˉr2(ξb) = 1p 2(ξb) = 10.8 = 1.25 a23a24a25a26 a114 ρ(ξgg) = ρ(ξgb) = [ˉr2(ξg)ˉr1(ξ0)]?1 = 0.81 ρ(ξbg) = ρ(ξbb) = [ˉr2(ξb)ˉr1(ξ0)]?1 = 0.72 ρ(ξg) = ρ(ξb) = [ˉr1(ξ0)]?1 = 0.9 a243a244a37a199a38a39a115a116a117a57a58a59 (5.4.9) a60a118 a253 a191 a117 a119a120a121a122a123 a48a165 5.2.3 a37 a124a125 a66a189a190a54a50a14 a120 a121 a115a116 a126 (5.4.3) a127a128 a124a125 a243a244a37a199a38a39 a253 a189a190a54a50 a114 q(ξgg) = 0.05, q(ξgb) = 0.22, q(ξbg) = 0.18, q(ξbb) = 0.3, q(ξg) = 0.3, q(ξb) = 0.6 a129 a243a244a37a199a38a39 a114 pi?(ξg) = q(ξg)ρ(ξ g) = 0.30.9 = 0.33 pi?(ξb) = q(ξb)ρ(ξ b) = 0.60.9 = 0.67 pi?(ξgg) = q(ξgg)ρ(ξ gg) = 0.050.81 = 0.061 pi?(ξgb) = q(ξgb)ρ(ξ gb) = 0.220.81 = 0.272 pi?(ξbg) = q(ξbg)ρ(ξ bg) = 0.180.72 = 0.25 pi?(ξbb) = q(ξbb)ρ(ξ bb) = 0.30.72 = 0.417 c?a171a172a173 39 September 6, 2005 5.6a105a106a107a108a130a103 a98a99a100 a101a102a103a104a25a105a106a107a108a109a110a25a111a103a112 a131 a197 pi?(ξgg) + pi?(ξgb) + pi?(ξbg) + pi?(ξbb) = 1 pi?(ξg)+ pi?(ξb) = 1 pi?(ξgg) + pi?(ξgb) = pi?(ξg) pi?(ξbg) + pi?(ξbb) = pi?(ξb) §5.6 a212a213a33a34a132a133 a102a255a0 a30 a83a41a156 a61a134(5.4.3)a151a49(5.2.3)a135 a50 Q(z) = qz = summationdisplay ξ∈Ξ q(ξ)z(ξ) = Tsummationdisplay t=1 summationdisplay ξt∈Ft q(ξt)z(ξt) = Tsummationdisplay t=1 summationdisplay ξt∈Ft pi?(ξt)ρ(ξt)z(ξt) = Tsummationdisplay t=1 E?[ρtzt] a61a134 Q(z) = Tsummationdisplay t=1 E?[ρtzt], ?z = (z1,...,z2) ∈Rk (5.6.1) a241a136a92a137a138a234 a255a0 a30 a83a41a156a150 a235a236 a253a227 a235a139 a140a141 a80a142 a238a150 a92 a252 a130a3 a17 a222 a91a231 a96 a3 a255a0 a30 a83a41a156a150 a228a95a150 a120 a253a143a144a119 a135 q(z) = Tsummationdisplay t=1 E?[ρtzt], ?z ∈M(p) (5.6.2) a163 5.6.1 ( a145a146a170 a223a147 a64a54)a48a165 3.3.1 a37 a120a121a148a149 a135a48 a187a188t a52 a67 a63 a189a190a135a167a150a48 a187a1880 a151 a187a188t a152a153 a118 a24 l a154 “down”, a129a155a189a190a54a50 a114 parenleftbigg u? ˉr ˉr(u?d) parenrightbigglparenleftbigg ˉr?d ˉr(u?d) parenrightbiggt?l a187a188 t a23a24 a39 ρt = (ˉr)?t a193 a186 a64a52 a253 a117(5.4.3) a135a243a244a37a199a38a39 a114 parenleftbiggu? ˉr u?d parenrightbigglparenleftbiggˉr?d u?d parenrightbiggt?l a117 a119 a48 a187a188 t a156 parenleftBigg t l parenrightBigg a63 a189a190a48 a187a188 0 a151 ta152a153a157a156 l a154 down a135a158 tsummationdisplay l=0 parenleftBigg t l parenrightBiggparenleftbigg u? ˉr u?d parenrightbigglparenleftbiggˉr?d u?d parenrightbiggt?l = parenleftbiggu? ˉr u?d + ˉr?d u?d parenrightbiggt = 1 a164a196a159 t a135 a187a188 t a160a156a189a190a52a243a244a37a199a38a39 a152 a151a161 a119 1 a253 a117 a119 a145a146a170a162a163 a207a208 a193a164a165a209 a210 a52a135 a62 a63a166 a64 a147a225a167 a48a168 a169a170a171a172 a153 a135a173 a205 a115 a126a174a175 a64a54a55a56a176a64a54 a253 September 6, 2005 40 c?a171a172a173 a98a99a100 a101a102a103a104a177a105a106a107a108a109a110a177a111a103a112 5.7a103a178a56a179 a180a181 a67 a63a182 a170a183 a147 a135 a155a184 a52a168 a169 a114a185a186 a184 a135 a149 a223a187a114 T a135 a188a189 a54a50 a114 K a253 a185a223a147 a48 a187a188 T a52 a174a175 a114 max{uT?ldl ?K,0} a190a191a192a119a187a188 0 a151 T a152a153down a118 a24 a52a154a193 l; a48 a155a190a187a188 a52 a174a175 a114 0 a253 a204 a126a194 a170 (5.6.2) a135a195 a149 a223a147 a48 a187a188 0 a52a54a50 a114 E?[ρTzT] = Tsummationdisplay l=0 parenleftBigg T l parenrightBigg 1 ˉrT parenleftbiggu? ˉr u?d parenrightbigglparenleftbiggˉr ?d u?d parenrightbiggT?l max{uT?ldl ?K,0} (5.6.3) a182a196 a193a145a146a170 a223a147 a64a54 a194 a170 a253 §5.7 a133a197a73a198 a199a228 a141 a80a142 a238a150 a92 a252a150a200a201 a120 a85a201 ( a2024.2)a135 a8 a232a234a124a125a92a203a20 a255a0 a30 a83a41a156 a67a204a205 a253 a206 a82 a239 a141 a80a142 a238z ∈Rk a135a207a208a209 qu(z) = max q∈Q {qz} ql(z) = min q∈Q {qz} a112a30Q = {q : q ≥ 0a210 a98 a124a125a92a203 a157a158}a253 a245a211z ∈M(p),a144a145qz a206a85a209q ∈Qa241a250a212 a150a213a214 a130a3 q(z)a118a70qu(z)a120ql(z)a214a130a3q(z). a245a211 a116 a234 a255a0 a30 a83a41a156 a235a236 a124a125a92a203a135a215 a201 a8a235a236 a98 qu(z) = max pi?∈Π Tsummationdisplay t=1 E?[ρtzt] ql(z) = min pi?∈Π Tsummationdisplay t=1 E?[ρtzt] a112a30Πa241a85a209 a255a0 a30 a83a41a156a150 a80 a21a253 §5.8 a216 a133a217 a93pi a241 a199a228a218a219 a30a7a220 a150 a103 a104a41a156 a135Ea241a103 a104a41a156a85a150a228a95 a69 a11a253a199a228a68a94a221a222 a30 a150 a64a54a223 a241a82 a239a91a231k q ∈M(p) a113a50 q(z) = Tsummationdisplay t=1 E[kqtzt], ?z ∈M(p) (5.8.1) a116 a234a103 a104a41a156pi a135 a200 a134 a8 a111a224 q(z) = summationdisplay ξ∈Ξ pi(ξ)kq(ξ)z(ξ), ?z ∈M(p) (5.8.2) c?a95a96a97 41 September 6, 2005 5.9a225a226 a98a99a100 a101a102a103a104a177a105a106a107a108a109a110a177a111a103a112 a227a228a229a230a231a21a89a90 a140 a232ξ t(t ≥ 1)a233 a49 a82a234 a229a230j a213a232a235 a82 a11a236a237ξ t+1 ? ξt a102 a112a238a239 a253a240a89a90a241a242 a243a244a231a98 z(ξt) = ?pj(ξt) z(ξt+1) = pj(ξt+1) + xj(ξt+1), ?ξt+1 ? ξt z(ζ) = 0, ?a112 a227ζ ∈ Ξ a245 a234(5.8.2)a50 0 = ?pi(ξt)kq(ξt)pj(ξt) + summationdisplay ξt+1?ξt pi(ξt+1)kq(ξt+1)[pj(ξt+1) + xj(ξt+1)] a246 kq(ξt)pj(ξt) = summationdisplay ξt+1?ξt pi(ξt+1 | ξt)kq(ξt+1)[pj(ξt+1)+xj(ξt+1)], j = 1,...,J, ?ξt, ?t ≥ 0. (5.8.3) a112a30kq(ξ0) = 1. a200 a134 a8 a111a224 kqtpjt = Et[kq,t+1(pj,t+1 + xj,t+1)], j = 1,...,J, ?t ≥ 0. (5.8.4) a245 a234a82 a228a237a238 a135 a200 a134 a8 a111a224 kqt = Et[kq,t+1rj,t+1], j = 1,...,J, ?t ≥ 0. (5.8.5) a143a144a247 a135a245a211a248 a232 a82 a239a249 a209a82 a228a254a255a0a237a238 ˉr t+1 a243a229a230 a135a215 kqt = ˉrt+1Et[kq,t+1], (5.8.6) a249 a220a250a251 a221a222 a30 a243a252a253a254a255a0 a134a1a238 kq(ξ) = q(ξ)pi(ξ) (5.8.7) a2a3a4a232(5.8.2) a5 a6z a98e(ξ)a246a7a8a9a10a2a11a12a13 a135 a249 a220a250a251 a221a222 a5 a243a252a253a254a12a14a15a16a17a18a19a20 a21a22a23a243a24a25a253 a203 a10 §5.9 a26a27 ? a244a28a243a253a203 a12a14a24a25a253 a203a135 a29a30 a5 a31a17a18 a135 a32a12a14a252a253a254a33a34a35 a135a250a36a37a38a39a40 a10a232a252a253a41a242 a229a230a42a14a29a30 a5 a31a17a18a43a44a45 a39a40a135a46 a12a47a48a29a30 a5 a31a17a18a7a49a50a51a14a52a53a54a244a28a243a229a230a243 a253 a203 a34a35a8a9a55a56a57a58a232a59a229a60a61 a5 a252a253a254a43a44a12a62a63a243a64a65a58a66a67a68 a38 a7a14a52a69a34a70a71a232a15 a16a17a18a0a243 a39a72a73a74a39a72a75 a58a47a76a58a76a42a14a15a16a17a18a77a14a29a30 a5 a31a17a18a78a48 a39a40 a10 September 6, 2005 42 c?a79a80a81 a82a83a84 a85a86a87a88a89a90a91a92a93a94a95a89a96a87a97 5.9 a98a99 ? a100 a42a252a253a254a14a101a102a68a33a103a104a58a232a2a105a106a107a0a58a108a252a253a243a244a28a232a235a109a24a25a14a2a109a20a243a244a28a56a110 a111a112a12a56a113a10a114a115a116a68(deflator)a12a2a109a252a253a254a243a102a68a10 a100a117 a61a118a58a119Duffie,a252a120a243a252a253a254a121 a62a63a243a56a122(a112a12a101a102a68)a58a123a124a125a126a48a114a115a116a68a10 ? a127a128 “a232a235a129a42a130a248 a232a131 a100 a129a132a133a29a30a70a71a243a229a230a231a134” a112a12a135a136a243a58a137a138a2a139a243a229a230a112 a248 a232a58a140a2325.4a236a243a141a142 a5 a58a7a14a143a144a101a125a131 a100a145 a243a129a132(a29a30)a70a71a243a229a230( a146 a229a230a231a134a147a148). a112a12a14a149a150a133a29a30a70a71a33a21a22a24a25a253a151a58a111a12a252a120a129a109a114a115a116a68a48a232a152 a100 a24a25a131 a100a153 a151 a145 a244 a28a243a229a230a231a134a147a148a58a16a154a14a2a109a114a115a116a68a33a21a22a24a25a253a151a10a235a129a109a114a115a116a68a252a120a23a129a109a129a155 a33a13a12a112a122a243a29a30 a5 a31a17a18a243a156a134a10 c?a79a80a81 43 September 6, 2005 5.9a98a99 a82a83a84 a85a86a87a88a89a90a91a92a93a94a95a89a96a87a97 September 6, 2005 44 c?a79a80a81 a157a158a159 a160a161a162a163a164a165a166 §6.1 a167a168 ? a169a170a73a169a170a171 a134a243a172a245a7a173a174a44a53a175a55a173 a234a176 a48a177a58a178a179a180a181a182a183a184a185a186a187a172a245a10a2a188a189a58 a3 a100a190a191 a63a192a193a194a195a172a245a7a173a196a197a42 a191 a63a198a7a199a194a195a172a245a10a200a201a247a58a2a202a203a204a58a3 a100a205 a129a206 a207a208a70a209a210a154a243a172a245a198a108a194a195a10 ? a53a175a243a172a245a211a212a213a214a169a170a73a169a170a171 a134a253a151 a5 a53a175a243a215a132a216a217a31a10a135a218a219 a169a220 a58a137a138a221a222a223a12 a197 a100a169a170a146a169a170a171 a134a223a224a225a111a112a12a125a63a223a253a151a58 a46a226 a2a105a53a175a223a215a132a216a217a31a219a186a227a228a224a225a12 a229a10 ? a230a231a232a233S a234 a223a129a109a235a236a237a238a239a240{y t}Tt=0 a241 a48a12a17a18a242a243pi a0a223a129a109a229a58a244 i) yt a245 a52a246a247 F t a12a7a242a223a248 ii) Et(yτ) = yt, ?