確率的な順序関係と多段決定問題の最適政策の単調性について

全文

(1)商経学叢第58巻第3号 2012年3月. 確率的な順序関係と多段決定問題の最適政策の単調性について. 中. 井. 概要確率的な順序関係は,信 [7]で. . 達*. 頼性理論をはじめとして決定問題でよく用いられている。. は,評価と関連する状態をもとに決定を行う逐次決定問題を扱い,不完備情報の多. 段決定問題として,最適政策や最適値に関する性質を求めた。また,[8,10]で. は,メンテ. ナンスを考慮した多段決定問題において,最適政策や不完備情報における最適値の単調性を扱った。これらの問題ではマルコフ連鎖にしたがった状態推移を考えたが,いずれの場合にも状態を表す確率変数の順序関係や確率的凸性と密接に関連する。ここでは,これらの問題を確率的な順序関係などとの観点から整理し,とくに最適政策の持つ単調性を中心に考える。. Abstract. Stochastic. decision problems. as a sequential problem. In [7] , properties. decision problem. is also considered. [8, 10], a sequential maintenance, value.. order relations. decision problem. ratio ordering. to the relationship. to these stochastic. Convexity,. lem, Markov Eff-..91. El. Decision. observable. Markov. to outcomes. decision. process.. is set up which takes into account. these properties. such as likelihood. sequential. of optimal policy and optimal value are treated. and observe some monotonic. Stochastic. role in analyzing. where a state is closely related. as a partially. For both problems,. キーワード. play an important. properties. of optimal. convexity.. a partial. order relations. In this paper, we focus. order relations.. Dynamic. Programming,. Sequential. D ecision. Process. 2012* 1 )] 23 11. * 千葉市稲毛区弥生町1-33千. In. policy and optimal. depend on the stochastic. and stochastic. This. 葉大学教育学部,e-mail:t-nakai@faculty. 99 (649)一. .chiba-u.jp. Prob-.

(2) . 1. . ͸. ͡. Ί. ʹ. ֬཰తͳॱংؔ܎͸ɼ৴པੑཧ࿦Λ͸͡Ίͱ͢Δܾఆ໰୊ͰΑ͘༻͍ΒΕɼෆ‫׬‬උ৘ใ ͷଟஈܾఆ໰୊ʹ͓͍ͯ΋ֶशϓϩηεͱີ઀ʹؔ࿈͢Δ໬౓ൺॱংͷ΋ͱͰ͍Ζ͍Ζͳ ੑ࣭͕‫ٻ‬ΊΒΕ͍ͯΔɻ[7] ʹ͓͍ͯɼධՁͱؔ࿈͢Δঢ়ଶΛ΋ͱʹɼࢧग़Λܾఆ͢Δஞ࣍ ܾఆ໰୊Λѻ͍ɼෆ‫׬‬උ৘ใͷଟஈܾఆ໰୊ͱͯ͠ɼ࠷ద੓ࡦ΍࠷ద஋ʹؔ͢Δੑ࣭Λ‫ٻ‬Ί ͨɻ·ͨɼ[8, 10] Ͱ͸ɼϝϯςφϯεΛߟྀͨ͠ଟஈܾఆ໰୊ʹ͓͍ͯɼ࠷ద੓ࡦʹؔ͢Δ ୯ௐੑͱෆ‫׬‬උ৘ใͷ৔߹ͷ࠷ద஋ͷ୯ௐੑΛѻͬͨɻ͍ͣΕͷ৔߹ʹ΋Ϛϧίϑ࿈࠯ʹ ͕ͨͨ͠ঢ়ଶͷਪҠΛߟ͕͑ͨɼঢ়ଶΛද֬͢཰ม਺ͷॱংؔ܎΍֬཰తತੑ (stochastic. convexity) ͸ɼ࠷ద੓ࡦ΍࠷ద੓ࡦͷ΋ͱͰͷ࠷ద஋ͷੑ࣭ͱີ઀ʹؔ࿈͢Δ͜ͱ͕Θ͔ͬ ͍ͯΔɻ͜͜Ͱ͸ɼ[7, 8, 10] ͳͲͰѻͬͨ໰୊Λ֬཰తͳॱংؔ܎ͳͲͱͷ‫͔఺؍‬Β੔ཧ ͠ɼ࠷ద੓ࡦͷ࣋ͭ୯ௐੑΛத৺ʹߟ͑ɼෆ‫׬‬උ৘ใͷଟஈܾఆ໰୊ͱͯ͠ͷ࠷ద஋ͷ࣋ ͭੑ࣭ʹ͍ͭͯ΋؆୯ʹ৮ΕΔɻ. 2 2.1. ֬཰తͳॱংؔ܎ͱ֬཰తತੑɾԜੑ. ֬཰తͳॱংؔ܎. X ͱ Y Λ 2 ͭͷ֬཰ม਺ͱ͢Δͱ͖ɼ‫ج‬ຊతͳ֬཰తͳॱংؔ܎ͱͯ͠ɼͭ͗ͷΑ͏ͳ ΋ͷ͕஌ΒΕ͍ͯΔɻ ఆٛ 1 ೚ҙͷ u ∈ (−∞, ∞) ʹରͯ͠ɼP (Y > u) ≤ P (X > u) ͱͳΔͱ͖ɼX ͸ Y ΑΓ ֬཰ॱং (usual stochastic order) ͷҙຯͰେ͖͍ͱ͍͍ɼX ≥ST Y ͱද͢ɻ ఆٛ 2 ֬཰ີ౓ؔ਺ f (x) ͓Αͼ g(x) Λ࣋ͭ 2 ͭͷ֬཰ม਺ X ͱ Y Λߟ͑Δɻx ≥ y ͱ ͳΔ೚ҙͷ x ͱ y ʹରͯ͠ɼf (y)g(x) ≤ f (x)g(y) ͱͳΔͱ͖ɼX ͸ Y ΑΓ໬౓ൺͷҙຯ Ͱେ͖͍ͱ͍͍ɼX ≥LRD Y ͱද͢ɻ ෆ‫׬‬උ৘ใͷܾఆ໰୊ͰϕΠζֶशΛߟ͑Δͱ͖ʹ͸ɼTP2 ͷੑ࣭ΛԾఆ͢Δ͕ɼ͜Ε ͸͜ͷ֬཰తॱংؔ܎ʹ‫ͮ͘ج‬΋ͷͰ͋Δɻ ఆٛ 3 ֬཰ີ౓ؔ਺ f (x) ͓Αͼ g(x) Λ࣋ͭ 2 ͭͷ֬཰ม਺ X ͱ Y Λߟ͑ɼ͜ΕΒͷ֬཰ ม਺ͷ෼෍ؔ਺Λ F (x) ͱ G(x) ͱ͢Δɻx ≥ y ͱͳΔ೚ҙͷ x ͱ y ʹରͯ͠ɼF (y)G(x) ≥. F (x)G(y) ͱͳΔͱ͖ɼX ͸ Y ΑΓ‫ނ‬ো཰ (hazard rate) ͷҙຯͰେ͖͍ͱ͍͍ɼX ≥HR Y ͱද͢ɻ͜͜ͰɼF (x) = 1 − F (x) Ͱ͋Δɻ. .

(3)

(4) . ͭ͗ʹɼt∗ = sup{t : F (t) > 0} ͱ͢Δͱ͖ɼฏ‫ۉ‬༨໋ؔ਺ (mean residual life function) Λͭ͗ͷΑ͏ʹఆٛ͢Δɻ. m(t) =. ⎧ ⎪ ⎨ E[X − t|X > t],. t < t∗ ͷͱ͖. ⎪ ⎩ 0. ͦͷଞ. ఆٛ 4 ֬཰ີ౓ؔ਺ f (x) ͓Αͼ g(x) Λ࣋ͭ 2 ͭͷ֬཰ม਺ X ͱ Y Λߟ͑Δɻ೚ҙͷ t ʹ ରͯ͠ɼmX (t) ≥ mY (t) ͳΒ͹ɼX ͸ Y ΑΓฏ‫ۉ‬༨໋ͷҙຯͰେ͖͍ͱ͍͍ɼX ≥M RL Y ͱද͢ɻ ͜ͷͱ͖ɼ࣍ͷੑ࣭͕੒Γཱͭɻ ิ୊ 1 2 ͭͷ֬཰ม਺ X ͱ Y ʹରͯ͠ɼX ≥LRD Y ͳΒ͹ X ≥HR Y Ͱ͋ΓɼX ≥HR Y ͳΒ͹ X ≥M RL Y Ͱ͋Δɻ ͱ͘ʹɼϚϧίϑաఔͷਪҠ๏ଇ P = (ps (t))s,t∈(−∞,∞) ʹ͍ͭͯɼ೚ҙͷ s < s , t ≤ t ͓ Αͼ u < v ͱͳΔ s, s , t, t , u, v ʹରͯ͠ pu (s)pv (t )−pu (t)pv (s ) ≥ pv (s)pu (t )−pv (t)pu (s ) ͱ͢Ε͹ɼͭ͗ͷΑ͏ͳੑ࣭Λ࣋ͭɻͨͩ͠ɼT (s) Λঢ়ଶ͕ s ͷͱ͖ͭ͗ͷঢ়ଶΛද֬͢ ཰ม਺ͱ͢Ε͹ɼEs [u(T (s))] =. . ∞ −∞. ps (t)u(t)dt Ͱ͋Δɻ. ิ୊ 2 s < s ͳΒ͹ɼs ʹؔ͢Δඇ૿Ճತؔ਺ u(s) ʹର͠ɼE[u(T (s))] ≤ E[u(T (s ))] Ͱ ͋Δɻ ͱ͜ΖͰɼX ͱ Y Λ 2 ͭͷ֬཰ม਺ͱ͢Δͱ͖ɼ͜ΕΒͷ֬཰ม਺ͷؒͷॱংؔ܎Λ࣍ ͷΑ͏ʹఆٛ͢Δ͜ͱ͕Ͱ͖Δɻ͜ΕΒͷதͰ࠷ॳͷఆٛ͸ɼఆٛ 1 ͷ֬཰తॱংؔ܎ͱ ಉ஋Ͱ͋Δɻ. (1) ೚ҙͷ૿Ճؔ਺ u(s) ʹରͯ͠ɼE[u(X)] ≥ E[u(Y )] Ͱ͋Δͱ͖ɼX ≥ST Y ͱද͢ɻ (stochastic order) (2) ೚ҙͷ૿Ճ (‫ݮ‬গ) ತ (convex) ؔ਺ u(s) ʹରͯ͠ɼE[u(X)] ≥ E[u(Y )] Ͱ͋Δͱ͖ɼ X ≥ICX (≥DCX )Y ͱද͢ɻ(increasing (decreasing) convex order) (3) ೚ҙͷ૿Ճʢ‫ݮ‬গʣԜ (concave) ؔ਺ u(s) ʹରͯ͠ɼE[u(X)] ≥ E[u(Y )] Ͱ͋Δͱ ͖ɼX ≥ICV (≥DCV )Y ͱද͢ɻ(increasing (decreasing) concave order) ิ୊ 3 2 ͭͷ֬཰ม਺ X ͱ Y ʹରͯ͠ɼX ≥M RL Y ͳΒ͹ X ≥ICX Y Ͱ͋Δɻ. .

(5) . 2.2. ֬཰తತੑͱԜੑ. Shaked and Shanthikumar [12] ʹ͕ͨͬͯ͠ɼs Λύϥϝʔλͱ͢Δ֬཰ม਺ྻ {X(s)|s ∈ (−∞, ∞)} ʹରͯ͠ɼ֬཰తತੑͱԜੑΛͭ͗ͷΑ͏ʹఆٛ͢Δɻ (1) {X(s)|s ∈ (−∞, ∞)} ͕ SI Ͱ͋Δͱ͸ɼ೚ҙͷ૿Ճؔ਺ u(s) ʹରͯ͠ɼE[u(X(s))] ͕ɼs ͷ૿Ճؔ਺ͱͳΔ͜ͱΛ͍͏ɻ(stochastically increasing). (2) {X(s)|s ∈ (−∞, ∞)} ͕ SICX Ͱ͋Δͱ͸ɼ೚ҙͷ૿Ճತؔ਺ u(s) ʹରͯ͠ɼ E[u(X(s))] ͕ɼs ͷ૿Ճತؔ਺ͱͳΔ͜ͱΛ͍͏ɻ(stochastically increasing and convex) (3) {X(s)|s ∈ (−∞, ∞)} ͕ SICV Ͱ͋Δͱ͸ɼ೚ҙͷ૿ՃԜؔ਺ u(s) ʹରͯ͠ɼ E[u(X(s))] ͕ɼs ͷ૿ՃԜؔ਺ͱͳΔ͜ͱΛ͍͏ɻ(stochastically increasing and concave) (4) {X(s)|s ∈ (−∞, ∞)} ͕ SD Ͱ͋Δͱ͸ɼ೚ҙͷ૿Ճؔ਺ u(s) ʹରͯ͠ɼE[u(X(s))] ͕ɼs ͷ‫ݮ‬গؔ਺ͱͳΔ͜ͱΛ͍͏ɻ(stochastically decreasing). (5) {X(s)|s ∈ (−∞, ∞)} ͕ SDCX Ͱ͋Δͱ͸ɼ೚ҙͷ૿Ճತؔ਺ u(s) ʹରͯ͠ɼ E[u(X(s))] ͕ɼs ͷ‫ݮ‬গತؔ਺ͱͳΔ͜ͱΛ͍͏ɻ(stochastically decreasing and convex) (6) {X(s)|s ∈ (−∞, ∞)} ͕ SDCV Ͱ͋Δͱ͸ɼ೚ҙͷ૿ՃԜؔ਺ u(s) ʹରͯ͠ɼ E[u(X(s))] ͕ɼs ͷ‫ݮ‬গԜؔ਺ͱͳΔ͜ͱΛ͍͏ɻ(stochastically decreasing and concave) ͭ͗ʹɼs1 ≤ s2 ≤ s3 ≤ s4 Ͱ s1 +s4 = s3 +s2 ͷͱ͖ɼXi = X(si ) ͱ͓͚͹ (i = 1, 2, 3, 4)ɼ. SICX ΍ SICV ΑΓऑ͍֓೦Ͱ͋Δ SICX(sp) ͱ SICV(sp) Λͭ͗ͷΑ͏ʹఆٛͰ͖Δɻ (1) {X(s)|s ∈ (−∞, ∞)} ͕ SICX(sp) Ͱ͋Δͱ͸ɼmax{X2 , X3 } ≤ X4 Ͱ͋Γ (a.s.)ɼ X2 + X3 ≤ X1 + X4 Ͱ͋Δ͜ͱΛ͍͏ɻ(stochastically increasing and convex in sample path sense) (2) {X(s)|s ∈ (−∞, ∞)} ͕ SICV(sp) Ͱ͋Δͱ͸ɼX1 ≤ max{X2 , X3 } Ͱ͋Γ (a.s.)ɼ X2 + X3 ≥ X1 + X4 Ͱ͋Δ͜ͱΛ͍͏ɻ(stochastically increasing and concave in sample path sense) .

(6)

(7) . ิ୊ 4 (Shaked and Shanthikumar [12]). (1) {X(s)|s ∈ (−∞, ∞)} ͕ SICX(sp) ͳΒ͹ɼSICX Ͱ͋Δɻ (2) {X(s)|s ∈ (−∞, ∞)} ͕ SICV(sp) ͳΒ͹ɼSICV Ͱ͋Δɻ ྫ 1 X(μ) Λਖ਼‫ن‬෼෍ N (μ, σ 2 ) ͱ͢Δɻ{X(μ)|μ ∈ (−∞, ∞)} ͸ SICX(sp) Ͱ͋Γ. SICV(sp) Ͱ͋Δɻ ิ୊ 5 ([12]). (1) {X(s)|s ∈ (−∞, ∞)} ͕ SICX(sp) Ͱ͋Γɼu(·) Λ૿Ճತؔ਺ͱ͢Δɻ͜ͷͱ͖ɼ {u(X(s))|s ∈ (−∞, ∞)} ΋·ͨ SICX(sp) Ͱ͋Δɻ (2) {X(s)|s ∈ (−∞, ∞)} ͕ SICV(sp) Ͱ͋Γɼu(·) Λ૿ՃԜؔ਺ͱ͢Δɻ͜ͷͱ͖ɼ {u(X(s))|s ∈ (−∞, ∞)} ΋·ͨ SICV(sp) Ͱ͋Δɻ ྫ 2 X(μ) Λਖ਼‫ن‬෼෍ N (μ, σ 2 ) ͱ͢ΔɻY (μ) = eX(μ) ͱ͓͚͹ɼu(x) = ex ͕૿Ճತؔ਺ ͔ͩΒ {Y (μ)|μ ∈ (−∞, ∞)} ͸ SICX(sp) Ͱ͋Δɻ͕ͨͬͯ͠ɼY (μ) ͸ର਺ਖ਼‫ن‬෼෍Ͱ͋ Γɼର਺ਖ਼‫ن‬෼෍͸ SICX(sp) Ͱ͋ΓɼSICX Ͱ͋Δɻ. s ∈ (0, ∞) Ͱఆٛ͞Εͨؔ਺ u(s) ͕ɼ೚ҙͷ s < t ͱ 0 < λ < 1 ͱͳΔ λ ʹରͯ͠ u(sλ t1 − λ) ≥ (≤)λu(s) + (1 − λ)u(t) ͱͳΔͱ͖ɼ͜ͷؔ਺ u(s) Λ P Ԝؔ਺ (P ತؔ਺) ͱ‫͏ݴ‬ɻ u(x) Λ x ͷ P ತؔ਺ͱ͢Ε͹ɼw(y) ≡ u(ey ) ͸ɼ. w(λ log a + (1 − λ) log b) = u(eλ log a+(1−λ) log b ) ≤ λu(elog a ) + (1 − λ)u(elog b ) = λw(a) + (1 − λ)w(b) ͳͷͰɼy ͷತؔ਺Ͱ͋Δɻ͍ͬΆ͏ɼX(s) Λີ౓ؔ਺͕ fs (t) = φlog s,σ2 (log t) t. √ 1 e− 2πσt. (log t−log s)2 2σ 2 =. ͷର਺ਖ਼‫ن‬෼෍ؔ਺ͱ͢Δɻ͜͜Ͱɼφμ,σ2 (x) Λਖ਼‫ن‬෼෍ N (μ, σ 2 ) ͷີ౓ؔ. ਺ͱ͠ɼY (s) Λਖ਼‫ن‬෼෍ N (s, σ 2 ) ʹ͕ͨ͠͏ؔ਺ྻͱ͢Δɻ͜ͷͱ͖ɼ. E[u(X(aλ b1−λ ))] = = =. . ∞. 0 ∞ −∞ ∞ −∞. φλ log a−(1−λ) log b,σ2 (log t) u(elog t )dt t φλ log a−(1−λ) log b,σ2 (x)u(ex )dx φλ log a−(1−λ) log b,σ2 (x)w(x)dx. ͱͳΔɻͱ͜ΖͰɼ{Y (s)} ͸ SICX ΑΓɼE[u(X(aλ b1−λ ))] = E[w(Y (λ log a−(1−λ) log b))]. ≤ λE[w(Y (log a))]+(1−λ)E[w(Y (log b))] ͱͳΔɻ͜͜ͰɼE[w(Y (log a))] = E[u(X(a))] .

(8) . ͓Αͼ E[w(Y (log b))] = E[u(X(b))] Ͱ͋ΔɻΑͬͯɼE[u(X(aλ b1−λ ))] ≤ λE[u(X(a))] +. (1 − λ)E[u(X(b))] ͱͳΔɻE[u(X(s))] ͸ x ͷ P ತؔ਺Ͱ͋Δɻ ఆٛ 5 u(s) Λ೚ҙͷ૿Ճ P ತ (P Ԝ ) ؔ਺ͱ͢ΔɻE[u(X(s))] ͕ s ͷ૿Ճ P ತ (P Ԝ ) ؔ ਺ͱͳΔͱ͖ɼ{X(s)}s∈(0,∞) Λ SIPCX(SIPCV)(stochastically increasing and P-convex. (P-concave)) ͱ͍͏ɻ. 3. ެ‫ڞ‬౤ࢿϞσϧʹ‫ͮ͘ج‬ஞܾ࣍ఆϞσϧ. ফ๷‫׆‬ಈ΍‫׆࡯ܯ‬ಈͱ͍ͬͨެ‫ڞ‬αʔϏεʹର͢Δࢧग़Λɼຖ೥౓ͷ༧ࢉͷൣғ಺Ͱߦ ͏ɻ͜ΕΒͷެ‫ڞ‬αʔϏεʹରͯ͠ɼ࣮ࡍͷઃඋ΍ࢪઃ͋Δ͍͸ਓһͱɼ͜ͷαʔϏεʹ ରͯ͠ຬ଍͢Δ͔ͱ͍͏͜ͱͷ͍͋ͩʹ͸ؔ࿈͕͋Δ͜ͱ͸͔֬Ͱ͋Δ͕ɼ͔ͱ͍ͬͯઃ උ΍ࢪઃɼਓһ͕ଟ͘ͳͬͨͱ͜ΖͰɼੜ‫ڥ؀׆‬΍‫ࡁܦ‬ঢ়‫͕Ͳͳگ‬มԽ͢Δ͜ͱͰɼ͜Ε ΒͷαʔϏεʹର͢Δཁ‫૿͕ٻ‬Ճ͠ɼຬ଍Λ‫͍ͯ͡ײ‬Δॅຽͷׂ߹͕௿Լ͢Δ͜ͱ΋͋Δɻ ͦ͜Ͱɼੜ࢈෺΍αʔϏεʹରͯ͠ຬ଍Λ‫͍ͯ͡ײ‬Δɼ͋Δ͍͸ॆ଍͍ͯ͠Δͱ‫͍ͯ͡ײ‬ Δॅຽͷׂ߹ΛΞ΢τΧϜͷ 1 ͭͷࢦඪͱͱΒ͑ɼ͜ͷࢦඪ͸֬཰తʹਪҠ͢Δঢ়ଶʹΑͬ ͯ΋มԽ͢Δͱ͢Δɻ·ͨɼ༧ࢉΛ௥Ճͯ͠ࢧग़͢Δ͜ͱͰɼঢ়ଶ͕มԽ͠ɼͦͷ݁ՌΞ ΢τΧϜͷࢦඪͰ͋Δॅຽͷׂ߹ͷมԽΛଅ͢͜ͱ͕Ͱ͖Δͱ͢Δɻ ͜ͷϞσϧΛঢ়ଶۭ͕ؒ (−∞, ∞) ͷϚϧίϑաఔͱߟ͑ɼ͜ͷঢ়ଶͱ͜ͷαʔϏεʹ ରͯ͠ຬ଍Λ‫͍ͯ͡ײ‬Δॅຽͷׂ߹͸ɼີ઀ʹؔ܎͢Δ΋ͷͱ͢Δɻঢ়ଶΛද͢஋ s ͕େ ͖͘ͳΕ͹ঢ়ଶ͕ྑ͘ͳΔͱ͢Δɻ·ͨɼ͜ͷঢ়ଶ͸ܾఆʹ͔͔ΘΒͣɼϚϧίϑաఔʹ ͕ͨͬͯ͠ਪҠ͢Δɻ͜ͷͱ͖ɼ‫ܭ‬ը‫Ͱ಺ؒظ‬རಘΛ࠷େԽ͢Δ࠷ద੓ࡦͱ࠷ద੓ࡦʹ͠ ͕ͨͬͨͱ͖ʹಘΒΕΔ࠷ద஋ʹ͍ͭͯߟ͑Δɻ·ͨɼ[7] ͳͲͱಉ༷ʹɼܾఆʹΑΓঢ়ଶ ΛมԽͰ͖Δͱ͢Δɻ ঢ়ଶۭؒΛ (−∞, ∞) ͱ͢Δɻঢ়ଶ͕ s ͷͱ͖ɼܾఆ x ΛऔΕ͹ɼঢ়ଶΛ s + x ͱͰ͖Δ. (x ≥ 0)ɻ͜ͷͱ͖ͷܾఆʹର͢Δඅ༻Λ C(x) ͱ͢Δɻu(s) Λ࠷‫ޙ‬ͷ‫ظ‬ͷঢ়ଶ͕ s ͷͱ͖ ͷऴ୺རಘͱ͠ɼu(s) ͸Ԝؔ਺ͱ͢Δɻ·ͨɼঢ়ଶ͸ਪҠ๏ଇΛ P = (ps (t))s,t∈(−∞,∞) ͱ ͢ΔϚϧίϑաఔʹ͕ͨͬͯ͠ਪҠ͢ΔͱԾఆ͢Δɻ. v(s) = maxx≥0 {−C(x) + u(s + x)} ͱ͢Δͱ͖ɼu(s) ͕ s ͷ૿Ճؔ਺Ͱ͋Ε͹ɼv(s) ΋ ૿Ճؔ਺Ͱ͋Δ͜ͱ͸໌Β͔Ͱ͋Δɻ ิ୊ 6 v(s) = maxx≥0 {−C(x) + u(s + x)} ͱ͢ΔɻC(x) ͕ತؔ਺ͷͱ͖ɼu(s) ͕Ԝؔ਺. .