τ ≥ t a101 a5Et a12pi a0Ft a234 a223a249a25a132a250a35a251a10a252a253 a169a220 a58ii) a75a254 a52 ii’) Et(yt+1) = yt, ?t < T. ? a169a170a146a169a170a171a134a223a224a225a100a255a105a229a0a189a228a129a109a12a14a29a30a5a31a17a18a58a1a129a109a12a14a15a16a17a18a73a2a254 a254a10 ? a135a218a127a2 a228 a205a3 a129a109a24a25a4 a205 a129a109 a169a170a146a169a170a171 a134a131 a100a153 a151 a145 a223a129a132a133a29a30a70a71a10 §6.2 a5a6a7a8a9a5a6 ? a205 a42a130t a10a11 a223a12 a169a170j a223a13a14a15a197a100a147a148a16a17a223a18a28a205a3a109a42a130τ < ta75 a52a172a225x jτ a58a111 a205 a42a130t a75 a52p jt+xjt a10a172a225x jτ a19a20a21 a173a22a6a129a132a133a29a30a70a71a58a101 a205 a42a130ta223 a254a23 a48(ρ τ/ρt)xjτ, a101 a5ρτ a12a205 τ a223a24a25a47a251a10a51a42a130 0 a9a42a130 ta197a100a169a170 a152a26a223a224a225a14a42a130 -ta186a27a28a29a33a243 a238a58a15 a2 a120a48a30a13a14a15a197 a100 a147a148a223a42a130 -ta18a28a121a42a130ta210a31a223a18a28 a19a20a21 a173a22a6a129a132a133a29a30 a70a71a223 a254a23 a210 a73 a10 ? a145 a212a32a58 a169a170j a205 a24a25ξ t(t ≥ 1) a223a224a225g j(ξt) a108 a2 a120a48 gj(ξt) = pj(ξt) + [ρ(ξt)]?1 tsummationdisplay τ=1 ρ(ξτ)xj(ξτ) a101 a5ξτ a12ξt a205τ a223a33a34a35a36a10a205 a42a1300a223a224a225 a75 a52 a254 a151a228 gj(ξ0) = pj(ξ0) a37a202a9x j(ξ0) = 0,a234a255 a212a7a38a129a39a40 gj(ξt) = pj(ξt) + [ρ(ξt)]?1 tsummationdisplay τ=0 ρ(ξτ)xj(ξτ), t = 0,...,T (6.2.1) 45 6.2 a41 a42a43a44a45 a41 a42 a82a46a84 a47a48a49 a41 a42a43a50 a148a51a24a25a52a53a58 a234 a212a7a54a39a40 gjt = pjt + ρ?1t tsummationdisplay τ=0 ρτxjτ, t = 0,...,T (6.2.2) a55 a169a170 j a205 a42a130 ta223a224a225 a75 a52a30 a169a170a205 a42a130 t a56 a101a173a31a57a109a42a130a223a172a225 a205 a42a130 ta223 a254a23 a19 a195 a234 a30 a169a170a205 a42a130ta223 a254 a151a10 ? a169a170j a205 a42a130ta223a58a59a60a61a12a14a42a130-0a186a27a28a29a243a238a223a224a225a228 djt = ρtgjt = ρtpjt + tsummationdisplay τ=0 ρτxjτ, t = 0,...,T (6.2.3) a55 a169a170j a205 a42a130ta223a24a25a224a225 a75 a52a42a130ta223a24a25 a254 a151a121a51a42a1300a9a42a130ta223a24a25a172a225a210 a73 a10 ? a62(6.2.3)a100 dj,t+1 ?dj,t = ρt+1(pj,t+1 + xj,t+1)?ρtpj,t, t = 0,...,T ?1 (6.2.4) a55a24a25a224a225 a205 a129a132a63a223a237a64 a75 a52 a190 a31a172a225a223a24a25 a23 a195 a234 a24a25 a254 a151a223a237a64a10 ? a12a52(a65 a24a25a223)a224a225a58 a62(6.2.2)a100 gj,t+1 ? ˉrt+1gj,t = xj,t+1 + pj,t+1 ? ˉrt+1pj,t, t = 0,...,T ?1 (6.2.5) a55a66a132a67a186a27a28a29a34a35a223a224a225 a205 a129a132a63a223a237a64 a75 a52a132a67a223a18a28a187a51a132a68 a254 a151 a205 a132a67a223 a254a23 a10 ? a69a70 a32a58a137a138a129a109 a169a170a71a205a10a72 a42a130a131 a100 a174a73a172a225a58 a46a226 a125a223a224a225 a205 a174 a10a72 a42a130 a75 a52 a254 a151a58 a205 a10a72 a42a130 a75 a52a172a225a58a125a223a24a25a224a225 a205 a174 a10a72 a42a130 a75 a52a24a25 a254 a151a58 a205a10a72 a42a130 a75 a52a24a25a172a225a228 gjt = braceleftBigg pjt, t < T, xjT, t = T, djt = braceleftBigg ρtpjt, t < T, ρTxjT, t = T. ? a169a170a171 a134a147a148a223a224a225 a73 a24a25a224a225a223 a2 a120a74a75a52a12 a169a170 a223 a2 a120a55 a169a170a171 a134a147a148h a205 a42a130ta223a224a225g t(h)a75 a52a30a147a148 a205 a42a130 t a56 a101a173a31a57a42a130a223a18a28 a205 a42a130 ta223 a254a23 a19 a195 a234 a30 a169a170a205 a42a130ta223 a254 a151a58a55 gt(h) = ptht + ρ?1t tsummationdisplay τ=1 ρτzτ(h,p), t = 0,...,T (6.2.6) a101a76a77 a2 summationtext0 τ=1 = 0,a78a79a80 a13a58g 0(h) = p0h0. ? a169a170a171 a134a223a24a25a224a225 a79a80 a224a225a223a24a25 a23 a58a55 dt(h) = ρtgt(h) = ρtptht + tsummationdisplay τ=1 ρτzτ(h,p), t = 0,...,T (6.2.7) a62(6.2.6),a100 gt+1(h)? ˉrt+1gt(h) = (xt+1 + pt+1)ht ? ˉrt+1ptht, (6.2.8) a51a111 dt+1(h)?dt(h) = ρt+1(pt+1 + xt+1)ht ?ρtptht (6.2.9) September 6, 2005 46 c?a81a82a83 a82a46a84 a47a48a49 a41 a42a43a50 6.3a44a45 a41 a42a43a50 §6.3 a8a9a5a6a7a84 a85a86 6.3.1 a87a88a89a87a88a90a91a92a93a94 a58a59a60a61a95a96a97a98a99a100a101a102 a94a103 a58a104 E?t[djτ] = djt, ?τ ≥ t, ?j E?t[dτ(h)] = dt(h), ?τ ≥ t, ?h a169 a228 (1) a62(5.5.2)a58 pjt = (ˉrt+1)?1E?t[pj,t+1 + xj,t+1] a255a72 a122a113a173a24a25a47a251ρ t a58a105 ρtpjt = ρt+1E?t[pj,t+1 + xj,t+1] (6.3.1) a62 a52E? t(ρtpjt) = ρtpjt a58 a234 a212a106a39a40 E?t[ρt+1(pj,t+1 + xj,t+1)?ρtpjt] = 0 a62(6.2.4)a58a234 a212a48 E?t[dj,t+1 ?dj,t] = 0. a62 a52E? t[dj,t] = dj,t a58 a234 a212a40a48 E?t[dj,t+1] = dj,t, ?t < T. (2)a205(6.3.1)a255a72a122a113a173hjt a58a105 ρtpjthjt = ρt+1E?t[(pj,t+1 + xj,t+1)hjt], j = 1,...,J. a234 a212 a245 a52j a107a73 a105 ρtptht = ρt+1E?t[(pt+1 + xt+1)ht]. a62 a52E? t(ρtptht) = ρtptht a58 a234 a212a106a39a40 E?t[ρt+1(pt+1 + xt+1)ht ?ρtptht] = 0. a62(6.2.9)a58a234 a212a48 E?t[dt+1(h)?dt(h)] = 0. a62 a52E? t[dt(h)] = dt(h) a58 a234 a212a40a48 E?t[dt+1(h)] = dt(h), ?t < T. ? a62 a52E? 0 a79a80a245 a52pi? a223a133a249a25a132a250a58 a2 a1816.3.1 a108a109 a204 E?[djτ] = dj0 = pj0, ?τ E?[dτ(h)] = d0(h) = p0h0, ?τ a110 a79a80 a13a58a143a144 a169a170a146a169a170a171 a134a147a148 a205a3 a109a42a130a223a24a25a224a225 a245 a52a29a30a76a31a17a18a223a132a250 a75 a52a101 a42a130-0 a254 a151a111 c?a79a80a81 47 September 6, 2005 6.4 a41 a42a43a50 a82a46a84 a47a48a49 a41 a42a43a50 ? a12a52a71a205a10a72a42a130a131a100a174a73a172a225a223a169a170 (a77a137j)a33a13a58a2a1816.3.1a40a48 E?t[ρτpjτ] = ρtpjt, ?τ ≥ t, τ < T, E?t[ρTxjT] = ρtpjt, ?t < T. a234a112a113 a129a109a212a251a0a189a24a25 a254 a151a131 a100 a229a31a136a111 ? a114 6.3.2 a115a116 4.2.8 a98a58a87a88 1 (a117a118a119a88) a94 a58a59a60a61a120 d1(ξg) = d1(ξb) = 0.9×1 = 0.9 d12 = 0.9 a87a88 2(a121a118a119a88) a94 a58a59a60a61a120 d2(ξg) = 0.9×0.9 = 0.81 d2(ξb) = 0.9×0.8 = 0.72 d2(ξgg) = d2(ξgb) = ρ2(ξgg)·x2(ξgg) = 0.9×0.9×1 = 0.81 d2(ξbg) = d2(ξbb) = ρ(ξbg)·x2(ξbg) = 0.8×0.9×1 = 0.72 a115a116 5.5.1 a98a122a123a124a125a96a97a98a99a100a101a228 pi?(ξg) = 13, pi?(ξb) = 23, pi?(ξgg) = 581, pi?(ξgb) = 2281, pi?(ξbg) = 14, pi?(ξbb) = 512 a126a127a128 a87a129a130a131a87a88a94 a58a59a60a61a132a133a134 a103 a99a135 ( a93) a111 §6.4 a5a6a7a84 a85a86 6.4.1 a136a137a87a88a89a87a88a90a91a92a93a94 a60a61a138a139a140a141 a94a142a143 a95a144a145a100a101a146 a94a103 a58a104 Et[gjτkqτ] = gjtkqt, ?τ ≥ t, ?j Et[gτ(h)kqτ] = gt(h)kqt, ?τ ≥ t, ?h a169 a228 (1) a62(5.8.4)a73(5.8.6)a100 kqtpjt = Et[kq,t+1(pj,t+1 + xj,t+1)] (6.4.1) kqt = ˉrt+1Et[kq,t+1] (6.4.2) a147a76a58 Et[kq,t+1(pj,t+1 + xj,t+1 ?pjtˉrt+1)] = 0 (6.4.3) a62(6.2.5)a58a234 a212a106a39a40 Et[kq,t+1(gj,t+1 ? ˉrt+1gjt)] = 0 September 6, 2005 48 c?a79a80a81 a82a46a84 a47a48a49 a41 a42a43a50 6.4 a41 a42a43a50 a147a148E t(kq,t+1ˉrt+1gjt) = ˉrt+1gjtEt(kq,t+1) = gjtkqt,a100 Et(kq,t+1gj,t+1) = kqtgjt (2)a205(6.4.3)a255a72 a122a113a173h jt a154 a245a149 j a107a73 a58a105 Et[kq,t+1(pt+1 + xt+1 ? ˉrt+1pt)ht] = 0 a62(6.2.7)a58a234a212a106a39a148 Et[kq,t+1(gt+1(h)? ˉrt+1gt(h))] = 0 a225a150(6.4.2)a58 a62a234 a212a106a105 Et[kq,t+1gt+1(h)] = kqtgt(h), ?t < T. a62 a149 E0 a79a80 a133a249a151a132a250a58 a2 a1816.4.1 a108a109 Et[gjτkqτ] = gj0 = pj0, ?τ, Et[gτ(h)kqτ] = g0(h) = p0h0, ?