(9)

(10) . ͳΒ͹ɼv(s) ΋Ԝؔ਺Ͱ͋Δɻͨͩ͠ɼC(x) ͸૿Ճؔ਺ͱ͢Δɻ ূ໌: v(s) = C(x∗ )+u(s+x∗ ) ͓Αͼ v(t) = C(x∗∗ )+u(t+x∗∗ ) ͱ͓͘ɻ0 ≤ λx∗ +(1−λ)x∗∗ Ͱ͋Γɼ೚ҙͷ λ (0 < λ < 1) ͱ s < t ʹରͯ͠ u(λs + (1 − λ)t) ≥ λu(s) + (1 − λ)u(t) ͩ ͔Β C(x) ʹؔ͢ΔԾఆΛ༻͍ͯ. v(λs + (1 − λ)t) = max{−C(x) + u(λs + (1 − λ)t + x)} x≥0. ≥ −C(λx∗ + (1 − λ)x∗∗ ) + u(λs + (1 − λ)t + λx∗ + (1 − λ)x∗∗ ) ≥ −(λC(x∗ ) + (1 − λ)C(x∗∗ )) + λu(s + x∗ ) + (1 − λ)u(t + x∗∗ ) = λv(s) + (1 − λ)v(t) ͱͳΔɻ͕ͨͬͯ͠ɼu(s) ͸Ԝؔ਺ͱͳΔɻ2 ঢ়ଶ͸ɼϚϧίϑաఔʹ͕ͨͬͯ͠ਪҠ͠ɼਪҠ๏ଇΛ P = (ps (t))s,t∈(−∞,∞) ͱ͢Δɻ ֤‫ͱ͝ظ‬ͷܾఆΛ x ≥ 0 ͱ͢Δɻ͜ͷͱ͖ɼ‫ܭ‬ը‫ ͕ؒظ‬n Ͱɼঢ়ଶ͕ s ͷͱ͖ɼ࠷େԽ໰ ୊ʹ͓͍ͯɼ࠷దʹৼΔ෣ͬͯಘΒΕΔ૯‫ظ‬଴རಘΛ un (s) ͱ͢Ε͹ɼঢ়ଶ͕Ϛϧίϑա ఔʹ͕ͨͬͯ͠ਪҠ͢Δ͔Βɼ࠷దํఔࣜ͸ͭ͗ͷΑ͏ʹͳΔɻ͜͜ͰɼT (s) Λঢ়ଶ͕ s ͷͱ͖ͭ͗ͷঢ়ଶΛද֬͢཰ม਺ͱ͢Ε͹ɼE[un−1 (T (s))] =. ∞. −∞ ps (t)un−1 (t)dt. un (s) = max{−C(x) + E[un−1 (T (s + x))]}, x≥0. Ͱ͋Δɻ. (1). ͜͜Ͱɼ. u1 (s) = max{−C(x) + E[u(T (s + x))]} x≥0. ͱ͢Δɻͨͩ͠ɼu(s) ͸ s ͷ૿Ճؔ਺ͱ͠ɼC(x) ͸ x ͷ૿Ճؔ਺ͱ͢Δɻ·ͨɼu(s) ͕Ԝ ؔ਺ͱͳΔ໰୊Λߟ͑Δͱ͖͸ɼC(x) ͸ತؔ਺ͱ͢Δɻ ਪҠ๏ଇ͕ (ps (t))s,t∈(−∞,∞) ͔ͩΒɼ֬཰ม਺ྻ {T (s)|s ∈ (−∞, ∞)} ʹରͯ͠ɼͭ͗ ͷԾఆΛઃ͚Δɻ Ծఆ 1 t ʹؔ͢Δ૿ՃԜؔ਺Λ u(t) ͱ͢Ε͹ɼE[u(T (s))] ͸ s ʹؔ͢Δ૿ՃԜؔ਺ͱͳͬ ͍ͯΔɻ͢ͳΘͪɼ֬཰ม਺ྻ {T (s)|s ∈ (−∞, ∞)} ͸ɼSICV Ͱ͋Δɻ ิ୊ 7 un (s) ͸ɼs ʹؔ͢Δ૿Ճؔ਺Ͱ͋Δɻ ূ໌: n ʹؔ͢Δ‫ؼ‬ೲ๏Λ༻͍Δɻu0 (s) = u(s) ͔ͩΒɼu0 (s) ͸૿ՃԜؔ਺Ͱ͋Δɻun−1 (s) ͕૿ՃԜؔ਺ͱԾఆ͢ΔͱɼԾఆ 1 ΑΓ E[un−1 (T (s+x))] ΋·ͨ s ʹؔ͢Δ૿Ճؔ਺ͰͰ͋. .

(11) . ΔɻE[un−1 (T (s+x))] ͕ s ͷ૿Ճؔ਺ͳͷͰɼun (s) = maxx≥0 {−C(x)+E[un−1 (T (s+x))]} ΋ s ͷ૿Ճؔ਺Ͱ͋Δɻ2 ͞Βʹɼิ୊ 6 ΑΓɼun (s) ͕Ԝؔ਺ͱͳΔɻ ิ୊ 8 Ծఆ 1 ͷ΋ͱͰɼun (s) ͸Ԝؔ਺Ͱ͋Δɻ ‫ܭ‬ը‫ ͕ؒظ‬n Ͱ͋Γɼঢ়ଶ͕ s ͷͱ͖ͷɼ࠷దͳܾఆΛ x∗n (s) ͱ͢Δɻ ੑ࣭ 1 Ծఆ 1 ͷ΋ͱͰɼx∗n (s) ͸ s ʹؔͯ͠‫ݮ‬গ͢Δɻ ূ໌: n ʹؔ͢Δ‫ؼ‬ೲ๏Λ༻͍Δɻn = 1 ͷ৔߹͸ɼҰൠͷͱ͖ͱಉ͡Α͏ʹಋ͚Δɻt ≥ s ͱ͢Δɻn(> 1) ͷͱ͖ɼx∗n (s) = x∗ ͱ͓͚͹ɼ(1) ࣜΑΓ. un (s) = max{−C(x) + E[un−1 (T (s + x))]} = −C(x∗ ) + E[un−1 (T (s + x∗ ))] x≥0. (2). ͱͳΔɻ0 < x∗ ≤ x ͱͳΔ೚ҙͷ x ʹରͯ͠ɼෆ౳ࣜ. −C(x) + E[un−1 (T (t + x))] ≤ −C(x∗ ) + E[un−1 (T (t + x∗ ))] ͕੒Γཱͯ͹ɼx∗n (s) = x∗ ≥ x∗n (t) ͱͳΔ͜ͱ͕ࣔ͞ΕΔɻ. (2) ࣜΑΓɼ೚ҙͷ x ≥ 0 ʹରͯ͠ −C(x) + E[un−1 (T (s + x))] ≤ −C(x∗ ) + E[un−1 (T (s + x∗ ))]. (3). ͔ͩΒɼ. −C(x) + C(x∗ ) ≤ E[un−1 (T (s + x∗ ))] − E[un−1 (T (s + x))]. (4). ͱͳΔɻ͍ͬΆ͏ɼԾఆ 1 ΑΓɼE[un−1 (T (s + x))] ͸ s ʹؔ͢Δ૿ՃԜؔ਺Ͱ͋Δɻ(t +. x∗ ) − (s + x∗ ) = (t + x) − (s + x) Ͱ͋Γ 0 < x∗ ≤ x ͔ͩΒɼ E[un−1 (T (t + x))] − E[un−1 (T (s + x)) ≤ E[un−1 (T (t + x∗ ))] − E[un−1 (T (s + x∗ ))] ͱͳΔɻ(4) ࣜͱ͜ͷෆ౳͔ࣜΒɼ(3) ͕ࣜಋ͔Εɼ͜ͷੑ࣭͕੒Γཱͭɻ2 Ծఆ 2 t ≥ s ͷͱ͖೚ҙͷԜؔ਺ u(s) ʹରͯ͠ɼE[u(T (t))] − E[u(T (s))] ≤ u(t) − u(s) Ͱ͋Δɻ. .

(12)

(13) . un (s) ͕૿ՃԜؔ਺͔ͩΒɼԾఆ 2 ΑΓɼs < t ʹରͯ͠ E[un (T (t))] − E[un (T (s))] ≤ un (t) − un (s) ͱͳΔɻ s < t ͷͱ͖ɼ೚ҙͷ n ≥ 1 ʹରͯ͠ E[un (T (t))] − E[un (T (s))] ͱ E[un−1 (T (t))] − E[un−1 (T (s))] ͷؔ܎Λߟ͑Δɻx∗ = x∗n (t) ͱ͓͚͹ɼ un (t) − un (s) = −C(x∗ ) + E[un−1 (T (t + x∗ ))] − max{−C(x) + E[un−1 (T (s + x))]} x≥0. ∗. ≤ E[un−1 (T (t + x ))] − E[un−1 (T (s + x∗ ))] ͱͳΔɻ͍ͬΆ͏ɼิ୊ 8 ΑΓɼԾఆ 1 ͷ΋ͱͰ E[un−1 (T (s))] ͸ s ʹؔ͢Δ૿ՃԜؔ਺ Ͱ͋Δɻ(t + x∗ ) − (s + x∗ ) = t − s Ͱ͋Γ s < t, 0 < x∗ ͔ͩΒɼ. E[un−1 (T (t + x∗ ))] − E[un−1 (T (s + x∗ ))] ≤ E[un−1 (T (t))] − E[un−1 (T (s))] ͱͳΔɻ͜ΕΒͷෆ౳͔ࣜΒ. un (t) − un (s) ≤ E[un−1 (T (t))] − E[un−1 (T (s))] ͱͳΔɻ ิ୊ 8 ΑΓɼun (s) ͕Ԝؔ਺ͳͷͰɼԾఆ 2 ΑΓ E[un (T (t))]−E[un (T (s))] ≤ un (t)−un (s) ͱͳΔɻ͜ΕΒͷෆ౳͔ࣜΒɼ೚ҙͷ n ≥ 1 ʹରͯ͠. E[un (T (t))] − E[un (T (s))] ≤ E[un−1 (T (t))] − E[un−1 (T (s))]. (5). ͱͳΔɻ ੑ࣭ 2 Ծఆ 2 ͷ΋ͱͰɼxn (s) ͸ n ʹؔͯ͠‫ݮ‬গ͢Δɻ ূ໌: n ʹؔ͢Δ‫ؼ‬ೲ๏Λ༻͍Δɻn = 1 ͷ৔߹͸ɼҰൠͷͱ͖ͱಉ͡Α͏ʹಋ͚Δɻn(> 1) ͷͱ͢Δɻs ≤ t ͷͱ͖ɼx∗n (s) = x∗ ͱ͓͚͹ɼ೚ҙͷ 1 < x∗ < x ʹରͯ͠ɼ. −C(x) + E[un−1 (T (s + x))] ≤ −C(x∗ ) + E[un−1 (T (s + x∗ ))] Ͱ͋Δɻ͍ͬΆ͏ɼ(5) ࣜΑΓ. E[un−1 (T (s + x∗ ))] − E[un−1 (T (s + x))] ≤ E[un (T (s + x∗ ))] − E[un (T (s + x))], ͔ͩΒɼ. −C(x) + E[un (T (s + x))] ≤ −C(x∗ ) + E[un (T (s + x∗ ))] . (6).

(14) . ͱͳΔɻ͜ͷ͜ͱ͔Βɼn ʹؔ͢Δ‫ؼ‬ೲ๏ΑΓɼx∗ ≤ x∗n+1 (s) ͱͳΔɻ2 ͕ͨͬͯ͠ɼ೚ҙͷ n ≥ 1 ʹରͯ͠ɼx∗n (s) ≤ x∗n+1 (s) ͳͷͰɼԾఆ 2 ͷ΋ͱͰ x∗n (s) ͷ. n ʹؔ͢Δ୯ௐੑ͕ࣔ͞ΕΔɻ ͱ͜ΖͰɼ࠷ద੓ࡦʹ͕ͨͬͨ͠ͱ͖ͷ࠷ద஋ un (s) ͷ n ʹؔ͢Δ୯ௐੑʹ͍ͭͯߟ͑ Δɻ‫ج‬ຊతʹɼެతαʔϏεʹର͢Δࢧग़͸ɼকདྷͷຬ଍౓΍ॆ଍౓ʹΑΔ‫ظ‬଴ޮ༻͕‫ݱ‬ ࣌఺ʹൺ΂ͯѱ͘ͳͬͨͱͯ͠΋ɼ͜ΕΒͷαʔϏεΛଧͪ੾Δ͜ͱ͸Ͱ͖ͣɼଓ͚ͯߦ ͏ඞཁ͕͋Δɻ͕ͨͬͯ͠ɼঢ়ଶͷؔ਺Ͱ͋Δޮ༻ؔ਺ͱਪҠ๏ଇʹΑͬͯ͸ɼun (s) ͸ n ʹؔͯ͠૿Ճ͢Δ͜ͱ΋͋Ε͹ɼ‫ݮ‬গ͢Δ͜ͱ΋ߟ͑ΒΕΔɻͱ͜ΖͰɼ೚ҙͷ s ʹର͠ ͯ un−1 (s) ≤ un−2 (s) ͳΒ͹ɼE[un−1 (T (s + x))] ≤ E[un−2 (T (s + x))] ͱͳΔͷͰɼ. un (s) = max {−c(x) + E[un−1 (T (s + x))]} x≥0. un−1 (s) = max {−c(x) + E[un−2 (T (s + x))]} x≥0. ΑΓɼun (s) ≤ un−1 (s) ͱͳΔ͜ͱ͕Θ͔Δɻ൓ରʹɼ೚ҙͷ s ʹରͯ͠ un−1 (s) ≥ un−2 (s) ͳΒ͹ɼun (s) ≥ un−1 (s) ͱͳΔɻ͕ͨͬͯ͠ɼ‫ؼ‬ೲ๏Λ༻͍Ε͹ɼn = 1 ͷͱ͖ͷੑ ࣭ʹΑͬͯɼun (s) ͷ n ʹؔ͢Δ୯ௐੑ͕ఆ·Δɻ͢ͳΘͪɼn = 1 ͷͱ͖͸ɼu1 (s) =. max {−c(x) + E[u(T (s + x))]} Ͱ͋Γɼu0 (s) = u(s) ͔ͩΒɼu1 (s) ≥ u0 (s) Ͱ͋Ε͹ un (s) x≥0. ͸ n ʹؔ͢Δඇ‫ݮ‬গؔ਺Ͱ͋Γɼu1 (s) ≤ u0 (s) Ͱ͋Ε͹ un (s) ͸ n ʹؔ͢Δඇ૿Ճؔ਺ͱ ͳΔ͜ͱ͕Θ͔Δɻ ͱ͜ΖͰɼu(s) ͕ s ʹؔ͢Δತؔ਺ͱԾఆ͢Δͱ͖ɼಉ༷ͷੑ࣭͕ಘΒΕΔɻ͍·ɼঢ় ଶ͕ s ͷͱ͖ɼܾఆ x ʹରͯ֬͠཰ม਺ T (s + x) ʹରͯ͠ɼE[T (s + 0)] ≥ s Ͱ͋Ε͹ɼ ΠΣϯηϯ (Jensen) ͷෆ౳ࣜΑΓ E[u(T (s))] ≥ u(s) ͱͳΔͷͰɼ. u1 (s) ≥ −c(0) + E[u(T (s + 0))] = E[u(T (s))] ≥ u(s) = u0 (s) ΑΓɼu1 (s) ≥ u0 (s) ͱͳΔ͜ͱ͕Θ͔Δɻ͕ͨͬͯ͠ɼun (s) ͸ n ʹؔ͢Δඇ‫ݮ‬গؔ਺ͱ ͳΔɻ͜ͷ৔߹͸ɼ௥Ճͷࢧग़Λ͠ͳ͘ͱ΋ɼ‫ظ‬଴ޮ༻͸‫ࡏݱ‬ͷॆ଍౓΍ຬ଍౓ʹΑΔޮ ༻ΑΓେ͖͘ͳΔ৔߹Ͱ͋Δɻ͜ͷ͜ͱ͸ɼެతͳαʔϏε͸ঢ়ଶ͕ྑ͘ͳΔ܏޲ʹ͋ͬ ͯ΋ɼ͋Δ͍͸ѱ͘ͳΔ܏޲Λ࣋ͭʹͯ͠΋ɼ͍ͣΕͷ৔߹ʹ΋αʔϏε͸ଓ͚ͯߦ͔ͳ ͯ͘͸ͳΒͣɼ͜Ε͕௨ৗͷ࠷దఀࢭ໰୊ͳͲͱҟͳ͍ͬͯΔ఺Ͱ͋Δɻ. .

(15)

(16) . 4. ෦෼‫؍‬ଌՄೳͳϚϧίϑաఔʹ͓͚Δஞܾ࣍ఆ໰୊. ͜Ε·Ͱ͸ɼ࠷ద੓ࡦͷ࣋ͭ୯ௐੑΛϚϧίϑܾఆաఔͷ৔߹ʹ͍ͭͯߟ͖͑ͯͨɻෆ ‫׬‬උ৘ใͷϚϧίϑաఔͷͱ͖ɼ͜ͷஞ࣍ࢧग़ϞσϧͰ࠷ద੓ࡦͷ࣋ͭ୯ௐੑΛࣔ͢͜ͱ ͸ࠔ೉Ͱ͋Δ͕ɼ࠷ద੓ࡦʹ͕ͨͬͨ͠ͱ͖ͷ૯‫ظ‬଴རಘͷ࣋ͭੑ࣭ʹ͍ͭͯߟ͑͜ͱ͕ Ͱ͖Δɻ͜͜Ͱ͸ɼNakai[7] ʹ͕ͨͬͯ͠؆୯ʹ݁ՌΛ·ͱΊΔɻ·ͨɼϕΠζͷఆཧʹ ֶ͕ͨͬͨ͠शϓϩηεΛߟ͑Δ͜ͱ͔Βɼ໬౓ൺॱংʹ‫͍ͯͮج‬ղੳ͢Δɻ. 4.1. ෦෼‫؍‬ଌՄೳͳϚϧίϑաఔͱ৘ใ. ͭ͗ʹɼঢ়ଶΛ௚઀‫؍‬ଌͰ͖ͳ͍ɼ෦෼‫؍‬ଌՄೳͳϚϧίϑ࿈࠯ʹ͓͚Δଟஈܾఆ໰୊ Λߟ͑Δɻ‫؍‬ଌͰ͖ͳ͍ঢ়ଶʹؔ͢Δ৘ใ͸ɼঢ়ଶۭؒ (−∞, ∞) ্ͷ֬཰෼෍ μ ͱͯ͠ ද͠ɼ৘ใશମͷू߹Λ S ͱ͢ΔɻS ʹ‫·ؚ‬ΕΔ৘ใͷ͍͋ͩʹɼ൒ॱংΛ μ ≥LRD ν ʹ Αͬͯఆٛ͢Δɻ͍ͬΆ͏ɼt > s ͷͱ͖ T (t) ≥LRD T (s) ͱԾఆ͢Δɻ͜ͷͱ͖ɼิ୊ 9 ͕ಘΒΕΔɻ ิ୊ 9 μ ≥LRD ν ͳΒ͹ (μ, ν ∈ S)ɼx ͷඇ‫ݮ‬গͳඇෛؔ਺ h(x) ʹରͯ͠ɼEμ [h(X)] ≥. Eν [h(X)] ͱͳΔɻ ঢ়ଶ s ରͯ͠ɼ͜ͷঢ়ଶʹґଘ͢Δ֬཰ม਺ Ys Λ৘ใϓϩηεͱ͢Δɻ͢ͳΘͪɼͦΕ ͧΕͷঢ়ଶʹؔ͢Δ৘ใΛ֬཰ม਺ Ys Λ௨ͯ͠ಘΔ͜ͱ͕Ͱ͖Δ‫؍‬ଌաఔͱ͢Δɻֶशϓ ϩηε͸ϕΠζֶशʹ͕ͨͬͯ͠ղੳ͢Δ͜ͱ͔ΒɼԾఆ 3 Λઃ͚Δɻ͜ͷԾఆ͸ɼNakai. [6] ʹ͕ͨͬͯ͠ҰൠԽͰ͖ɼଟஈܾఆ໰୊΁Ԡ༻Ͱ͖Δɻ Ծఆ 3 s ≤ t ͳΒ͹ɼYt ≥LRD Ys Ͱ͋Δ (s, t ∈ (−∞, ∞))ɻ Ծఆ 3 ʹ͓͍ͯɼYs ≥LRD Yt ͱ͔ͨ͠Βɼ֬཰ม਺ Ys ͸ s ͷ஋͕খ͘͞ͳΔʹ͕ͨͬ͠ ͯখ͞ͳ஋ΛͱΓɼs ͕େ͖͘ͳΔʹ͕ͨͬͯ͠ྑ͘ͳΔɻਪҠ๏ଇʹؔ͢ΔԾఆ͔Βɼঢ় ଶΛද͢ s ͕େ͖͘ͳΕ͹ɼΑΓྑ͍ঢ়ଶʹਪҠ͢Δ֬཰͸େ͖͘ͳΔɻ ֬཰աఔͷ‫؍‬ଌͰ͖ͳ͍ঢ়ଶʹؔͯ͠ɼ֬཰ม਺ {Ys }s∈(−∞,∞) Λ‫؍‬ଌ͢Δ͜ͱʹΑͬ ͯɼϕΠζͷఆཧΛ༻ֶ͍ͯशΛߦ͏ɻͦͷ‫ޙ‬ɼঢ়ଶ͸ਪҠ͠৽͍͠ঢ়ଶʹͳΔͱߟ͑Δɻ ΋ͪΖΜɼ͜ͷॱংΛม͑ͯ΋ಉ͡Α͏ʹղੳͰ͖Δɻy Λ‫؍‬ଌͨ͠ͱ͖ɼࣄ‫ޙ‬৘ใΛ μy ͱ͠ɼͦͷ‫Ͱޙ‬ਪҠ๏ଇ P ʹ͕ͨͬͯ͠ঢ়ଶ͕ਪҠ͠ɼͭ͗ͷ৽͍͠ঢ়ଶʹؔ͢Δ৘ใΛ. μy ͱ͢Δɻ͜͜Ͱɼࣄલ৘ใ͕ μ ͷͱ͖ɼਪҠ‫ޙ‬ͷࣄ‫ޙ‬৘ใΛ μ ͱ͢Δɻ .