τ. a62 a149 E[gjτkqτ]a80 a224a225g jτ a223a152a130-0 a254 a151a58 a234 a212a153 a220 a143a144 a169a170a146a169a170a171 a134a147a148 a205 a143a144a152a130a223a224a225a223 a152a130-0 a254 a151 a75 a149 a30 a169a170a146a169a170a171 a134a147a148a223a152a130 -0 a254 a151a154 c?a79a80a81 49 September 6, 2005 6.4 a41 a42a43a50 a82a46a84 a47a48a49 a41 a42a43a50 September 6, 2005 50 c?a79a80a81 a157a155a159 a156a157a158a159a162a160a161a160a161 §7.1 a167a168 a162 a149 a186a27a223 a169a170a2a254 a219 a3 a109 a169a170a163a146a169a170a171 a134a164a223a165a166a24a167a121a30 a169a170 a70a71a223 a74a168a72a169a170a171a172 a111 a135a218 a205a173 a132a211a201a223a174a175a176a58a148a131 a100 a150a132a250a177a150a0a189a223a177a150a178a68a223a103a181a179a180a213a110a109 a245 a170 a111 §7.2 a181a182a183a184 ? a129a109a186a27a185a247 c ∈ Rk+1 a106a173a186a40a129a109Ft a106a242a178a68 ct a223 T + 1 a187a171{c0,...,cT}a58a125a205a230a231 s ∈ S a176a223a59a25a148c(s) = [c0(s),...,cT(s)]. ? a129a109a103a181a179a223a177a150a178a68 u : Rk+1 → R a241 a148a131 a100a230a231a188a189 a223 a118a190 a191a192a193a194a58a137a138a4 a205 a129a109a178 a68V : RT+1 →R a73S a234 a223a129a109a195a196a242a243pi, a197 a105a12?c,cprime ∈Rk+1 a100 u(c) ≥ u(cprime) ? Ssummationdisplay s=1 pisV (c(s)) ≥ Ssummationdisplay s=1 pisV (cprime(s)). (7.2.1) ? a132a250a177a150a0a189a76a223a195a196pi a241 a148a198a199a195a196a111 a230a231 a156S a234 a223 a3 a109a106a242a178a68a106a186a40a131 a100 a195a196a242a243pi a223S a234 a223a129a109a235a236a237a238a111 a245a149 pi a223a132a250a200a148E,a140a132a250a177a150a178a68a148 E[V (c)] = Ssummationdisplay s=1 pisV (c(s)) (7.2.2) a201a111(7.2.1)a106a0a189a148 u(c) ≥ u(cprime) ?E[V (c)] ≥E[V (cprime)] (7.2.3) ? a178a68V a80a173 a132a186a27a223Von Neumann-Morgenstern (VNM)a177a150a178a68a154 Va223a129a109a202a150a203a212 a80 a152 a233 a106a246a203a212a228 V (y) = Tsummationdisplay t=0 δtυ(yt), ?y = (y0,...,yT) ∈RT+1 (7.2.4) a101a76υ : R → R a80 a129a109a152 a233 a112a237a223a204a206a177a150a178a68a58 δ a80 a129a109a152 a233 a112a237a223a24a25a147a251a58 0 < δ a205 a206a202δ < 1. ? a131a100 a152 a233 a106a246a223VNMa177a150a178a68a223a132a250a177a150 a80 E[V (c)] = Tsummationdisplay t=0 summationdisplay s∈S pi(s)δtυ(ct(s)) (7.2.5) a234 a212a106a207a129a208a39a40 E[V (c)] = Tsummationdisplay t=0 δtE[υ(ct)] (7.2.6) 51 7.3a90a91a209a210 a82a211a84 a212a213a214a215a49a47a48a216a217 a62 a149 ct a80Ft a106a242a223a218(7.2.5)a219 a106a39a40 E[V (c)] = Tsummationdisplay t=0 summationdisplay ξt∈Ft pi(ξt)δtυ(c(ξt)) (7.2.7) a220a76pi(ξ t) = summationtext s∈ξt pi(s)a80a221 a151ξ t a223a195a196a111 ? a173a222 a186a27a185a247a223a223a224a223 a222 a250a177a150a0a189a223a178a181a64a74a75 a149 a28 a222 a186a27a185a247a223a178a181a64a111 §7.3 a225a226a227a228 ? a229a230a222 a250a177a150a178a231(7.2.2)a223a232a233a234 a241 a148 a80 a96a97a235a236 a94 a218a237a238a239a240a241a223a224a242a27a185a247a223 a222 a250a243a244 a242a27a185a247a245a246a218a55 E[V (c)] ≤ V (E(c)), ?c ∈Rk+1 a220a76E(c)a0a247a248a249a250 a2a251a252a173a222 a242a27a185a247 [c 0,E(c1),...,E(cT)]. ? a232a233a234 a241 a148 a80 a96a97a98a99 a94 a218a237a238 E[V (c)] = V (E(c)), ?c ∈Rk+1 ? a232a233a234 a241 a148 a80a253a254 a96a97a235a236 a94 a218a237a238 E[V (c)] < V (E(c)), ?a255 a250a0 a251 a242a1a185a2c ? (1)a248a249a232a233a234a80 a165a166a3a4 a252?a239a240a241a252VNMa177a150a178a231V a80a5a252 a111 (2)a248a249a232a233a234a80a165a166a76a251a252?a239a240a241a252VNMa177a150a178a231V a80a6a251a252 a111 (3)a248a249a232a233a234a80a7a8a165a166a3a4a252?a239a240a241a252VNMa177a150a178a231V a80a7a8a5a252 a111 a9a10 (1) ?) a11Jensena12a13a14a15a16a17 ?) a11a5 a178a231 a252 a0a18 a15a16a17 ... ? V a19a6a251a252 a218 a15 V (y) = Tsummationdisplay t=0 αtyt, ?y ∈RT+1 a220a20α t > 0,?t. a21a22a23 a147a24 a19 a152a25 a12a26a252a27 a203a218 a230αt = δt. §7.4 a28a29a30a31a32a33a31a32 ? a173a222a34a35 a20a162 a149 a242a1 a252 a9a36a0a37a38a39a40a41 a252a42 a151a43 a168a44a45a42 a151 a168a44a17 ? a9a36j a45k a252 a248 a222 a40a41r j,t+1 a45rk,t+1 a46 a25 a252a42 a151a43 a168a44a19a47a48 a249a40a41 a252a49a50a252a42 a151 a222a51a52 a53a54a55 a252a42 a151 a222a51a252a49a50 a218 a15 Covt(rj,t+1,rk,t+1) = Et(rj,t+1rk,t+1)?Et(rj,t+1)Et(rk,t+1) September 6, 2005 52 c?a56a57a58 a59a211a60 a212a213a214a215a61 a62a63 a216a217 7.5a212a213a214a215a61 a62a63 a216a217 ? a40a41rj,t+1 a45 a54a198a64 a46 a25 a252a42a65 a43a66 a44a19rj,t+1 a252a42a65 a66 a44 a218a67a68 Vart(rj,t+1) = Et(r2j,t+1)?E2t(rj,t+1) a69a70 a252a71a72a44 a67a68σ t(rj,t+1) = radicalbigVar t(rj,t+1). §7.5 a73a74a75a76a77a78a79a80a81 ? a232a233a234a252a222a51a82a83a84 a231a85a86a87a88 summationdisplay s∈S pisV (c(s)) = summationdisplay ξt∈Ft summationdisplay s∈ξt pisV (c(s)), t = 0,...,T a89a90a242a1 a21a221a65ξt a252a91a92a82a83 a68 summationdisplay s∈ξt pis?tV (c(s)) (7.5.1) a220a20? tV (c(s))a93 a247 V a94a95t a249a26a96(a15a97a98t a242a1) a252 a223a99a231 a17a47a100a101 a218a237a238 VNM a82a83a84 a231V a12a229a97 a25a85a102 a251a252a103 a218a242a1 a21a104a105a97a98a252a91a92a82a83a106a107a108a109a110a230a97a98a252 a242a1 a17 a111 a83a42a65 a222a51 a218a85a112(7.5.1)a87a88 pi(ξt)E[?tV|ξt] (7.5.2) a220a20? tV a93 a247a113a114a68? tV (c(s))a252a115a116a26a96a17 ? a211.6a117 a20a118a55a119a120a121a242a1-a9a36a122a123a124a125a126a127 max u(c) s.t. c(ξ0) = w(ξ0)?p(ξ0)h(ξ0) c(ξt) = w(ξt) + z(h,p)(ξt), ?ξt ∈Ft,t = 1,...,T a128a99a129a121a220a248a130a131a132 a251a42a65 pj(ξt) = summationdisplay ξt+1?ξt [pj(ξt+1)+ xj(ξt+1)] ?ξt+1u? ξtu , ?j, ?ξt, t < T a220a20? ξtu = ?u?c(ξt) a19 a242a1 a21ξt a252a91a92a82a83 a218a112a220 a83(7.5.2)a232a133a16a134 pj(ξt)E[?tV|ξt] = summationdisplay ξt+1?ξt [pj(ξt+1) + xj(ξt+1)]pi(ξt+1)pi(ξ t) E[?t+1V|ξt+1] = summationdisplay ξt+1?ξt [pj(ξt+1) + xj(ξt+1)]pi(ξt+1)pi(ξ t) summationdisplay s∈ξt+1 pi(s) pi(ξt+1)?t+1V (c(s)) = summationdisplay ξt+1?ξt summationdisplay s∈ξt+1 [pj(ξt+1) + xj(ξt+1)] pi(s)pi(ξ t) ?t+1V (c(s)) = summationdisplay s∈ξt pi(s|ξt)[pj,t+1(s) + xj,t+1(s)]?t+1V (c(s)) = E[(pj,t+1 + xj,t+1)?t+1V|ξt], ?j, ?ξt, t < T (7.5.3) c?a56a57a58 53 September 6, 2005 7.6 a135a136 a137a138a139 a61 a62a63 a216a217 a59a140a60 a212a213a214a215a61 a62a63 a216a217 a141a53 a221a65a142a143 a218a144 a14 a85a145a87a68 pj,tEt[?tV ] = Et[(pj,t+1 + xj,t+1)?t+1V], ?j, t < T (7.5.4) a111 a83 a40a41a218a85a86a112a144 a14 a87a88 Et[?tV ] = Et[rj,t+1?t+1V], ?j, t < T (7.5.5) ? a146 a0 a21a147 a248 a97a98t(t < T)a221a65a148a21 a9a36(a240a149a9a36a122a123) a229a230 a248 a222a150a151a152 a40a41 ˉr t+1. a112(7.5.5) a70 a109a153a151a152 a9a36 a16a134 ˉrt+1 = Et[?tV ]E t[?t+1V ] , ?t < T (7.5.6) ? a154a155a156 a99a9a36j a252a157a158a159a160a161a162a163a164Et[rjt+1]? ˉrt+1 a252a93a165a14a166 rj,t+1 a167 ?t+1V a46 a25 a252a42a65 a43a66 a44 a68 Covt(rj,t+1,?t+1V ) = Et(rj,t+1?t+1V )?Et(rj,t+1)Et(?t+1V ) a89a90a168 a11(7.5.5)a169(7.5.6)a16a134 a9a36j a252 a248a170a40a41 a252a42a65 a170 a51 a68 Et(rj,t+1) = ˉrt+1 ? ˉrt+1Covt(rj,t+1,?t+1V )E t[?tV] (7.5.7) a144 a14a171a19a172 a170a9a36 a34a35 a20a173a174a175a176a177a178a179a180a181a182 a14a17 a54 a100a101 a168 a147 a248a9a36j a252a42a65 a248a170 a151a152a22 a183a88a184a185 a109a153 a9a36 a252 a248a170a40a41 a167 a97a98ta169 t + 1 a242a1a25 a252a91a92 a133a232a186 a46 a25 a252a42a65 a43a66 a44a252a187 a114a168a184a185a188a189 a171a19 a248a170 a150a151a152 a40a41ˉr t+1 a17a7a8a190a100 a168 ?t+1V Et[?tV ] a12a19 a170 a51a82a83a154a252a91a92 a133a232a186a168 a191 a149 a46 a25a69 a44 a248a249 a42a65a192 a186 a17 ? a237a238a232a233a234a252 a242a1 a21a147 a249 a97a98a193a19 a250a0 a251a252 a168a194a195 a91a92a82a83?tV a21a147 a249 a97a98a196a19 a250a0 a251a252 a168 a90 a97a197a109 a242a1 a252 a0a37a182 a14(7.5.7)a198a199a200 a182a201a0a37a168 a15a147 a249a9a36 a252 a248a170 a42a65 a170 a51 a40a41 a13a109 a248a170 a150a151a152 a40a41 a17 §7.6 a202a203a204a205a206a77a78a79a80a81 ? a207 a170 a91a92 a133a232a186 a107a108a109a110a208a97a98a252 a242a1a168 a47a209a16 a182 a14(7.5.6)a169(7.5.7)a21 a70 a83a210a211 a20 a12 a66a212 a17 a89a90a168a248a213 a209a83a97 a25a85a102a170 a51a82a83a84 a189(7.2.7). ? a21(7.2.7)a154 a168a242a1 a21a214a65ξt a252a91a92 a170 a51a82a83 a68 pi(ξt)δtυprime(c(ξt)) a109a19 a242a1-a9a36a122a123a124a125a126a127 a252 a248a130a131a132 a251a42a65 a88a68 pj(ξt)υprime(c(ξt)) = δ summationdisplay ξt+1?