(17) . ͜ͷͱ͖ɼू߹஋ؔ਺ h(x, s) ʹରͯ͠ɼఆٛ 6 ʹΑͬͯ୯ௐੑΛఆٛ͢Δɻ ఆٛ 6 ೚ҙͷ s ∈ ͱ x ∈ ʹؔ͢Δඇෛͷू߹஋ؔ਺ h(x) = (h(x, s))s∈(−∞,∞) ʹ ରͯ͠ɼ೚ҙͷ t ͱ s (s ≤ t ͔ͭ s, t ∈ (−∞, ∞)) ʹ͍ͭͯɼx < y ͳΒ͹ h(y) ≥LRD. h(x) (h(x) ≥LRD h(y)) ͱ͢Δɻ͢ͳΘͪ h(x, t) h(y, s) ≤ h(x, s) h(y, t) (h(x, t) h(y, s) ≥ h(x, s) h(y, t)) Ͱ͋Δɻ͜ͷͱ͖ɼؔ਺ h(x) Λ x ʹؔ͢Δ૿Ճؔ਺ ( ‫ݮ‬গؔ਺ ) ͱ͍͏ɻ ࣄલ৘ใ μ ͱࣄ‫ޙ‬৘ใ μy ͷ͍͋ͩʹ͸ɼͭ͗ͷ‫ج‬ຊతͳੑ࣭͕੒Γཱͭ (Nakai [6] ͳͲ)ɻ ิ୊ 10 μ ≥LRD ν ͳΒ͹ɼ೚ҙͷ y ʹରͯ͠ɼμy ≥LRD ν y ͓Αͼ μy ≥LRD ν y Ͱ͋Δɻ ೚ҙͷ μ ʹରͯ͠ɼμy ͱ μy ͸ y ʹؔ͢Δ૿Ճؔ਺Ͱ͋Δɻ ิ୊ 10 ͔Βɼࣄલ৘ใ μ ʹ͓͚Δॱংؔ܎͸ɼμy ͱࣄ‫ޙ‬৘ใ μy ʹରͯ͠อͨΕΔɻ͞ Βʹɼಉ͡ࣄલ৘ใ μ Ͱ͋Ε͹ɼ‫؍‬ଌͨ͠஋ y ͕େ͖͘ͳΕ͹ɼࣄ‫ޙ‬৘ใ μy ΋·ͨΑ͘ ͳΔɻ ෆ‫׬‬උ৘ใͷ࠷దܾఆ໰୊Λߟ͑ΔͨΊʹɼ͍͔ͭ͘ͷ४උΛ͢Δɻ͜͜Ͱɼ. μx (t) =. . ∞ 0. μ(s)ps+x (t)ds.. (7). ͱ͓͘ɻ͜Ε͸ɼࣄલ৘ใ͕ μ ͷͱ͖ɼܾఆ x Λͱͬͨ͋ͱͷɼঢ়ଶ্ۭؒͷࣄ‫ޙ‬෼෍Ͱ ͋Δɻ·ͨɼμ = μ0 Ͱ͋Δɻ ঢ়ଶશମͷू߹ S ʹ‫·ؚ‬ΕΔ֬཰෼෍ μ ͕. s < s , t < t ͱ s − s = t − t = c < 0 Λຬͨ͢೚ҙͷ s < s , t ≤ t ʹରͯ͠ɼ μ(t) μ(s) ≥ μ(s ) μ(t ) ͱͳΔͱ͖ɼ͜ͷ μ ͸ੑ࣭ (G) Λຬͨ͢ͱ͍͏͜ͱʹ͢Δɻ ྫ 3 ঢ়ଶ্ۭؒͷਖ਼‫ن‬෼෍ μ(s) =. (s−a)2 √ 1 e− 2σ 2 2πσ. ͸͜ͷੑ࣭Λຬ଍͢Δɻ. ิ୊ 11 μ ∈ S ͕ੑ࣭ (G) Λຬͨ͢ͱ͖ɼx > y ͳΒ͹ɼμx ≥LRD μy Ͱ͋Δɻ ิ୊ 12 ঢ়ଶશମͷू߹ S ʹ‫·ؚ‬ΕΔ֬཰෼෍ μ ͱ ν ͕ੑ࣭ (G) Λຬͨ͢ͱ͖ɼμ ≥LRD ν ͳΒ͹ɼ೚ҙͷ x(≥ 0) ʹରͯ͠ɼμx ≥LRD ν x Ͱ͋Δɻ Ծఆ 4 ೚ҙͷ s < s , t ≤ t ͓Αͼ u < v ͱͳΔ s, s , t, t , u, v ʹରͯ͠ pu (s)pv (t ) −. pu (t)pv (s ) ≥ pv (s)pu (t ) − pv (t)pu (s ) ͱ͢Δɻ .

(18)

(19) . ิ୊ 13 μ ∈ S ͕ੑ࣭ (G) Λຬͨ͢ͳΒ͹ɼμ ΋·ͨੑ࣭ (G) Λຬͨ͢ɻ ྫ 4 ਖ਼‫ن‬෼෍ʹΑΔਪҠ๏ଇ pv (s) =. (s−v)2 √ 1 e− 2σ 2 2πσ. ͸ɼԾఆ 4 ͷ৚݅Λຬ଍͢Δɻ. ֬཰ม਺ Ys ͷີ౓ؔ਺ fs (y) ͕ (s ∈ (−∞, ∞))ɼ೚ҙͷ s < s , t < t Ͱ s −s = t −t > 0 ͱͳΔ s, s , t, t ʹରͯ͠ɼੑ࣭. ft (y) fs (y) ≥ fs (y) ft (y) ͕੒ΓཱͭͱԾఆ͢Δɻ ิ୊ 14 μ ∈ S ͕ੑ࣭ (G) Λຬͨ͢ͳΒ͹ɼ೚ҙͷ y ʹରͯ͠ μy ΋·ͨੑ࣭ (G) Λຬ ͨ͢ɻ ͜͜Ͱɼঢ়ଶʹؔ͢Δ৘ใ͕ μ Ͱɼ௥Ճͯ͠ࢧग़ֹ͕ͨ͠ x ͷͱ͖ɼਪҠ‫ޙ‬ͷ৘ใ͸ μx Ͱ͋Δɻ ิ୊ 15 μ, ν ∈ S ͕ੑ࣭ (G) Λຬͨ͢ͱ͢Δɻμx ΋·ͨੑ࣭ (G) Λຬͨ͢ɻμ ≥LRD ν ͳΒ͹ɼ೚ҙͷ x(≥ 0) ʹରͯ͠ μx ≥LRD ν x Ͱ͋Δɻx > y ͳΒ͹ μx ≥LRD μy Ͱ͋Δɻ. 4.2. ஞ࣍ࢧग़Ϟσϧ–ෆ‫׬‬උ৘ใͷ৔߹. ෆ‫׬‬උ৘ใͷϚϧίϑաఔͷ৔߹ʹஞ࣍ࢧग़ϞσϧͰɼ࠷ద੓ࡦʹ͕ͨͬͨ͠ͱ͖ͷ૯ ‫ظ‬଴རಘʹ͍ͭͯߟ͑Δɻঢ়ଶʹؔ͢Δ৘ใ͸ɼ৘ใϓϩηεΛ௨ͯ͠ಘΒΕɼ4 અͷ෦ ෼‫؍‬ଌՄೳͳϚϧίϑաఔͰͷஞܾ࣍ఆ໰୊ͱͯ͠ఆࣜԽͰ͖Δɻ ‫؍‬ଌͰ͖ͳ͍ঢ়ଶʹؔ͢Δ৘ใ͸ɼঢ়ଶ্ۭؒͷ֬཰෼෍ͱͯ͠ද͞Εɼ৘ใϓϩηε ͔ΒಘΒΕͨ‫؍‬ଌ஋Λ΋ͱʹϕΠζͷఆཧʹֶ͕ͨͬͯ͠शΛߦ͏ɻ·ͨɼͦΕͧΕͷঢ় ଶ s (s ∈ (−∞, ∞)) ʹରͯ͠ɼ֬཰ม਺ Ys Λ‫؍‬ଌաఔͱ͢Δɻ‫؍‬ଌͰ͖ͳ͍ঢ়ଶʹؔ͢Δ ৘ใ͕ μ Ͱɼ‫ܭ‬ը‫ ͕ؒظ‬n ͷͱ͖ɼ࠷ద੓ࡦʹ͕ͨͬͯ͠ಘΒΕΔ૯‫ظ‬଴རಘΛ Vn (μ) ͱ ͢Ε͹ɼ࠷దੑͷ‫ݪ‬ཧΑΓɼͭ͗ͷ࠶‫ํؼ‬ఔ͕ࣜಘΒΕΔɻ. Vn (μ) = Eμ [Vn (μ|Y )] . Vn (μ|y) = max −c(x) + Vn−1 (μxy ). . x≥0. (8). ͜͜ͰɼV0 (μ) = Eμ [u(S)] ͱ͢Δɻࣄલ৘ใ͕ μ ͷͱ͖ɼ·ͣ࢝Ίʹ‫؍‬ଌ஋ y Λ‫؍‬ଌ͠ɼ ঢ়ଶʹؔ͢Δ৘ใΛϕΠζͷఆཧʹ͕ͨͬͯ͠ μy ͱվྑ͢Δɻܾఆ x ͷ͋ͱͰɼঢ়ଶ͕ s ͷͱ͖ɼਪҠ๏ଇ (ps+x (t))0≤s≤1 ʹ͕ͨͬͯ͠ 1 ‫ؒظ‬ਐΉɻ͜ͷΑ͏ʹɼ֬཰աఔͷঢ়ଶ. .