ξt [pj(ξt+1)+ xj(ξt+1)]pi(ξt+1)pi(ξ t) υprime(c(ξt+1)) a15 pj(ξt)υprime(c(ξt)) = δE[(pj,t+1 + xj,t+1)υprime(ct+1)|ξt] (7.6.1) September 6, 2005 54 c?a56a57a58 a59a140a60 a215a216a217a218a61 a62a63a219a220 7.7 a221a222a223a224 a225a226a227a61a228a229a230 a231a20υprime(c t+1)a93a232 a113a114a68 υprime(c(ξ t+1)) a252a115a116a26a96 a168 ξ t+1 ? ξt. a141a53 a101 a233 a252a214a65a142a143 a168 (7.6.1) a85a87a88 pjtυprime(ct) = δEt[(pj,t+1 + xj,t+1)υprime(ct+1)], ?j (7.6.1prime) a111 a83 a40a41a168a144 a14 a85a86a87a88 υprime(ct) = δEt[rj,t+1υprime(ct+1)], ?j, ?t < T. (7.6.2) a112a144 a14 a70 a83a109a234 a170 a150a151a152 a40a41a168 a208 ˉrt+1 = δ?1 υ prime(ct) Et[υprime(ct+1)], ?t < T (7.6.3) a11a109rj,t+1 a235 υprime(ct+1) a46 a25a236 a42a65 a43a66 a44 a68 Covt(rj,t+1,υprime(ct+1)) = Et[rj,t+1υprime(ct+1)]?Et(rj,t+1)Et[υprime(ct+1)] a89a90a168 a11(7.6.2) a235 (7.6.3)a16a134 Et(rj,t+1) = ˉrt+1 ?δˉrt+1Covt(rj,t+1,υ prime(ct+1)) υprime(ct) (7.6.4) a47a171a19a97 a25a85a102 a154 a236 a197a109a237 a1a236a9a36a0a37a182 a14a17 ? a238a239a240a241a242a19a151a152 a20a243a236a168a194a195(7.6.4) a198a199a200 a182a201a0a37a168(7.6.3) a198a199a200a234 a170 a150a151a152 a40a41 a13a109 a22a23 a89a24a236a244a189 a17 §7.7 a245a246a247a248a249a250a251a77a252a253a254 a197a109a237 a1a236a9a36a0a37a85 a83a109a156 a99 a207 a170 a91a92 a133 a240 a186a236 a71a72a44 a236 a154a255a17 a0a1(7.5.5) a235 (7.5.6)a168a208 Et[(rj,t+1 ? ˉrt+1)?t+1V ] = 0 (7.7.1) a91a92a82a83?t+1V a235 a234 a170a2a3a40a41r j,t+1 ? ˉrt+1 a46 a25a236 a42a65 a69a4a188a189a68 ρt = Covt(?t+1V,rj,t+1 ? ˉrt+1)σ t(?t+1V )σt(rj,t+1 ? ˉrt+1) =(7.7.1) ?Et(?t+1V )·Et(rj,t+1 ? ˉrt+1)σ t(?t+1V )σt(rj,t+1) = (7.5.6) ? Et(?tV )Et(rj,t+1 ? ˉrt+1)ˉr t+1σt(?t+1V )σt(rj,t+1) a11a109|ρt|≤ 1a168a208 σt parenleftbigg ? t+1V Et(?tV ) parenrightbigg ≥ |Et(rj,t+1)? ˉrt+1|ˉr t+1σt(rj,t+1) , ?j, ?t < T (7.7.2) a144 a14a100a101 a168 a97a98 t a235 t + 1a237 a1a25a236 a91a92 a133 a240 a186a236 a42a65a5a6 a186a7 a109a147a234 a9a36a236 sharpe a184a186 a167 a234 a170 a150a151a152 a40a41a236a184a114a236a8 a94 a114 a17 c?a56a57a58 55 September 6, 2005 7.8a9a10 a59a140a60 a215a216a217a218a61 a62a63a219a220 a11 a198(7.5.5)a94a109a104a105 a9a36a122a123a12a141a236 a234 a170a40a41a13a88a14a168a15a16 (7.7.2) a94a109a104a105 a9a36a122a123a12a141a236 a234 a170a40a41a13a88a14a168a89a90 a94a110a208a234 a170a40a41r t+1 a113a144a17 a255a16a134a207 a170 a91a92 a133 a240 a186a236 a42a65a5a6 a186a236 a234a18a154 a255 a10 σt parenleftbigg ? t+1V Et(?tV ) parenrightbigg ≥ sup rt+1 |Et(rt+1)? ˉrt+1| ˉrt+1σt(rt+1) (7.7.3) a19 a155a20a21 a236a22a23 a34a35 a189a1 a97 a168a144 a14a24a25 a121 a234a18a26a242a27a28 a236a29 a239 a10 a234 a66 a155 a168a30a31a32a33 a134a234a34a35 a22a23 a34a35a36 a189a236 a151a152a22 a183a69 a94 a7 a109a153a36 a189a40a41a236 a5a6 a186a168a89a16 a153a36 a189a236sharpe a184a186a37a7a168a15a16 a207 a170 a91 a92 a133 a240 a186a236 a42a65a5a6 a186a37a7a38a39 a234 a66 a155 a168a30a31a32a33 a134a237 a1 a5a6 a37a40a168a16a40a236 a237 a1 a5a6a41a208 a19 a240a241 a242a19a42 a231 a151a152 a3a4a236a43a85a44a45 a115 a7a236 a91a92 a133 a240 a186 a5a6 a168a89a68 a151a152 a3a4 a13a46a109a82a83a84 a189a236a47a186a168a7 a151a152 a3a4 a198a199a200 a19 a237 a1 a208a48a49 a236 a26a50a97 a168 a237 a1a236 a91a92a82a83a208a48a51 a236 a26a50 a168a89a16a40 a151a152 a3a4a52a53 a200 a94a12a46 a236 a237 a1a183a201a168 a237 a1a236 a91a92a82a83a44a54a48a49a17a240a241a242a19 a7 a151a152 a3a4a236 a47a234 a29a120a55a56a57a68 a19a234a18 a58a168a89a68a54 a167a59 a9a9a1a60a61a168a16a62 a21a63 a3 a198 a18a144a168 a191 a149a64a65 a193 a52a66a67a68a236 a151a152 a3a4 a17a47a18a69a214 a88a68 a22a23a70a37 a46 a58 a17 §7.8 a71a72 a7a8a190a100 a168a182 a14(7.5.7) a235 (7.7.3) a20a236 ?t+1VEt(?tV) a12a19a97a98t a235 t+ 1a237a73 a25a236 a91a92 a133 a240 a186 a17 a0a1(7.5.2)a168 a237a73a21ξt a91a92a82a83 a68pi(ξ t)E[?tV|ξt] a168a89a90 a21a214a65ξt a235 a231a24 a214a65ξt+1 a236 a237a73 a236 a91a92 a133 a240 a186(a68 a91a92a82a83 a236a74a186)a68 pi(ξt+1)E[?t+1V|ξt+1] pi(ξt)E[?tV|ξt] (7.8.1) a89a90a168 ?t+1V Et(?tV ) a75a76 a214a65a192 a186 a235 a21 a102a24a20a236 a42a65 a170 a51a17 a75a76 a42a65 a170 a51a77a19a142a143 a144a236 a214a27 a168a89a68r j,t+1 a19Ft+1 a85a78a236a168a79a80 a240 a170 a51a81a82a208 Covt(rj,t+1,?t+1V ) = Et(rj,t+1?t+1V )?Et(rj,t+1)Et(?t+1V ) = Et[Et+1(rj,t+1?t+1V )]?Et(rj,t+1)Et[Et+1(?t+1V )] = Et[rj,t+1Et+1(?t+1V )]?Et(rj,t+1)Et[Et+1(?t+1V )] = Covt[rj,t+1,Et+1(?t+1V)] a83a64a236a168 a208 σt parenleftbigg ? t+1V Et(?tV ) parenrightbigg = σt parenleftbiggE t+1(?t+1V ) Et(?tV ) parenrightbigg a75a76 a192 a186 a100a101a83a84 a236a85 a72 a17 a17 September 6, 2005 56 c?a56a57a58 a86a87a88 a89a90a91a92a93a94a95 CAPM §8.1 a96a97a98a99 1a166 Hilberta100 a25 a167a101 a50 a102H a19a103a96a100 a25a168Ha144a236a104a105 a19 a15H×H a134Ra236a234a18a84 a189a168a67 a211·a168a106a107a154a108a197a109 a243a110a111 ?x,y,z ∈H,a,b ∈R, (1) a94a112 a243a111 x·y = y ·x, (2) a113 a243a243a111 x·(ay + bz) = a(x·y) + b(x·z), (3) a114a115a116a243a111 x·x ≥ 0,a62x·x = 0 ? x = 0. a101 a50a208a97a196a112 a68a117a118a105a119a149a120a105a121 a122a123a121 a101 a50 a236 a103a96a100a124a112 a68a104a105a125a126a121 a101 a50a127 a99a236a128a129(a119a149a130) bardblxbardbl = √x·x a231a131 a208a238a154a132a35 a243a110a111 ?x,y ∈H, (1) a133a134 a85a135a136a111 bardblx + ybardbl≤bardblxbardbl+bardblybardbl (2) Cauchy-Schwarza85a135a136a111 |x·y|≤bardblxbardblbardblybardbl a122a123a121a137a189a236 a103a96a100a124a112 a68a138a128a139a118a125a126a121 a208 a121a137a189a168a85a86a122a123a140a141 a192a142 a235a143a144 a243a121 a102{x n}∞n=1 ?H, a ∈H. a238a239 limn→∞bardblxn ?abardbl = 0, a82a112a145a108{xn}∞n=1 a146a147 a109aa168a67a211 limn→∞xn = a, a119 xn → a(n →∞) a128a62a90 a97a112{xn} ∞n=1 a68H a148 a236 a146a147 a120a149a168a16 a112aa68{xn}∞n=1 a236a150a151a121 a145a108{xn}∞n=1 a112 a68Cauchy a149a168 a238a239 ?ε > 0,?N = N(ε) ∈N, ? m,n ≥ N, |xn ?xm| < ε a152a137 a103a96a100a124Ha112 a68a153a154a155a177a168 a238a239H a148 a236 a147a18Cauchya108a193 a140a141 a109H a148 a236a156a121 a238a239 a101 a157 a100a124a158 a231 a101 a157 a127 a99a236a137a189 a110a159 a88a236a152a137 a103a96a100a124 a153a160a161a236a168 a82a112a162a18 a101 a157 a100a124 a153a154 a155a177a121 a160a161a236 a101 a157 a100a124a112 a68 a112 a68Hilbert a125a126. 2a166 a170a163 a101 a157 a164a165 a234 a170a166a167a168a169a168 a26S a68a170a171a172a173a174a168a196a83Sa93a232sharp(S). a175a122a176a177(a119 a97a981a236a237a73a178a179) a100a124RS a153a234a18Hilberta100a124 a121 (1) a180a181 a101 a157 x·y = summationdisplay s∈S xsys 57 8.1a182a183a184a185 a186a187a188 a189a190a191a192 a219a220a193CAPM (2) a170a163 a101 a157 x·y = E[xy] = summationdisplay s pisxsys a231 a148pis a68Sa144a236a192 a186a78a68a121 a170a163 a101 a157 a127 a99a236a194a68 bardblxbardbl = radicalbig E[x2] = radicalbig Var[x] + (E[x])2 3a166a116a195a103a96 a102x,y ∈H, Ha68 a101 a157 a100a124a166 a112x a167 y a153a196a197a177a168a67a211x⊥ya168a238a239x·y = 0a121 a103a96 a174{z 1,...,zn}?Ha112 a68a196a197a198a199a168 a238a239zi⊥zj,? i negationslash= j. a238a239bardblzibardbl = 1,? ia168a82a116a195 a188a200{z 1,...,zn}a112a201 a117a202a196a197a198a199a121 a203a204 a116a195 a188a200a153a231a205a206a207a236a117a202a196a197a173a121 a208a209 8.1.1 Pythagorean a210 a211a212a213a214a215a216a180a217a111 a218a219{z 1,...,zn}?Ha220 a196a197a198a199a221a222 vextenddoublevextenddouble nsummationdisplay i=1 zivextenddoublevextenddouble2 = nsummationdisplay i=1 bardblzibardbl2. a166a223a111a224a225a226a227 a101 a157a228 a232 a221a229a227 a116a195 a122a123a230a231a232 a233a234a111a235a236a79a237a238 a103 a239a240a241a236 a116a195a242 a200 a234 a122a153 a113 a243a243a4a236a121 a166a223a111a102{z 1,...,zn} a153 a116a195a242 a200a221z i negationslash= 0,? i. a102a244λ i ∈Ra245 a231 nsummationdisplay i=1 λizi = 0 a79a246{λ 1z1,...,λnzn}a247 a153 a116a195a242 a200a221a79a122 a241 a244 nsummationdisplay i=1 λ2ibardblzibardbl2 =vextenddoublevextenddouble nsummationdisplay i=1 λizivextenddoublevextenddouble2 = bardbl0bardbl2 = 0 a248a249λ i = 0, ? i. a250a251z1,...,zn a113a252 a243a253a121 4a232a116a195a254a255a112a103 a239x ∈H a116a195 a246a0 a100a124Z ?Ha221a1a2x⊥Z a221a3a4 x·z = 0, ? z ∈ Z a3a4Z = span{z 1,...,zn} a221 a82x⊥Z ? x⊥zi,? i a174Z⊥ = {x ∈H : x⊥Z} a112a201Z a5 a196a197a6a221a230a7a0 a100a124Z a116a195a5a8 a244 a103 a239 a159 a241 a5 a174a9a221a10a153 Ha5 a0 a100a124 a121 a208a209 8.1.2 a11a12a13 a217a111 a14 Hilbert a125a15H a16a17a18a19 a151a20a21a125a15Z a22a17a18 a139a118 x ∈Ha221a23a24a25a26 xz ∈ Z a22 y ∈ Z⊥ a27a28 x = xz + y. a166a223 a29a30{z1,...,zn} a153 a116a195a242 a200a221 Z = span{z1,...,zn}defines braceleftBigg z = nsummationdisplay i=1 λizi : λi ∈R, i = 1,...,n bracerightBigg September 6, 2005 58 c?a31a32a33 a186a187a188 a189a190a191a192a34a35 a193CAPM 8.1 a182a183a184a185 Z Z⊥ x xZ a368.1: a37a38a39a40 a122a123 xz = nsummationdisplay i=1 x·zi zi ·zizi, y = x?x z a41a171xz ∈ Z, y ·zj = parenleftBigg x? nsummationdisplay i=1 x·zi zi ·zizi parenrightBigg ·zj = parenleftbigg x? x·zjz j ·zj zj parenrightbigg ·zj = 0 a248a249y⊥z j, ? j. a250a251y ∈ Z⊥ a42a166xz a153a43a44 a5 a221a45x = xz 1 +y1 = xz2 +y2, xz1,xz2 ∈ Z, y1,y2 ∈ Z⊥. a46a47Pythagorean a122 a48a244 bardbly2bardbl2 = bardblxz1 ?xz2bardbl2 +bardbly1bardbl2 bardbly1bardbl2 = bardblxz1 ?xz2bardbl2 +bardbly2bardbl2 a250a251bardblxz1 ?xz2bardbl2 = 0a221a49a251xz1 = xz2 a233a234 8.1.3 a218a219 Z a220 Hilbert a125a15Ha16a210a19 a151a20a216a21a125a15a221a222H a50a51a52a53 H = Z + Z⊥, a54a55Z ∩Z⊥ = {0} (a196a197a51a52a13 a217) a254a255a56 a48 a148a5xz a57a201xa58Z a59a5 a196a197 a11a12 a221a60a136x = xz +y a57 a201xa253a246Z a5a61a195a62a63 a136a64 a3a4 a254a255a65 a241a253a246a66a163a67a157a221 a46 x·zi zi ·zi = E[xzi] E[Z2i ] a250a251 xz = nsummationdisplay i=1 E[xzi] E[Z2i ]zi a59a68 a7a69 a252a70 a71a72a73a74a75a221a76 a56a77 a78a239 a5a79 a221a80a248a78a239 a5a79 a49a81a82a83 a5 a72a73 a61a84a59a68 a64 c?a85a86a87 59 September 6, 2005 8.1a88a89a90a91 a92a93a94 a95a96a97a98a34a35a99CAPM a100 8.1.4 a101a102 Hilbert a103 a15R2, a54a104a105a106 a107a108a109a110 (1 4, 3 4) a111a13 a221a112 Z = span{(1,1)},x = (1,2). a222 a113a114 a11a12 xz = (1,2)·(1,1)(1,1)·(1,1)(1,1) = 74(1,1) = (74, 74) 5a232 Riesza228a115a56a48 a208a209 8.1.5 a218a219F : H→R a220Hilbert a103 a15H a116a16a117a118 a205a206a119a120a221a222a23a24a25a26 a16a121 a122 k f ∈Ha27a28 F(x) = kf ·x, ? x ∈H (8.1.1) a123a223 a29 a3a4F a84 a238a124a125a221 a46a65kf = 0a230a126a232a127a45F a128 a237a238a124a125a221 a30N = {x ∈H : F(x) = 0} a128F a5 a238a129a130a221N⊥ a128N a5a61a131a132 a221 a46 a244H = N + N⊥, N⊥ negationslash= {0}. a65z ∈ N⊥, z negationslash= 0, F(z) = 1. a235a236x ∈Ha126a133a241 x = [x?F(x)z] + F(x)z a134a135x?F(x)z ∈ N a221 a47 a246z ∈ N⊥ a221a244 z ·x = z ·[F(x)z] a30 kf = zz ·z a46 kf ·x = F(x)(z ·z)z ·z = F(x) a136a42a123a223k f a84 a43a44 a5 a221a3a4a137 a58kf a138 kprimef a139a140 (8.1.1)a221a46 (kf ?kprimef)·x = 0, ?x ∈H a250a251a141 a244k f = kprimef. (8.1.1) a142a5a143 a239k f a57a128Riesz a144. a145a146a228a115 a56 a48a80a147a148a149a150a151 a5 a135a152 a58 a246 a29 a3a4a123a153 a5a154a155a8a156 a241 a5a157a158a159a160 a129a130a244a161a162 a5 a69 a252a56a163 a124a125a221a164a165a10a44 a56 a126a166a167a168a169 a170a171a127a248a0a172a173a228a115a64 6a232 Riesza174a5a175a176 a64 a177a178a179Hilberta129a130RS a59a5 a69 a252 a124a125F. (1) a3a4RS a5 a67a180 a128a181a182 a67a180a183 a46a184F a5a174kf a185a186a187 a232a188a189 a59 a183k f a5a190s a169 a62 a239 a128 kfs = F(es) a191 a142es a128 a74a192a246a193a194s a5a195a196a143 a239a232a246 a84 a183 a47F a5 a69 a252a252 a183 F(x) = F parenleftBiggsummationdisplay s xses parenrightBigg = summationdisplay s kfsxs = kf ·x September 6, 2005 60 c?a31a32a33 a92a93a94 a95a96a97a98a34a35a99CAPM 8.1a88a89a90a91 (2) a3a4RS a5 a67a180 a128 a66a197a67a180a183 a46a184kf a247 a185a186a187 a232a248a249F a126a133a241 F(x) = summationdisplay s piskfsxs = summationdisplay s ksxs a191 a142ks = piskfs a183a229a227(1)a183a244k s = F(es),a250a251kfs = F(es)pis . Hilberta129a130a5a235a236a198a199a0a129a130a183a58a191a67a180a42a200a201a128Hilberta129a130a232a248a249Riesza228a115a56a48a126a192a227 a246Hilberta129a130 a5 a198a199a0a129a130a232a246 a84 a183a3a4Z a84Hilberta129a130Ha5 a198a199a0a129a130a183F a84Z a59a5 a69 a252 a124a125a183 a46 a137 a58 a43a44 a5kf ∈ Z a183a245 a231F(z) = k f ·z, ? z ∈ Z. a3a4Z = span{z 1,...,zn},a46 a69 a252 a124a125F : Z →R a5a174kf a126a175a176a3a42a29 a30F a58Z a5a202a143 a239 a5a79a128 wi = F(zi), i = 1,...,n a46kf a203 a139a140 a72a73a240 wi = kf ·zi, i = 1,...,n a47 a246k f ∈ Z, a244k f = summationtextn j=1 ajzj. a204a205a59a68 a183a244 wi = nsummationdisplay j=1 aj(zj ·zi), i = 1,...,n a206 a84a207 a244na169a208a209a210a 1,...,an a138 na169a72a73a5a69a252a72a73a240a183a227a211a212a72a213a126a214a184a63 a183 a250a251 a126 a184 a231k f. a100 8.1.6 a112 Z = span{(1,1)}?R2 a183 a106 a107a215a108a109a110 (1 4, 3 4) a111a216a16a104a105a106 a107a183 F : Z →R F(z) = 2z1, ? (z1,z2) ∈ Z. a108a217 a121 a122 (1,1) a218a53 Z a16a219 a183 a144 kf a50a220a221a53 kf = a(1,1) a54a55a ∈Ra222a13 a232a108a217 F(1,1) = 2, a19 2 = a(1,1)·(1,1) = a(14 + 34) = a a223a = 2,k f = (2,2). 7a232a66a197a174 a224a225a226a241a129a130(a224a225a227) a84Hilberta129a130RS(a207 a244a66a197a67a180) a5 a0a129a130a183 a250a251 a10a228a229 a247a84Hibert a129a130a232a248a249a183Riesza228a115 a56 a48a126a192a227a246 a56 a152 a58 a224a225a227M a59a5 a69 a252 a124a125a183a230a231 a160a232a233a234a235Ma59a5 a236a169a69 a252 a124a125 a29 a66a197a124a125 a138 a154a155a56a163 a124a125a64a230a231a177a237a238a239a240a183a241a240 a58 a42a44a242a243a237a238a64 a56 a152a66a197a67a180 a5a244a245 a83a246pi a65a128a204 a48a247 a5a248a249a244a245 a83a246a250a3a4 a204 a48a247 a5a251a252a207a253 a66a197a254a255a0 a115a183a164a165pi a1a84 a66a197a254a255 a5a244a245 a83a246 (a80a2 a8a253a204 a48a247a3a74a75) a104a105 a4 a119E : M→R, z →E[z]. c?a31a32a33 61 September 6, 2005 8.1a88a89a90a91 a92a93a94 a95a96a97a98a34a35a99CAPM a10 a5Riesza174ke ∈Ma84 a139a140 a42 a68a5 a43a44 a154a155 a29 E[z] = E[kez], ? z ∈M. a126a2556 a142a5 a72a213 a175a176a174ke. a3a4a5a6a7 a154a155a58M a142 a183 a46ke = e := (1,...,1) ∈RS. a3a4a5a6a7 a154a155a8a58M a142 a183 a46ke a9 a2e a58Ma5a61a131a254a255 a250a188a189 a59 a183 a47 a2 E[(e?ke)z] = E[ez]?E[kez] = E[z]?E[kez] = 0, ? z ∈M a10e?k e⊥M. a11 e = (e?ke)+ ke a248a249 k e a84 ea58Ma59a5a61a131a254a255 a250a12 e ∈ M a13 a183 e?k e ∈ M a64a10 (e?k e)·(e?ke) = 0 a183 a250a251 ke = e. a100 8.1.7 a14 a23a24a15a16a17a18a19a20a183a21a16a22a23a183 a54a24a25 a215x 1 = (1,1,0), x2 = (0,1,1). a26 a16a19a20 a16 a109a110 a215 1 3. a108a217E[x 1] = 13 + 13 = 23, E[x2] = 13 + 13 = 23 a183 a19a27a28a29 2 3 = E[kex1] 2 3 = E[kex2] a108a217 k e ∈M a183?a22a23 a29a30 (h1,h2) a27a28 ke = h1x1 + h2x2 = (h1,h1 + h2,h2) a31 a116a32a33a34a27a28a29 a183 a28 2 3 = 1 3h1 + 1 3(h1 + h2) 2 3 = 1 3(h1 + h2) + 1 3h2 a52 a28 h1 = h2 = 2 3 a183a217a35 ke = parenleftbigg2 3, 4 3, 2 3 parenrightbigg . 8a250a56a163a174 a56 a152 a58Ma59a5a154a155a56a163 a124a125 q a5 Riesz a174 a57 a128a13a36a144kq a183a10a84 a139a140 a42 a68a5 M a142a5 a43a44 a154 a155 a29 q(z) = E[kqz], ? z ∈M a126a2556 a142a5 a72a213 a175a176kq. September 6, 2005 62 c?a31a32a33 a92a93a94 a95a96a97a98a34a35a99CAPM 8.1a88a89a90a91 a3a4 a8 a137 a58a37a38(a39a37a38)a183a46 a137 a58a40a41a61(a61)a5 a193a194 a163a41a143 a42q = (q 1,...,qs)a43a44 q(z) = summationdisplay s qszs, ? z ∈M a178a179a255 a244a245a45a46a47 a243a48 a5 a193a194 a163a41a143 a42 q pi = parenleftbiggq 1 pi1,..., qs pis parenrightbigg a46 q(z) = E bracketleftBigq piz bracketrightBig a250a251a253 E bracketleftBigparenleftBigq pi ?kq parenrightBig z bracketrightBig = 0, ? z ∈M a49 q pi ?kq⊥M. a11 q pi = parenleftBigq pi ?kq parenrightBig + kq a10k q a84 qpi a58Ma59a5a61a131a254a255 a250 a56a163a174a84 a43a44 a5 a183 a8a50a51a52a84a53 a198a199a250a3a4 a51a52a84a8 a198a199 a5 a183 a46 a137 a58a54 a169a193a194 a163a41a143 a42a250a12a255 a244a245a45a46a47 a243a241a183 a8a253 a206a55 a143 a42 a58 a224a225a227 a59a253 a74a75 a5a254a255 a183a56a57 a254a255a128a56a163a174 kq. a58a51a52a84 a198a199 a5 a183 a46 a137 a58 a43a44 a5 a193a194 a163a41a143 a42 q a183a56 a56a163a174kq a9 a2 q pi. a3a4q a84a59a60a154a155a56a163 a124a125a183 a46 q(z) = E bracketleftbigg? 