(20) . ͸৽͍͠ঢ়ଶͱͳΓɼ͜ͷ৽͍͠ঢ়ଶʹؔ͢Δ৘ใ͸ μxy ͱͳΔɻͦΕҎ߱ɼ࠷ద੓ࡦʹ͠ ͕ͨͬͯಘΒΕΔ࢒Γ‫ܭ‬ը‫Ͱؒظ‬ͷ૯‫ظ‬଴རಘ͸ Vn−1 (μxy ) Ͱ͋ΔɻΑͬͯɼn ʹؔ͢Δ‫ؼ‬ ೲ๏Λ༻͍Ε͹ɼ3 અͷԾఆͷԼͰͭ͗ͷੑ࣭͕ಘΒΕΔɻ ੑ࣭ 3 μ, ν ∈ S ͕ੑ࣭ (G) Λຬͨ͢ͱ͖ɼμ ≥LRD ν ͳΒ͹ɼVn (μ) ≥ Vn (ν) Ͱ͋Δɻ. μ ≥LRD ν Ͱ͋Ε͹ɼิ୊ 10 ΑΓ‫؍‬ଌ஋ y ʹରͯ͠ɼμy ≥LRD ν y Ͱ͋Γɼิ୊ 12 ͔ Βɼܾఆ x ʹରͯ͠ɼμx ≥LRD ν x Ͱ͋Δɻ͜ΕΒͷࣄ‫ޙ‬৘ใʹؔ͢Δ୯ௐੑ͔Βɼ೚ҙ ͷܾఆ x ͱ‫؍‬ଌ஋ y ʹରͯ͠ɼμ ≥LRD ν ͳΒ͹ɼμxy ≥LRD ν xy Ͱ͋Γɼ͜ͷ͜ͱ͔Βੑ ࣭ 3 ͕ n ʹؔ͢Δ‫ؼ‬ೲ๏ʹΑͬͯࣔ͞ΕΔɻ ͜ͷΑ͏ʹɼෆ‫׬‬උ৘ใͷϚϧίϑաఔͷ৔߹ʹஞ࣍ࢧग़ϞσϧͰɼ࠷ద੓ࡦʹ͕ͨͬ͠ ͨͱ͖ͷ૯‫ظ‬଴རಘʹؔ͢Δ୯ௐੑΛ‫ٻ‬ΊΔ͜ͱ͕ग़དྷΔɻ͔͠͠ɼ͜ͷ৔߹ͷ࠷ద੓ࡦ ʹؔ͢Δ୯ௐੑΛ‫ٻ‬ΊΔ͜ͱ͸՝୊ͱͳ͍ͬͯΔɻ. 5. ϝϯςφϯεΛߟྀͨ͠ଟஈܾఆ໰୊. ͍·ɼࣗಈं΍ిԽ੡඼ͳͲ͕࣌ؒͱͱ΋ʹྼԽ͍ͯ͘͠ͱ͖ɼͲͷΑ͏ʹϝϯςφϯ εΛ͢Δ͔Λܾఆ͢ΔϞσϧΛߟ͑Δɻ͜͜Ͱ͸ɼ੡඼ͷঢ়ଶΛ s ∈ (0, ∞) ʹΑͬͯද͠ɼ ঢ়ଶΛද͢஋ s ͕େ͖͘ͳΕ͹ঢ়ଶ͕ྑ͘ͳΔͱ͢Δɻ·ͨɼ͜ͷঢ়ଶ͸ܾఆʹ͔͔ΘΒ ͣɼϚϧίϑաఔʹ͕ͨͬͯ͠ঢ়ଶ͕ਪҠ͢Δɻ͜ͷͱ͖ɼ‫ܭ‬ը‫Ͱ಺ؒظ‬རಘΛ࠷େԽ͢ Δ࠷ద੓ࡦͱ࠷ద੓ࡦʹ͕ͨͬͨ͠ͱ͖ʹಘΒΕΔ࠷ద஋ʹ͍ͭͯߟ͑Δɻ·ͨɼ[7] ͳͲ ͱಉ༷ʹɼܾఆʹΑΓঢ়ଶΛมԽͰ͖Δͱ͢Δɻ ঢ়ଶۭؒΛ (0, ∞) ͱ͢Δɻঢ়ଶ͕ s ͷͱ͖ɼܾఆ α ΛऔΕ͹ɼঢ়ଶΛ αs ͱͰ͖Δ (α ≥ 1)ɻ ͜ͷͱ͖ͷܾఆʹରԠ͢Δඅ༻Λ C(α) ͱ͢Δɻu(s) Λ࠷‫ޙ‬ͷ‫ظ‬ͷঢ়ଶ͕ s ͷͱ͖ͷऴ୺ རಘͱ͢Δɻ·ͨɼঢ়ଶ͕֬཰తʹਪҠ͢Δ৔߹Λߟ͑Δͱ͖ʹ͸ɼঢ়ଶ͸ਪҠ๏ଇΛ. P = (ps (t))s,t∈(0,∞) ͱ͢ΔϚϧίϑաఔʹ͕ͨͬͯ͠ਪҠ͢Δͱ͢Δɻ s ∈ (0, ∞) Ͱఆٛ͞Εͨؔ਺ u(s) ͕ɼ೚ҙͷ s < t ͱ 0 < λ < 1 ͱͳΔ λ ʹରͯ͠ u(sλ t1−λ ) ≥ (≤)λu(s) + (1 − λ)u(t). (9). ͱͳΔͱ͖ɼ͜ͷؔ਺ u(s) Λ؆୯ͷͨΊʹ P Ԝؔ਺ (P ತؔ਺) ͱ‫͏ݴ‬ɻ ิ୊ 16 u(s) Λ P Ԝؔ਺ (P ತ ؔ਺) ͱ͢Δɻs < t, s < t ͱͳΔ s, t, s , t ʹରͯ͠ɼ. u(t ) − u(s ) ≤ (≥)u(t) − u(s).. (10) .

(21)

(22) . Ͱ͋Δɻ ূ໌:. s < t < s < t (s, t, s , t ≥ 0) ͱ͢Δɻ೚ҙͷ 0 < λ < 1 ʹରͯ͠ s ≤ s¯ =. s) + (1 − λ)u(¯ s) ͔ͩΒɼ sλ t1−λ ≤ t ͱ͓͚͹ɼλu(s) + (1 − λ)u(t) ≤ u(sλ t1−λ ) = λu(¯ (u(¯ s) − u(s))/(1 − λ) ≥ (u(t) − u(¯ s))/λ ͱͳΔɻlog t − log s > 0 ͳͷͰɼs¯/s = (t/s)1−λ ͓Αͼ t/¯ s = (t/s)λ ΑΓɼ(u(¯ s) − u(s))/(log s¯ − log s) ≥ (u(t) − u(¯ s))/(log t − log s¯) ͱͳ Δɻ͜ͷෆ౳͔ࣜΒ s < t < s < t ͱͳΔ೚ҙͷ s, t, s , t ʹରͯ͠ɼ. u(s ) − u(t) u(t ) − u(s ) u(t) − u(s) ≥ ≥ log t − log s log s − log t log t − log s ͱͳΔɻ. s < t < s < t ͷͱ͖ɼs/t = s /t ͔ͩΒɼlog t − log s = log t − log s > 0 ͳͷͰɼෆ ౳ࣜ (u(t) − u(s))/(log t − log s) ≥ (u(t ) − u(s ))/(log t − log s ) ΑΓ (10) ͕ࣜಋ͔ΕΔɻ ͍ͬΆ͏ɼs < s < t < t ͷͱ͖ (s, t, s , t ≥ 0)ɼs/s = t/t ͔ͩΒɼಉ༷ʹͯ͠. u(t ) − u(t) ≤ u(s ) − u(s) ͢ͳΘͪ (10) ͕ࣜಋ͔ΕΔɻ2 v(s) = maxα≥1 {−C(α) + u(αs)} ͱ͢Δͱ͖ɼu(s) ͕ s ͷ૿Ճؔ਺Ͱ͋Ε͹ɼv(s) ΋૿ Ճؔ਺Ͱ͋Δ͜ͱ͸໌Β͔Ͱ͋Δɻ ิ୊ 17 v(s) = maxα≥1 {−C(α) + u(αs)} ͱ͢ΔɻC(α) ͕ P Ԝؔ਺ͷͱ͖ɼu(s) ͕ P ತ ؔ਺ͳΒ͹ɼv(s) ΋ P ತؔ਺Ͱ͋Δɻͨͩ͠ɼC(α) ͸૿Ճؔ਺ͱ͢Δɻ ূ໌: ೚ҙͷ λ (0 < λ < 1) ͱ s < t ʹରͯ͠ u(sλ t1−λ ) ≤ λu(s) + (1 − λ)u(t) ͔ͩΒ C(α) ʹؔ͢ΔԾఆΛ༻͍ͯ. v(sλ t1−λ ) = max{−C(αλ α1−λ ) + u(αλ α1−λ sλ t1−λ )} α≥1. = max{−C(αλ α1−λ ) + u((αs)λ (αt)1−λ )} α≥1. ≤ max{−λC(α) − (1 − λ)C(α) + λu(αs) + (1 − λ)u(αt)} α≥1. ≤ λ max{−C(α) + u(αs)} + (1 − λ) max{−C(α) + u(αt)} α≥1. α≥1. = λv(s) + (1 − λ)v(t) ͱͳΔɻ͕ͨͬͯ͠ɼ೚ҙͷ 0 < λ < 1 ͱ s < t ʹରͯ͠ɼv(sλ t1−λ ) ≤ λv(s) + (1 − λ)v(t) ͱͳΔɻ2. .