1υ ?0υz bracketrightbigg , ? z ∈M. a191 a142 ?1υ ?0υ a84a207a253 a66a197a254a255a0a115E[υ(c)]a56a191 a59a60a61a62a128 a67a63 a5a204 a48a247 a5a64a65a66a204a245a143 a42a250a57 a143 a42 a58 a224a225a227M a59a5a254a255 a9 a2 a56a163a174kq a64a3a4a51a52a84 a198a199 a5 a183a57 a64a65a66a204a245a143 a42 a1 a9 a2 a56a163a174kq a183a206 a80 a207a253 a67a63 a61a62a5a8a253a204 a48a247 a59a67a68 a64 a58 a60 a68Ebracketleftbigparenleftbigqpi ?kqparenrightbigzbracketrightbig= 0 a142a65z = ke a183a44 E bracketleftBigq pi bracketrightBig = E[kq] a248a249a183a3a4a193a194 a163a41a143 a42 q a84a61a5 a56a69a70a183 a46a56a163a174a5 a66a197 a84a40a41a61a5 a250a3a4a5a6a7 a154a155a58 a224a225 a227 a142 a183 a46a477a209ke = ea183a250a251 E[kq] = E[kqke] = q(ke) = q(e) = 1ˉr a100 8.1.8 a14 a23a24a15a16a17a18a19a20a183a109a110a215 (1 3, 1 3, 1 3) a183a21a16a22a23a183 a36a71 a215 p 1 = 1, p2 = 43 a183 a24a25 a215 x1 = (1,1,0), x2 = (0,1,1). a215a72a13a36a144 a183 a101a102 a22a23 a36a71a27a28 ? ? ? 1 = E[kqx1] 4 3 = E[kqx2] c?a85a86a87 63 September 6, 2005 8.1a88a89a90a91 a92a93a94 a95a96a97a98a34a35a99CAPM a108a217 k q ∈M a183?a22a23 a29a30 (h1,h2) a27a28 kq = h1x1 + h2x2 = (h1,h1 + h2,h2) a33a34a27a28a29a19 ? ?? ?? 1 = 13h1 + 13(h1 + h2) 4 3 = 1 3(h1 + h2) + 1 3h2 ? ?? ? ?? h1 = 23 h2 = 53 a73 kq = parenleftbigg2 3, 7 3, 5 3 parenrightbigg . 9a250a59a79-a72a74a64a75a154a155 a44 a154a155 a57 a128a59a79 - a72a74 a64a75a154a155 a183a3a4 a8 a137 a58 a191a10 a154a155a43a44a207a253 a74a75 a5a163a41 a138 a74a75 a5 a66a197a183 a76a77a242 a5 a72a74a250a78a49 a58 a76 a56a5a163a41 a138 a66a197a42a79a242a80a72a74a64 a30ε = span{ke,kq} a84a47 a66a197 a174ke a138 a56a163a174kq a156a67a5Ma5 a0a129a130a64 a81a82 8.1.9 a26a16 a24a25 a35a83a84 - a27a85a86a87a24a25a88 a89a90 a88 a91a92a24 ε a55 a250 a123a93 a29a94a95 a135 a154a155z ∈Ma58εa59a96a61a131a97a98(a99 a2a66a197a67a180) z = zε + epsilon1, zε ∈ ε, epsilon1 ∈ ε⊥ a47 a2epsilon1 a61a131 a2k e a138 kq a183a253 E[epsilon1] = E[keepsilon1] = 0, q(epsilon1) = E[kqepsilon1] = 0 a49epsilon1 a253 a70a66a197 a138 a70 a163a41 a183a100a101z a138 zε a253 a74a75a102a66a197 a138 a74a75a102 a163a41 a250a49a44a103a183 a47 a2epsilon1 a61a131 a2ε,a214a104 E[epsilon1] = 0a183a253 Cov(epsilon1,zε) = E[epsilon1zε]?E[epsilon1]E[zε] = 0 a100a101 Var(z) = Var[zε] + Var(epsilon1) ≥ Var(zε) a105a106epsilon1 negationslash= 0a183 a59a107 a102 a8 a9 a68a128a40a41a8 a9 a68 a250a108a109 a59a79 -a72a74a64a75a154a155 a44 a56a58ε a142 a250 a110a111a183a230a231 a203 a123ε a142 a102a112a169 a154a155 a44 a56a84a59a79-a113 a74 a64a75a154a155 a250a114a45a115a238 a8a116 a183 a46 a137 a58a154a155z ∈ ε, a117 a8a84a59a79-a113 a74 a64a75a154a155 a183a100a101a137 a58a118 a44 a154a155z prime ∈M a43a44 E[zprime] = E[z], q(zprime) = q(z), Var(zprime) < Var(z) a119a120 a141a121a122 a102a123a93a183zprime a58εa59 a102 a61a131a97a98zprimeε ∈ ε a139a140 E[zprime] = E[zprimeε], q(zprime) = q(zprimeε), Var(zprime) ≥ Var(zprimeε) a2 a84 E[zprimeε] = E[z], q(zprimeε) = q(z), Var(zprimeε) < Var(z) September 6, 2005 64 c?a31a32a33 a92a93a94 a95a96a97a98a34a35a99CAPM 8.1a88a89a90a91 zε z ke kq ε a368.2: ε a123a124 a100a101 E[kq(z ?zprimeε)] = q(z ?zprimeε) = q(z)?q(zprimeε) = 0 E[ke(z ?zprimeε)] = E[z ?zprimeε] = E[z]?E[zprimeε] = 0 a49a101z ?zprimeε⊥ε. a11a47 a2z ?zprimeε ∈ εa183 a253z = zprimeε a183a206a125Var(zprimeε) < Var(z)a126a127 a64 a105a106k e a138 kq a84a128 a69 a122 a102a183a49a137 a58γ negationslash= 0a43a44kq = γke,a46a59a79-a113 a74 a64a75a154a155a129εa84 a44a130a131 a69a64 ke a138 kq a84a128 a69 a122 a102? a132a253 a224a225a133a134 a253 a74a75a102a66a197a135a136 (= 1 γ). a188a189 a59 a183 a95 a135a133a134a102a135a136 r a9 a2a191 a154a155z a125a163a41q(z)a102a137a183a253 E(r) = E bracketleftbigg z q(z) bracketrightbigg = E[z]q(z) = E[kez]E[k qz] a10 E[kez] E[kqz] = 1 γ, ? z ∈M ? kq = γke. a105a106a5a6a7 a154a155a58 a224a225a227M a142 a183a138 braceleftBigg ke = e ke a125kq a128 a69 a122 ? a60a139a56a163(a49a112a169a123a153j a102a66a197a135a136E[rj] a9 a2a5a6a7a135a136 ˉr). a58 a60a139 a56a163a140 a183k e = e, kq = 1ˉre a183a191 a142e = (1,...,1) ∈RS. a188a189a59 a183 E[erj] = E[rj] = E[kerj] ? j ? E[(ke ?e)rj] = 0 ? j. a108e ∈M, k e ?e ∈M a183 a253ke = e. a12ke a138 kg a69a128 a69 a122 a183ε a128a141 a139 a107 a250 a105a106a142 a253 a236a169a69a143a144a123a153a183a138a224a225a227M a84a141 a139 a107 a250a145a101a183a105a106k e a138 kg a69a128 a69 a122 a183a138a224a225a227 Ma125a59a79a146a113 a74 a64a75a154a155a129 εa74a75a183a49a112a169a154a155a147a84a59a79-a113 a74 a64a75a154a155 a64 c?a148a149a150 65 September 6, 2005 8.1a88a89a90a91 a92a93a94 a95a96a97a98a151a152a99CAPM re ke rqk q ε a1538.3: k e a154kq a155a156a157a158 10a250a64a75 a135a136 a154a155 a102a135a136= a154a155÷a154a155 a102 a163a41 a250 a64a75 a135a136 a1a84a64a75a154a155 a102a135a136 a66a197 a174ke a138a159 a163a174kq a102a135a136a160a232a128 re = keE[k q] , rq = kqE[k2 q] a105a106k e a138kq a128 a69 a122 a183a138r e = rq a183 a64a75 a135a136 a129a128a195 a63 a129{re}. a145 a141 a103a183a105a106a5a6a7 a154a155a161M a142 a183a138r e = ˉr. a214 a140 a114a162k e a138 kq a69a128a69a122a183a138a64a75a135a136a129a128a167 re a138 rq a102a131a69a163 rλ = re + λ(rq ?re) a191 a142?∞ < λ < +∞. a188a189a164a183 a95 a141a64a75 a135a136a126a133 a67 a163 αke + βkq q(αke + βkq) = αreE[kq] + βrqE[k2q] αE[kq] + βE[k2q] = αE[kq]αE[k q] + βE[k2q] re + βE[k 2q] αE[kq] + βE[k2q]rq = (1?λ)re + λrq = re + λ(rq ?re) a191 a142λ = βE[k 2 q] αE[kq]+βE[k2q], α,β ∈R. a100 8.1.10 a14a165 a15a16a166a167a168a19a20a169a15a16 a36a71 a83a215 1 a170 a114a171 a22a23a183a22a23a172a173a215 r1 = (3,0,0), r2 = (0,6,0), r3 = parenleftbigg6 7, 3 7, 9 7 parenrightbigg a174a175a176 a105 a177a178a179a180a181a182a183 a86a87a116a184 a215a185a186a72 r e a169 r q a187 September 6, 2005 66 c?a85a86a87 a92a93a94 a95a96a97a98a151a152a99CAPM 8.1a88a89a90a91 a108a217a188a189a35a190a191 a170 a183a192a193a194 a24a25 a183a195a196a197a180a183a198a199a183 ke = e = (1,1,1); a188a189a190a191a200a201a202a203a204 a36a144 kq a35a205a19a20a109a110a206a207a208a170 a19a20 a36a71 q pi a183a199a19a20 a36a71 a35 a27a28a29 ? ??? ??? 1 = 3q1 1 = 6q2 1 = 67q1 + 37q2 + 97q3 a170a209a210a211 q1 = 13, q2 = 16, q3 = 12. a73 kq = qpi = parenleftbigg 1, 12, 32 parenrightbigg ke a169kq a170a36a71 a215 q(ke) = qke = 13 + 16 + 12 = 1 q(kq) = qkq = 13 + 112 + 34 = 76 a172a173a215 re = keq(k e) = (1,1,1), rq = kqq(k q) = parenleftbigg6 7, 3 7, 9 7 parenrightbigg = r3 a212a185 r 3 a183 a86a87 a172a173a183a213 r 1 a169r 2 a214 a35a183a212a215a91a175 a214 a183 r e a169r q a170a117a215a116a187 a64a75 a135a136r λ a102a216a197 a128 E[rλ] = E[re] + λ(E[rq]?E[re]) a113 a74 a128 Var[rλ] = Var[re] + 2λCov(re,rq ?re) + λ2Var(rq ?re) a211a212a74 a128 σ[rλ] = radicalbig Var[rλ] a105a106k e a84 a5a6a7a102a183a138r e = ˉr E[rλ] = ˉr + λ(E[rq]? ˉr) Var[rλ] = λ2Var[rq] σ[rλ] = |λ|σ(rq) a217a2k q a84a253 a6a7a102( a161 a114a162k q a138 ke a69a128a218a122a140)a183a253 E[k2q] = [E[kq]]2 + Var(kq) > [E[kq]]2 a100a101 E(rq) = E[kq]E[k2 q] < 1E[k q] = ˉr c?a148a149a150 67 September 6, 2005 8.1a88a89a90a91 a92a93a94 a95a96a97a98a151a152a99CAPM 0 λ E[rλ] E[re] E[rq] 1 (a) 0 λ σ[rλ] λ0 (b) 0 λ σ[rλ] a49 ˉr > E(rq) a219a137 a161a207a253 a79a242 a113 a74a102 a64a75 a135a136a163 (1) a105a106a5a6a7a154a155a161 a224a225a227 a142 a183a138a79a242 a113 a74 a64a75 a135a136 a1a84 a5a6a7a135a136 a187 (2) a105a106a5a6a7a154a155a8a161 a224a225a227 a142 a183a138 a132a253 a135a136 a59a253a40a41a220 a102 a113 a74a250a79a242 a113 a74 a64a75 a135a136a126a214a166a167 a99 a2λa79a242a80Var[r λ]a221a44 a183a217a2 Var[r λ]a84λ a102a222a223a125a210a183a79a242a80a224a225 min λ Var[rλ] a102a226 a141 a79a227a228λa183a229a217 a141a230 a130a231 a184a44 a163 λ0 = ?Cov(re,rq ?re)Var(r q ?re) a135a136a102a216a197a125a211a212a74a102 a99a232 a105 a140a233a132a234 a187 11a250a70a235a113 a74 a64a75 a135a136 a217a2 a64a75 a135a136 a129a236a141 a130a131 a218 a183 a95a237 a236a238 a8a239 a102 a64a75 a135a136a229a255a240a241 a204 re a242 rq a214a243a244a57a131a218 a187 a245a48a79a242 a113 a74 a64a75 a135a136a246a247a248a112a238 a64a75 a135a136 r λ a147a249a161a118a141a64a75 a135a136 r μ(a250rμ a251 rλ a102a70a235a113 a74 a64a75 a135a136) a43a44 a222a240a102a235 a113 a74 a251 a70a250a252a253a164a247a217 0 = Cov(rλ,rμ) = Var(re)+ (λ + μ)Cov(re,rq ?re) + λμVar(rq ?re) a228 a44 μ = ? Var(re) + λCov(re,rq ?re)Cov(r e,rq ?re)+ λVar(rq ?re) September 6, 2005 68 c?a148a149a150 a254a255a0 a1a2a3a4 a151a152a5CAPM 8.1a6a7a8a9 σ[rλ] E[rλ] ˉr cjkD3D0cjkD0A7cjkB1DFcjkBDE7 cjkCEDEcjkD0A7cjkB1DFcjkBDE7 (c) a10a11a12a13a14a15a16a17a18a19 σ[rλ] E[rλ] cjkD3D0cjkD0A7cjkB1DFcjkBDE7 cjkCEDEcjkD0A7cjkB1DFcjkBDE7 (d) a20a10a11a12a13a14a15a16a17a18a19 a108a109a247a142 a121 a160a21a22a23a24a25a247 μ a26a236 a159a27 a93a28a102a29a101a160a21a23a24a25a247a30a31a32a30 λ = λ 0 a247a33r λ a251a34a35 a113a36 a135a136 a187 a34a35 a113a36a37a38 a135a136a22 a249a161 a25a235 a113a36a37a38 a135a136 a187 a39a40a41a42a43a44a45 a161a46a47a48a49 a247a138a50a238 a37a38 a135a136( a34a35a51 a36a37a38 a135a136a245a246) a52 a25a235 a51 a36a37a38 a135a136 a251 a41a42a43a135a136(a33μ = 0). 12a29a53a54 a159a55 a56r λ a236(a57a34a35a51 a36 a135a136) a37a38 a135a136a247r μ a236a58 a25a235 a51 a36a37a38 a135a136a29 a59a60a61j a52 a44a45z j a161a37a38 a44a45a139a62εa164a63 a220a64a65a66(a67 a216a68a69a70): zj = zεj + epsilon1j, zεj ∈ ε, epsilon1j ∈ ε⊥. a217a24epsilon1 j a71 ke a242 kq a220a64 a247a72a73epsilon1 j a74 a25a216a68 a242 a25 a55a75 a163 E[epsilon1j] = E[epsilon1jke] = 0 q(epsilon1j) = E[epsilon1jkq] = 0 a76a73a247z j a71 zεj a74a77 a239a52 a55a75 a247a72a73j a52a78a79 a251 rj = rεj + ?epsilon1j a58a49rεj a251 a37a38a78a79 a247?epsilon1 j = epsilon1j q(zj). a67rλ a242rμ a243a244 a37a38a78a79a80a218 a247a81rε j a229a82 a234 a251 rεj = rμ + βj(rλ ?rμ) a76a73 rj = rμ + βj(rλ ?rμ) + ?epsilon1j a83a84a241a216a68a85 E(rj) = E(rμ) + βj[E(rλ)?E(rμ)] a241 a71 rλ a52 a235 a51 a36 a85( a86a87rλ a71 rμ a88 ?epsilon1j a22 a77a89a90 βj = Cov(rj,rλ)Var(r λ) c?a91a92a93 69 September 6, 2005 8.1a6a7a8a9 a254a255a0 a1a2a3a4a94a95 a5CAPM a39a40a41a42a43a44a45a96 a46a47a48a49 a247a81 E(rj) = ˉr + βj[E(rλ)? ˉr] a97a62a83a98 a250 a251a99 a100a101a102a103a98a247a104a82a105 a106 a107a108a60a61 a52 a42a43a109 a55a110a111 a112a24a113a60a61 a52a78a79 a71a114a115 a37 a38a78a79a52 a235 a51 a36a116a117 a238a103a98 a71 CAPMa52 a60a61a118a119a120a103a98a121a122a247a123a124a118a119 a78a79a125a126a117 a124 a52 rλ. a53a54a127 a55 a103a98a248a107 a237 a78a79r a128 a110a129 a106 E(r) = ˉr + β[E(rλ)? ˉr] β = Cov(r,rλ)Var(r λ) 13a29a128a130a131 a51 a36 a74a132 a78a79 a108a238 a78a79a250 a251 a236a128a130 - a51 a36 a74a132 a52 a247a39a40a22 a249 a96 a58 a104 a78a79 a247a133 a74a77 a239a52 a51 a36a134a135a136a52a137 a68a138 a33 a128a130- a51 a36 a74a132 a78a79a236 a96a139a127 a51 a36a140a141a52a142a143a144 a34 a136a145a137 a68 a78a79a116 a96 a233a146 a97a247 a128a130- a51 a36 a74a132 a78a79a236a137 a68 a78a79 a23a24a147a148 a136 a24 a34a35a51 a36a78a79a52a137 a68 a52a37a38a78a79a116 a30 a137 a68a149 a236 a41a42a43 a52a150a151 a247a152a24 ˉr > E(r q) a247 a128a130 a51 a36 a74a132 a78a79a236λ ≤ 0a52a37a38a78a79rλ. 14a29a60a61a118a119a120 a125a153a154ia52a150a1551a156a157w i1 a229a158a108a159a228 a110 wi1 = wi1M + wi1N a58a49wi1M ∈Ma236a125a153a154ia156a157a52 a229 a64a160a161 a159a247wi 1N ∈ N = M ⊥ a236 a22a229 a64a160a161 a159a29 a239a153 a247a162 a156a157 a229a159a228 a110 ˉw1 = ˉw1M + ˉw1N a163a164a165a166ma127 a27a251 a162 a156a157a52 a229 a64a160a161 a159 a106 m = ˉw1M a163a164a167a168r m a127 a27a251 a118a119a44a45ma245a169 a128a170 a55a75 q(m)(a171a172negationslash= 0). a152CAPMa247a118a119 a78a79rm a251 a37a38a78a79 a247a173a174 a171a172 a104a22 a236 a34a35a51 a36a78a79 a247a81 a249 a96a175a108 a37a38a78a79 rm0 a247a104 a71 rm a52 a235 a51 a36 a251 a25a247a59r m a242 rm0 a67 a24a53a54a127 a55 a103a98a247 a74 a176a177 8.1.11 a178a179 a163a164a167a168a180a181a182 - a183a184a185a186a187 a247a188 E(rj) = E(rm0) + βj[E(rm)?E(rm0)], ? j a189a190β j = Cov(rj,rm) Var(rm) . a97a98 a26a236a191a192 a163a164a193 a51a194 a29a39a40a41a42a43a44a45a96 a46a47a48a49 a247a81r m0 a251 a41a42a43 a78a79ˉra247a76a73a97a98 a110a251 E(rj) = ˉr + βj[E(rm)? ˉr], ? j a117 a82a105 a106 a50a108a60a61 a52 a42a43a109 a55 E(rj)? ˉr a71 a232 a115 βj a110a111 a112a247 a111 a112 a232 a115a251 a118a119 a78a79a52 a42a43a109 a55 (a250 a251 a163a164a195a196a197a102). β j a198 rj a248a118a119a78a79a52a199 a200a201 a115 a247 a198 a60a61a42a43 a52a202 a30a203a204 a116 September 6, 2005 70 c?a91a92a93 a254a255a0 a1a2a3a4a94a95 a5CAPM 8.1a6a7a8a9 a60a61a118a119a120 a51a194 a248a107 a87a78a79a128 a110a129 a106 E(r) = E(rm0) + β[E(rm)?E(rm0)], ? r a58a49β = Cov(r,rm)Var(rm) a198 a78a79 r a248a118a119a78a79a52a199 a200a201 a115 a116 a248a118a119 a78a79 a247 β = 1; a248a25a205 a51 a36a78a79 rm0, β = 0. a206rm0 a207 a250a208 a99 a100a167a168 a116 14a29a128a130- a51 a36a209a210 a125a153a211 a133 a74 a181a182 - a183a184a212a213 a247a39a40 a58 a132 a67a214 a115 u(c0,c1)a215 a75a216a217a218 a82a219 a251 u(c0,c1) = v0(c0) + f[E[c1],Var(c1)] a58a49v0 : R → R, f : R×R → R. a97a98a220a105a125a153a211a52a209a210 a198 a150a221a222 a159 a52 a31a248 a150a155 1 a223a224a225a226a52a209 a210 a32a227a228a24 a137 a68a229 a51 a36a116 a128a130- a51 a36a209a210a52a125a153a211 a198a230a231 a183a184a232a233a234 a247a39a40f a89 a24 a51 a36a215 a75a216a235 a116 a144a236a237a146 a59a238a239 a128a130 a51 a36a209a210 a106 (1) VNM a132 a67a214 a115a251a240a241 a52 (2) a60a61a44a45a229a150a1551a223a224a225a226a242 a76a243a244a245a246a159a247 (3) a60a61a44a45a229a150a1551a223a224a225a226a242 a76a248a249a159a247 a176a177 8.1.12 a178a179a250a251a252a253a254 a181a255a181a182 - a183a184a212a0a234a1 a230a231 a183a184a232a233a234a2 a188a181a3a4a163a164a167a168a180a181a182 - a183a184a185a186a187a116 a60 a106 a56ci 1 a5 a125a153a211ia52a128a170a150a1551a223a224a2 a59ci 1 a159a6 a110 a5 a222a64a160a161 a159a229a22 a222a64a160a161 a159 a106 ci1 = ci1M + ci1N (8.1.2) a58a49ci1M ∈M, ci1N ∈ N = M⊥. a7a8a60a105a50a9 a125a153a211a52a150a1551a223a224a52a222a64a160a161 a159ci 1M a96 a128a130- a51 a36a37a38 a44a45a10ε a49a2 a72 a5 a39 a40 a117 a198a11 a52a2 a123a12a162 a223a224a52a222a64a160a161 a159 a207 a96ε a49a116a134 a198 a2 a152a24 a132 a67a214 a115a198 a215 a75a216a217 a52a2 a34a13 a223a224a14 a15a23a98a16a17a18a19 (ci 0 = wi0 ?phi, ci1 = wi1 + hiX)a2 a76a73a162 a223a224a52a222a64a160a161 a159a23a24a162a20 a157a52a222a64a160 a161 a159 a2a21 a148a22a127 a27 a5 a118a119a44a45 a116 a72a23a118a119 a78a79 a5 a37a38a78a79a116 a5 a60a105ci 1M ∈ εa2 a59ci 1M a65a66a24εa2a74 ci1M = ci1ε + ci1I (8.1.3) a58a49ci1ε ∈ ε, ci1I ∈ ε⊥(εa96M a49a52 a245 a64a25). a26a60a27 a2a171a172a28ci1M negationslash∈ εa2 a81ci 1I negationslash= 0. a29a30 a175a108a9 a150a1551a223a224a225a226 ?ci1 := ci1ε + ci1N = ci1 ?ci1I (8.1.4) a152a24ci 1 ?wi1 ∈Ma2 ?ci1 ?wi1 = (ci1 ?wi1)?ci1I,a74 ?ci1 ?wi1 ∈M (8.1.5) a33 a223a224a225a226 ?ci1 a222a31a32a33a64a160a34 a85a29a152a24 ci 1I⊥ε = span{kq,ke}, q(ci1I) = E(kqci1I) = 0 (8.1.6) E(ci1I) = E(keci1I) = 0 (8.1.7) c?a91a92a93 71 September 6, 2005 8.2a35a36 a37t a38a39a40a41 a42a43a44a45a46 a47a48a49 a50 a39a51a52 a94a95a53CAPM a152(8.1.6) a74 q(?ci1 ?wi1) = q(ci1 ?wi1), a33 ?ci1 ?wi1 a71 ci1 ?wi1 a74a77a54 a52 a55a75 a29a72a23 a2 a152(8.1.5) a222a55 (ci0,?ci1) a207 a14 a15 a125a153a211 i a52 a16a17a18a19 a116 a152(8.1.7) a74 E(?ci1) = E(ci1). a152a24 ci1ε,ci1N,ci1I a56 a77a57 a80a2 a152 (8.1.7) a74 Cov(ci1ε,ci1I) = Cov(ci1I,ci1N) = 0 a76a73a152(8.1.4) a74 Cov(?ci1,ci1I) = 0a2 a24 a198 a152ci 1I negationslash= 0a74 Var(ci1) = Var(?ci1)+ Var(ci1I) > Var(?ci1) a97a62a60a105a58E(?ci 1) = E(ci1), Var(?ci1) < Var(ci1). a152a24 a125a153a211i a198 a215 a75a51 a36a59a60a52a2a61 ?ci1 a52a209a210a215 a75 a62a24ci 1 