(23) . 5.1. ϝϯςφϯεΛߟྀͨ͠ଟஈܾఆ໰୊. ঢ়ଶۭؒΛ (0, ∞) ͱ͠ɼঢ়ଶ͕ s ͷͱ͖ɼܾఆ α ΛऔΕ͹ɼ৽͍͠ঢ়ଶΛ αs ͱͰ͖Δ. (α ≥ 1)ɻ͜ͷͱ͖ͷඅ༻Λ C(α) ͱ͠ɼu(s) Λ࠷‫ޙ‬ͷ‫ظ‬ͷঢ়ଶ͕ s ͷͱ͖ͷऴ୺རಘͱ͢ Δɻ·ͨɼC(0) = 0 Ͱ͋ΓɼC(α) ͸૿Ճؔ਺ͱ͢Δɻ͜ͷͱ͖‫ܭ‬ը‫಺ؒظ‬ͷ૯རಘΛ࠷ େԽ͢Δ໰୊Λߟ͑Δɻ࢝Ίʹɼঢ়ଶ͕֬཰తʹਪҠ͠ͳ͍ͱ͢Δɻ ঢ়ଶ͕ s ͷͱ͖ɼn ‫ʹؒظ‬ΘܾͨͬͯఆΛߦ͍૯རಘΛ࠷େʹ͢Δ໰୊Ͱɼ࠷ద੓ࡦʹ ͕ͨͬͨ͠ͱ͖ʹಘΒΕΔ૯རಘΛ wn (s) ͱ͢Ε͹ɼ࠷దੑͷ‫ݪ‬ཧΑΓͭ͗ͷ࠷దํఔࣜ ͕ಘΒΕΔɻ. wn (s) = max{−C(α) + wn−1 (αs)}, α≥1. (11). ͜͜Ͱ w1 (s) = maxα≥1 {−C(α) + u(αs)} ͱ͢Δɻͨͩ͠ɼu(s) ͸ s ͷ૿Ճؔ਺ͱ͠ɼC(α) ͸ α ͷ૿Ճؔ਺ͱ͢Δɻ·ͨɼC(α) ͸ P Ԝؔ਺ͱ͢Δɻ ঢ়ଶ͕ s ͷͱ͖ɼα Λબ୒͠ (α ≥ 1)ɼඅ༻ C(α) Λࢧ෷ͬͯঢ়ଶ s Λ αs Ͱ͖Δɻ͜ͷ ͱ͖ɼͭ͗ͷੑ࣭͕੒Γཱͭɻ ิ୊ 18 wn (s) ͸ɼs ʹؔ͢Δ૿Ճؔ਺Ͱ͋Δɻ ิ୊ 19 u(s) ͕ P ತؔ਺ͳΒ͹ɼwn (s) ͸ P ತؔ਺Ͱ͋Δɻ ‫ܭ‬ը‫ ͕ؒظ‬n Ͱɼঢ়ଶ͕ s ͷͱ͖ͷ࠷ద੓ࡦΛ αn (s) ͱ͢Δɻ ิ୊ 20 u(s) ͕ P ತؔ਺ͳΒ͹ɼαn (s) ͸ s ʹؔͯ͠૿Ճ͢Δɻ ূ໌: s < t ͷͱ͖ɼ. wn (s) = max{−C(α) + wn−1 (αs)} = −C(α∗ ) + wn−1 (α∗ s) α≥1. ͱ͓͘ (1 ≤ α∗ )ɻα∗ ͕ s ʹର͢Δ࠷ద੓ࡦ͔ͩΒɼ೚ҙͷ α ≥ 1 ʹରͯ͠ −C(α∗ ) +. wn−1 (α∗ s) ≥ −C(α) + wn−1 (αs) Ͱ͋Δɻ೚ҙͷ α < α∗ ʹର͠ɼ −C(α∗ ) + wn−1 (α∗ t) ≥ −C(α) + wn−1 (αt). (12). Ͱ͋Ε͹ɼαn (t) ≥ α∗ = αn (s) ͱͳΔ͜ͱ͕൑Δɻ ೚ҙͷ α (≥ 1) ʹରͯ͠ wn (s) = −C(α∗ ) + wn−1 (α∗ s) ≥ −C(α) + wn−1 (αs) ͔ͩΒɼ. −C(α∗ ) + C(α) ≥ wn−1 (αs) − wn−1 (α∗ s) Ͱ͋Δɻα∗ > α, s < t ͔ͭ α∗ s/αs = α∗ t/αt ͳ .

(24)

(25) . ͷͰɼิ୊ 16 ΑΓ wn−1 (αt) − wn−1 (α∗ t) ≤ wn−1 (αs) − wn−1 (α∗ s) ͱͳΔɻ͜ΕΒͷෆ ౳͔ࣜΒɼ೚ҙͷ α < α∗ ʹରͯ͠ −C(α∗ ) + C(α) ≥ wn−1 (αt) − wn−1 (α∗ t) ͱͳΔ͜ͱ ͔Βɼ(12) ͕ࣜಋ͔ΕΔɻ2 ิ୊ 21 u(s) ͕ P ತؔ਺ͳΒ͹ɼαn (s) ͸ n ʹؔͯ͠૿Ճ͢Δɻ ূ໌: s < t ͷͱ͖ɼn > 1 ʹରͯ͠ α∗ = αn (s) ͱ͓͚͹ɼwn (s) = −C(α∗ ) + wn−1 (α∗ s) ͓Αͼ wn (t) ≥ −C(α∗ ) + wn−1 (α∗ t) ΑΓ wn (t) − wn (s) ≥ wn−1 (α∗ t) − wn−1 (α∗ s) ͱͳΔɻ. wn−1 (s) ͕ s ͷ P Ԝؔ਺Ͱ͋Γ. α∗ t α∗ s. =. t s. ͔ͩΒ (s < t, α∗ > 1)ɼ(10) ͔ࣜΒ wn−1 (α∗ t) −. wn−1 (α∗ s) ≥ wn−1 (t) − wn−1 (s) ͱͳΔɻ͜ΕΒͷෆ౳ࣜΑΓ wn (t) − wn (s) ≥ wn−1 (t) − wn−1 (s) ͕ಋ͚Δɻ. αn (s) = α∗ ͷͱ͖ɼα < α∗ ͱ͢Ε͹ɼ−C(α) + wn−1 (αs) ≤ −C(α∗ ) + wn−1 (α∗ s) ͱ ͳΔɻs < t ͷͱ͖ɼwn (t) − wn (s) ≥ wn−1 (t) − wn−1 (s) ͔ͩΒɼwn (αs) − wn (α∗ s) ≤. wn−1 (αs) − wn−1 (α∗ s) ͱͳΔɻ͕ͨͬͯ͠ɼ−C(α) + wn (αs) ≤ −C(α∗ ) + wn (α∗ s) ͱͳ Δɻ͜ͷ͜ͱ͔Βɼα∗ ≤ αn+1 (s) ͱͳΓɼ೚ҙͷ n ≥ 1 ͱ s > 0 ʹରͯ͠ αn (s) ≤ αn+1 (s) ͱͳΔɻ2. 5.2. Ϛϧίϑܾఆաఔͱͯ͠ͷଟஈܾఆ໰୊. ঢ়ଶ͕Ϛϧίϑաఔʹ͕ͨͬͯ͠ਪҠ͠ɼਪҠ๏ଇΛ P = (ps (t))s,t∈(0,∞) ͱ͢Δɻ֤‫ظ‬ ͝ͱͷܾఆΛ α ≥ 1 ͱ͢Δɻ͜ͷͱ͖ɼ‫ܭ‬ը‫ ͕ؒظ‬n Ͱɼঢ়ଶ͕ s ͷͱ͖ɼ࠷େԽ໰୊ʹ ͓͍ͯɼ࠷దʹৼΔ෣ͬͯಘΒΕΔ૯‫ظ‬଴རಘΛ vn (s) ͱ͢Ε͹ɼঢ়ଶ͕Ϛϧίϑաఔʹ ͕ͨͬͯ͠ਪҠ͢Δ͔Βɼ࠷దํఔࣜ͸ͭ͗ͷΑ͏ʹͳΔɻ͜͜ͰɼT (s) Λঢ়ଶ͕ s ͷͱ ͖ͭ͗ͷঢ়ଶΛද֬͢཰ม਺ͱ͢Ε͹ɼE[vn−1 (T (s))] =. ∞ 0. ps (t)vn−1 (t)dt Ͱ͋Δɻ. vn (s) = max{−C(α) + E[vn−1 (T (αs))]}, α≥1. (13). ͜͜Ͱɼ. v1 (s) = max{−C(α) + E[u(T (αs))]} α≥1. ͱ͢Δɻͨͩ͠ɼu(s) ͸ s ͷ૿Ճؔ਺ͱ͠ɼC(α) ͸ α ͷ૿Ճؔ਺ͱ͢Δɻ·ͨɼu(s) ͕. P ತؔ਺ ͱͳΔ໰୊Λߟ͑Δͱ͖͸ɼC(α) ͸ P Ԝؔ਺ͱ͢Δɻ .

(26) . T (s) Λঢ়ଶ͕ s ͷͱ͖ͭ͗ͷঢ়ଶΛද֬͢཰ม਺ͱ͢ΔɻਪҠ๏ଇ͕ (ps (t))s∈(0,∞) ͩ ͔Βɼ֬཰ม਺ྻ {T (s)|s ∈ (0, ∞)} ʹରͯ͠ɼͭ͗ͷԾఆΛઃ͚Δɻ Ծఆ 5 t ʹؔ͢Δ૿Ճ P ತؔ਺Λ u(t) ͱ͢Ε͹ɼE[u(T (s))] ͸ s ʹؔ͢Δ૿Ճ P ತؔ਺ ͱͳ͍ͬͯΔɻ͢ͳΘͪɼ֬཰ม਺ྻ {T (s)|s ∈ (0, ∞)} ͸ɼSIPCX Ͱ͋Δɻ ิ୊ 22 vn (s) ͸ɼs ʹؔ͢Δ૿Ճؔ਺Ͱ͋Δɻ ূ໌: n ʹؔ͢Δ‫ؼ‬ೲ๏Λ༻͍Δɻv0 (s) = u(s) ͔ͩΒɼv0 (s) ͸૿Ճؔ਺Ͱ P ತؔ਺Ͱ͋ Δɻvn−1 (s) ͕૿Ճؔ਺Ͱ P ತؔ਺ͱԾఆ͢ΔͱɼԾఆ 5 ΑΓ E[vn−1 (T (αs))] ΋·ͨ s ʹؔ ͢Δ૿Ճؔ਺Ͱ͋ΔɻE[vn−1 (T (αs))] ͕ s ͷ૿Ճؔ਺ͳͷͰɼvn (s) = maxα≥1 {−C(α) +. E[vn−1 (T (αs))]} ΋ s ͷ૿Ճؔ਺Ͱ͋Δɻ2 ͞Βʹɼิ୊ 17 ΑΓɼvn (s) ͕ P ತؔ਺ͱͳΔɻ ิ୊ 23 Ծఆ 5 ͷ΋ͱͰɼu(s) ͕ P ತؔ਺ͳΒ͹ɼvn (s) ͸ P ತؔ਺Ͱ͋Δɻ ‫ܭ‬ը‫ ͕ؒظ‬n Ͱ͋Γɼঢ়ଶ͕ s ͷͱ͖ͷɼ࠷దͳܾఆΛ αn∗ (s) ͱ͢Δɻ ੑ࣭ 4 Ծఆ 5 ͷ΋ͱͰɼu(s) ͕ P ತؔ਺ͳΒ͹ɼαn∗ (s) ͸ s ʹؔͯ͠૿Ճ͢Δɻ ূ໌: n ʹؔ͢Δ‫ؼ‬ೲ๏Λ༻͍Δɻn = 1 ͷ৔߹͸ɼҰൠͷͱ͖ͱಉ͡Α͏ʹಋ͚Δɻs < t ͱ͢Δɻn(> 1) ͷͱ͖ɼαn∗ (s) = α∗ ͱ͓͚͹ɼ(13) ࣜΑΓ. vn (s) = max{−C(α) + E[vn−1 (T (αs))]} = −C(α∗ ) + E[vn−1 (T (α∗ s))], α≥1. (14). ͱͳΔɻ1 ≤ α ≤ α∗ ͱͳΔ೚ҙͷ α ʹରͯ͠ɼෆ౳ࣜ. −C(α) + E[vn−1 (T (αt))] ≤ −C(α∗ ) + E[vn−1 (T (α∗ t))] ͕੒Γཱͯ͹ɼαn∗ (s) = α∗ ≤ αn∗ (t) ͱͳΔ͜ͱ͕ࣔ͞ΕΔɻ. (14) ࣜΑΓɼ೚ҙͷ α ≥ 1 ʹରͯ͠ −C(α) + E[vn−1 (T (αs))] ≤ −C(α∗ ) + E[vn−1 (T (α∗ s))]. (15). ͔ͩΒɼ. −C(α) + C(α∗ ) ≤ E[vn−1 (T (α∗ s))] − E[vn−1 (T (αs))]. . (16).