a2a117a71ci1 a52a34a13a63a64a65a116 §8.2 a66a67a68 t a69a70a71a72a73a74a75a76a77 a96a50a108a9 a57a78 a84a79 a143ξt a2a222a80a81 a108a9a108 a137 a60a61a118a119 a106 a59a96ξ t a52a82 a204a83 a5 a150a1550a52a82 a204 a2 a96ξ t a52a84a85a86a52a82 a204a83 a5 a150a1551a52a82 a204 a116 a96ξ t+1 a52 a108 a137 a44a45 a5 [p(ξt+1)+ x(ξt+1)]h(ξt). a125a153a211a87 a58a96 ξ t a88a89 a108a9a60a61a90a91 h(ξ t) a92a2a93a94 a96 ξ t a88 a58a84a85a86 ξt+1 a88a89 a108a9 a223a224a225a226 (c(ξt);c(ξt+1) : ξt+1 ∈ ξt). a171a172 a50a9 a125a153a211a52 a132 a67a214 a115 a5 a150a221a222 a159 VNM a132 a67a214 a115 a106 V i(y) = Tsummationdisplay t=0 (δi)tυi(yt), ? y = (y0,...,yT) ∈RT+1 (8.2.1) a58a49δi > 0. a171a172a95 a74 a125a153a211 a74a96a54 a52 a79 a143a97a98pi. a125a153a211ia61 a243 a137a223a224a225a226 ca52a137 a68 a132 a67 a5 E[V i(c)]a116 a96ξ t a52 a108 a137 a60a61a118a119 a2a125a153a211 ia52 a132 a67a214 a115 a5 υi(c(ξt)) + δiE[υi(ct+1)|ξt] a29a30 a243 a137 a60a61a118a119 a49a52 a108a9 a128a170{p,{h i},{ci}} a2a58a49p a5 a60a61 a55a75a99 a204 a2hi a229ci a159a100 a5 a125 a153a211ia52 a60a61a90a91a101a102a229 a223a224a225a226a116 a96ξ t a52 a108 a137 a60a61a118a119 a2a125a153a211 ia96ξt a52 a20 a157 a5 wi(ξt)+ [p(ξt) + x(ξt)]hi(ξ?t ) a96ξ t+1 ? ξt a52 a20 a157 a5 wi(ξt+1)?p(ξt+1)hi(ξt+1) a152a24summationtext i h i = 0 a2 a96ξ t a52 a162a20 a157 a5 ˉw(ξt) =summationtexti wi(ξt)a2 a96ξ t+1 ? ξt a52 a162a20 a157 a5 ˉw(ξt+1). {p(ξt),{hi(ξt)},{ci(ξt)},{ci(ξt+1)}}a80 a110 a58a96ξ t a52 a108 a137 a60a61a118a119 a52 a108a9 a128a170(a103 a60a105). September 6, 2005 72 c?a91a92a93 a47a48a49 a50 a39a51a52 a94a95a53CAPM 8.3a50 a39a51a52 a94a95 §8.3 a104a70a105a106a107a108 a109a110a59a60a105 a2 a111 a137 a60a61a118a119 a52 a53a54a127 a55 a103a98 a222a112a113a24 a96ξ t a52 a108 a137 a60a61a118a119 a2 a114a23 a5 a142a143 a53a54 a127 a55 a116 a96 ξ t a88a89 a52 a60a61a90a91 a52 a108 a137 a44a45a10a114 a5 a96 ξ t a52a115a116a117 a118a119 a2a120 a5 Mξt(p) a2 a104 a198 Rk(ξt) a52 a84a121a221a2 a245a98a122 a2 Mξt(p) = {z ∈Rk(ξt) : z(ξt+1) = [p(ξt+1)+ x(ξt+1)]h(ξt), ? ξt+1 ? ξt, a61a28h(ξt) ∈RJ} a96ξ t a52a115a116 a165a166a101a102a123a124q ξt : Mξt(p) →R a127a125 a5 qξt(z) = p(ξt)h(ξt), ? z ∈Mξt(p) a104 a61Mξt(p) a49a52 a50a9a44a45z a126 a127a58a108a9 a55a75 a2a127a128a47a129z a52 a60a61a90a91h(ξ t) a96ξ t a52a55a75 a116 Mξt(p) a198 Hilberta121a221a2a130a131a132a133 a58a134a135 a116a136a137 a138 y ·z = E(yz|ξt), ? y,z ∈Mξt(p) = summationdisplay ξt+1?ξt pi(ξt+1|ξt)y(ξt+1)z(ξt+1) a139a22Riesza82a219a127 a153a2a140 a96 a115a116 a101a102a141kq ξt ∈Mξt(p)a2a142 a85 qξt(z) = E(kqξtz|ξt), ? z ∈Mξt(p). a121a122a122 a2a140 a96a134a135 a116a136 a141ke ξt ∈Mξt(p)a142 a85 E(z|ξt) = E(keξtz|ξt), ? z ∈Mξt(p). a130a131 a96ξ t a140 a96a60a61a147a148a60a61a90a91a133 a74 a108 a137a143a144a145a146a147a2 a81 ke ξt = e = (1,...,1) ∈R k(ξt). a108a9a108 a137a146a147 a114 a5 a108 a137a142a143a148a149a146a147a2a130a131a150a140 a96a151a104a108 a137a146a147 a133 a74a77a54a152a153a75 a229 a77a54a152 a142 a143a137a154a134a155a156 a152 a142a143a157a158a2a159 a5 a96a139a127 a153a160 a229 a142a143a137a154a144a142a142a143a157a158a161a156 a152 a146a147a116 a162ε ξt ∈Mξt(p) a5 a142a143a148a149a141a163a2a159 a152a96ξ t a152 a108 a137a164a165 a118a119a166 a152 a148a149a146a147 a90 a110a152 a84a121a221a116 a167a111 a137a164a165 a118a119a108a168 a2 a173a174a169 εξt = span{kqξt,keξt} a108 a137 a127 a153 a149a229 a142a143a137a154 a149 a152a170a171 a5 rqξt = k q ξt qξt(kqξt), r e ξt = keξt qξt(keξt) a96ξ t a152 a108 a137a142a143a148a149 a170a171 a10 a198 a32rqξt a229reξt a152a172 a120 a106 rλ = reξt + λ(rqξt ?reξt), ?∞ < λ < +∞ a7 a94rλ a150 a198 a161a156a142a143a157a158 a170a171 a2a173a140 a96a108 a137a142a143a148a149 a170a171 rμ a2a142a174 rλ a229rμ a152 a142a143 a205 a157a158 a5a175 a116 c?a91a92a93 73 September 6, 2005 8.4a176a177a178a179a180a181a40 a50 a39CAPM a47a48a49 a50a39a51a52 a94a95a53CAPM a182r λ a229r μ a183a184 a108 a137a142a143a148a149 a170a171a172 a120 a2a222a174 a134a135 β a101a102a103a98 E(rj,t+1|ξt) = E(rμ|ξt) + βj(ξt)[E(rλ|ξt)?E(rμ|ξt)], ? j a151a166β j(ξt) = Cov(rj,t+1,rλ|ξt) Var(rλ|ξt) . a102a185a79 a143a186a187a2a142a143β a127 a153 a103a98 a110 a5 Et(rj,t+1) = Et(rμ) + βtj[Et(rλ)?Et(rμ)] §8.4 a188a189a190a191a192a193a71a104a194 CAPM a109a110a229 a144 a108a110 a29a30 a243 a137a164a165 a118a119 a128a170a195 a171a172 a118a119 a198a196 a246a197a198 a152 a2 a199a200a96a201a9a202 a78a203 a79 a143ξt a152 a108 a137a164a165 a118a119 a198 a197a198 a152 a195a128 a198 a2 a96ξ t+1 ? ξt a152 a162a20 a157a204a205 a96ξ t a152 a108 a137a206a207a208 a166 a2 a209a200 a140 a96a96ξ t a152 a164a165 a90a91?h(ξ t)a2a142a174 a104 a152 a108 a137a146a147a127a128 a162 a20 a157 a106 [p(ξt+1) + x(ξt+1)]?h(ξt) = ˉw(ξt+1), ? ξt+1 ? ξt a114?h(ξ t) a5a210a211a212 a191a192a213a214a2 a104a96ξ t+1 a152 a108 a137 a170a171 a5 rˉw(ξt+1) = [p(ξt+1) + x(ξt+1)] ?h(ξt) p(ξt)?h(ξt) a182a108 a137a146a147 a127 a153a215 a214a222a216 a110 rˉw(ξt+1) = ˉw(ξt+1)q ξt(ˉwt+1) a171a172a125a153a211 a152a132 a182 a214 a115 a5 V i(y) = Tsummationdisplay t=0 (δi)tυi(yt) a151a166VNM a132 a182 a214 a115 υi a5 a240a241 a214 a115 υi(yt) = ?(yt ?αi)2, yt < αi, ? t a128 a198 a2 a96ξ t a152 a108 a137a164a165 a118a119 a2a125a153a211i a152a132 a182 a214 a115 a5 ?[c(ξt)?αi]2 ?δiE[(ct+1 ?αi)2|ξt] = ?[c(ξt)?αi]2 ?δiVar(ct+1|ξt)?δi[E(ct+1|ξt)?αi]2. a173a174 a55a217a2 a96a111 a137a164a165 a118a119a166 a2 a118a119 a146a147 a5 a162a20 a157 a152 a222a218a160a161 a159 a2 a200a118a119 a170a171 a5 a148a149 a170a171 a195 a219a23 a85a131a220 a182 a24 a96ξ t a152 a108 a137a164a165 a118a119 a55a2 a108 a137 a170a171 rˉw,t+1 a5 a142a143a148a149 a170a171 a2 a209a23a104 a222a221 a5 a142a143β a127 a153 a103a98a166 a152a114 a29 a170a171 a195a171a172 a108 a137a143a144a145a146a147 a96a108 a137a206a207a208 a166 a2a173 a169a134a135 a191a192 a163a164a193 E(rj,t+1|ξt) = ˉr(ξt+1) + βj(ξt)[E(rˉw,t+1|ξt)? ˉr(ξt+1)] a151a166β j(ξt) = Cov(rj,t+1,rˉw,t+1|ξt) Var(rˉw,t+1|ξt) . a117 a220a105a108 a137a142a143a144a145 a109 a153 E(rˉw,t+1|ξt+1)?ˉr(ξt+1) a110a111 a112 a128βj(ξt). a222a185a79 a143a186a187a2a142a143a164a165 a118a119a120 a110 a5 Et(rj,t+1) = ˉrt+1 + βtj[Et(rˉw,t+1)? ˉrt+1]. September 6, 2005 74 c?a223a224a225 a47a48a49 a50 a39a51a52a226a227 a53CAPM 8.5 a228 a42a45a46a229a230 §8.5 a231a73a76a77a232a233 a111 a137a234a235 a152a236a237 a164a165 a90a91 a152 a61a127a238a2 (1) a239ξt a152a240a241 a164a165 a236a237 a166 a5a242 a20a243 a164a165 a90a91 ?h(ξ t) (2) a239 a243 a241 a164a165 a236a237 a166 a5a244 a168 a152 a164a165 a90a91a101a102 ?h a2a245a207a129 a152 a243 a241 a146a147 a5a242 a20a243a246 (pt+1 + xt+1)?ht ?pt+1?ht+1 = ˉwt+1, ? t < T. a247a168a101a102 a152 a140a239 a63a248a249a196 a246a197a198 a236a237a152a250a251 a2a252a253 a114 ?h a5a254 a116a255a0a1a2a213a214a3a4. a5a6a2a239 a201 a240 a7a8T ?1 a152 a79a9ξ T?1 a10 ?h(ξT?1) = ?h(ξT?1). a11 a100 a10 a130a131T = 1 a10 a173 a242 a20a243 a164a165 a90a91a12a13 a241a236a237 a164a165 a90a91a14 a54a10a15 a167a111 a241a236a237 a164a165 a90a91a14 a54a16 a13 a241a236a237 a164a165 a90a91a101a102 ?h a152a240a241a170a171a17 rm,t+1 = (pt+1 + xt+1) ?ht pt?ht = ˉwt+1 + pt+1?ht+1 pt?ht a18 a128a19a20 a166 a161a21a21a203a22 a152a23a24a10a170a171 rm,t+1 a240a25 a150a26a27 a9 a148a149 a170a171a10 a209a200 a150a28a29a30a27 a9 a164a165 a236a237 a31 a157a32 a166 a152a170a171 rˉw,t+1. §8.6 a33a34a35a36a37a38 a39a40 a194 CAPM a2398.1a41 a166a42a43a44 a241 CAPMa7 a10a45 a169 a250a251a236a237 a197a198 a195 a46 a2398.4a41 a42a43 a27 a9CAPMa7 a10 a252a253 a250a251a196a47 a197a198 a195 a48a13 a241a236a237 a202a197a198a7 a10a49a50a51a52a53 a27 a9 a164a165 a236a237 a31 a10 a247a54a55a56a57 a242a58 a243a59 a17 a245a239 a240a241 a206a207a208 a19 a152a60a61 a195 a162 ˉwM t+1 a17 ˉwt+1 a239Mξt(p)a19a152a60a61a10a62a63 ˉrˉw,t+1 = ˉw Mt+1 qξt(ˉwMt+1) a64a65 a30a66a67 a152a68 a182a69a70 a26a71a72a235a73 a10 a173a239a74a75 a7 ˉr ˉw,t+1 a17 a27 a9a76a77 a170a171a10 a199a200 a19a41a73a27 a9 a164a165 a236 a237 a31 a157a32a78a79 a16 c?a223a224a225 75 September 6, 2005