(27)

(28) . ͱͳΔɻ͍ͬΆ͏ɼԾఆ 5 ΑΓɼE[vn−1 (T (αs))] ͸ s ʹؔ͢Δ૿Ճؔ਺Ͱ P ತؔ਺Ͱ͋Δɻ. α∗ t/α∗ s = αt/αs Ͱ͋Γ 1 ≤ α ≤ α∗ ͔ͩΒɼ(10) ࣜͷΑ͏ʹ E[vn−1 (T (αt))] − E[vn−1 (T (αs)) ≤ E[vn−1 (T (α∗ t))] − E[vn−1 (T (α∗ s))] ͱͳΔɻ(16) ࣜͱ͜ͷෆ౳͔ࣜΒɼ(15) ͕ࣜಋ͔Εɼ͜ͷੑ࣭͕੒Γཱͭɻ2 Ծఆ 6 t ≥ s ͷͱ͖೚ҙͷ P ತؔ਺ u(s) ʹରͯ͠ɼE[u(T (t))] − E[u(T (s))] ≥ u(t) − u(s) Ͱ͋Δɻ. vn (s) ͕ s ͷ૿Ճؔ਺Ͱ P ತؔ਺͔ͩΒɼԾఆ 6 ΑΓɼs < t ʹରͯ͠ E[vn (T (t))] − E[vn (T (s))] ≥ vn (t) − vn (s) ͱͳΔɻ s < t ͷͱ͖ɼ೚ҙͷ n ≥ 1 ʹରͯ͠ E[vn (T (t))] − E[vn (T (s))] ͱ E[vn−1 (T (t))] − E[vn−1 (T (s))] ͷؔ܎Λߟ͑Δɻα∗ = αn∗ (s) ͱ͓͚͹ɼ vn (t) − vn (s) = −C(α∗ ) + E[vn−1 (T (α∗ t))] − max{−C(α) + E[vn−1 (T (αs))]} α≥1. ∗. ≥ E[vn−1 (T (α t))] − E[vn−1 (T (α∗ s))] ͱͳΔɻ͍ͬΆ͏ɼิ୊ 23 ΑΓɼԾఆ 5 ͷ΋ͱͰ E[vn−1 (T (s))] ͸ s ʹؔ͢Δ૿Ճؔ਺Ͱ. P ತؔ਺Ͱ͋Δɻα∗ t/α∗ s = t/s Ͱ͋Γ s < t, α∗ ≥ 1 ͔ͩΒɼ(10) ࣜΑΓ E[vn−1 (T (α∗ t))] − E[vn−1 (T (α∗ s))] ≥ E[vn−1 (T (t))] − E[vn−1 (T (s))] ͱͳΔɻ͜ΕΒͷෆ౳͔ࣜΒ. vn (t) − vn (s) ≥ E[vn−1 (T (t))] − E[vn−1 (T (s))] ͱͳΔɻ ิ୊ 23 ΑΓɼvn (s) ͕ P ತؔ਺ͳͷͰɼԾఆ 6 ΑΓ E[vn (T (t))] − E[vn (T (s))] ≥ vn (t) −. vn (s) ͱͳΔɻ͜ΕΒͷෆ౳͔ࣜΒɼ೚ҙͷ n ≥ 1 ʹରͯ͠ E[vn (T (t))] − E[vn (T (s))] ≥ E[vn−1 (T (t))] − E[vn−1 (T (s))] ͱͳΔɻ ੑ࣭ 5 Ծఆ 6 ͷ΋ͱͰɼu(s) ͕ P ತؔ਺ͳΒ͹ɼαn (s) ͸ n ʹؔͯ͠૿Ճ͢Δɻ. . (17).

(29) . ূ໌: n ʹؔ͢Δ‫ؼ‬ೲ๏Λ༻͍Δɻn = 1 ͷ৔߹͸ɼҰൠͷͱ͖ͱಉ͡Α͏ʹಋ͚Δɻn(> 1) ͱ͢Δɻs ≤ t ͷͱ͖ɼαn∗ (s) = α∗ ͱ͓͚͹ɼ೚ҙͷ 1 < α < α∗ ʹରͯ͠ɼ. −C(α) + E[vn−1 (T (αs))] ≤ −C(α∗ ) + E[vn−1 (T (α∗ s))] Ͱ͋Δɻ͍ͬΆ͏ɼ(17) ࣜΑΓ. E[vn−1 (T (α∗ s))] − E[vn−1 (T (αs))] ≤ E[vn (T (α∗ s))] − E[vn (T (αs))], ͔ͩΒɼ. −C(α) + E[vn (T (αs))] ≤ −C(α∗ ) + E[vn (T (α∗ s))]. (18). ∗ ͱͳΔɻ͜ͷ͜ͱ͔Βɼn ʹؔ͢Δ‫ؼ‬ೲ๏ΑΓɼα∗ ≤ αn+1 (s) ͱͳΔɻ͕ͨͬͯ͠ɼ೚ҙ ∗ ͷ n ≥ 1 ʹରͯ͠ɼαn∗ (s) ≤ αn+1 (s) ͳͷͰɼԾఆ 6 ͷ΋ͱͰ αn∗ (s) ͷ n ʹؔ͢Δ୯ௐੑ. ͕ࣔ͞ΕΔɻ2 ͜ͷଟஈܾఆ໰୊Ͱ͸ɼ࠷ద੓ࡦ͸Ϛϧίϑաఔͷঢ়ଶʹґଘ͢Δͱͱ΋ʹɼͦΕ·Ͱͷܾ ఆʹ΋ґଘ͢Δɻ͍ͬΆ͏ɼঢ়ଶ͸Ϛϧίϑաఔʹ͕ͨͬͯ͠ਪҠ͢ΔͷͰɼܾఆΛऔΔॱং ͕কདྷͷܾఆʹӨ‫͢ڹ‬Δɻྫ͑͹ɼα ͱ α Λ 2 ͭͷҟͳΔܾఆͱ͢Ε͹ (α, α ≥ 1, α = α )ɼ ঢ়ଶ͕ s ͷͱ͖ɼT (α T (αs)) ͸ɼ͸͡Ίʹܾఆ α ΛऔΓɼͦͷ͋ͱͰܾఆ α Λऔͬͨ͋ͱ ਪҠͨ͠ঢ়ଶΛද֬͢཰ม਺Ͱ͋Δɻ΋͠ɼT (s) ͕ྫ 2 ͷີ౓ؔ਺͕ fs (t) =. φlog s,σ2 (log t) t. ͷର਺ਖ਼‫ن‬෼෍ͱ͢Ε͹ɼ2 ͭͷ֬཰ม਺ T (α T (αs)) ͱ T (αT (α s)) ͸ɼ2 ͭͷܾఆ α ͱ α ʹରͯ͠ಉ͡෼෍Λද͕͢ɼҰൠͷ෼෍ͷ৔߹ʹ͸ඞͣ͠΋౳͘͠ͳΔͱ͸‫͍ͳ͑ݴ‬ɻ͞ Βʹɼ࠷ద੓ࡦ αn∗ (s) ͸ঢ়ଶ s ʹΑͬͯҟͳΓɼ͜ͷܾఆʹΑΔ‫ظ‬଴අ༻͸ E[C(αn∗ (T (s))] Ͱ͋Δɻ ऴ୺རಘ u(s) ͕ɼs ʹؔ͢Δ૿Ճؔ਺Ͱ P Ԝؔ਺ͱԾఆͯ͠΋ɼ͜͜Ͱ༻͍ͨ΋ͷͱಉ ༷ͷํ๏ʹΑͬͯ࠷ద੓ࡦͳͲͷ୯ௐੑΛࣔ͢͜ͱ͕Ͱ͖Δɻ͔͠͠ɼԾఆ 5 ͷ୅ΘΓʹɼ. {T (s)|s ∈ (0, ∞)} ͕ SIPCV ͱԾఆ͠ͳ͚Ε͹ͳΒͳ͍͕ɼSIPCV ͷੑ࣭Λ࣋ͭൺֱత؆ ୯ͳ֬཰ม਺ྻΛ‫ͱ͍ͩ͜͢ݟ‬͸ࠔ೉Ͱ͋Δɻ ิ୊ 23 ΑΓɼvn (s) ͸ s ʹؔ͢Δ૿Ճؔ਺Ͱ P ತؔ਺Ͱ͋ͬͨɻ࠷‫ʹޙ‬ɼؔ਺ vn (s) ͷ n ʹ ؔ͢Δੑ࣭Λߟ͑ͯΈΑ͏ɻv1 (s) = maxα≥1 {−C(α)+E[u(T (αs))]} ͳͷͰɼE[u(T (s))] ≤. u(s) ͳΒ͹ v1 (s) ≤ u(s) Ͱ͋Δɻn ʹؔ͢Δ‫ؼ‬ೲ๏Λ༻͍Ε͹ɼvn (s) ͕ n ʹؔͯ͠૿Ճ ͢Δ͜ͱ͸؆୯ʹ൑Δɻ͢ͳΘͪɼ(13) ͔ࣜΒ vn (s) ≤ vn−1 (s) ͱͳΔɻ. Nakai [8] Ͱ͸ɼܾఆʹΑͬͯঢ়ଶΛՃ๏తʹมԽͤ͞Δ͜ͱ͕Ͱ͖Δ৔߹ͷ෦෼‫؍‬ଌՄ ೳͳϚϧίϑܾఆաఔΛಉ͡Α͏ʹղੳ͠ɼ࠷ద੓ࡦͷ΋ͱͰͷ૯‫ظ‬଴རಘʹؔ͢Δ୯ௐ. .

(30)

(31) . ੑΛ‫ٻ‬Ί͍ͯΔɻ͔͠͠ɼ(9) ࣜͱಉ༷ͷੑ࣭͕ಘΒΕ͓ͯΒͣɼ࠷ద੓ࡦʹؔ͢Δ୯ௐੑ ʹ͍ͭͯ͸কདྷͷ՝୊Ͱ͋Δɻ ࣌ࠁ n Ͱঢ়ଶ͕ s ͷͱ͖ɼ࠷దܾఆ αn∗ (s) ͸ɼs ʹؔͯ͠૿Ճ͠ɼঢ়ଶ͕ྑ͘ͳΕ͹ͳ Δ΄ͲमཧΛ͢Δඞཁੑ͕૿͢ɻ͍ͬΆ͏ɼαn∗ (s) ͸ n ʹؔͯ͠૿Ճ͠ɼ࢒Γ‫ܭ‬ը‫͕ؒظ‬ ૿͑Ε͹૿͑Δ΄ͲमཧΛ͢Δඞཁੑ͕૿͢ɻ3 અͰ͸ɼঢ়ଶ͕֬཰తತੑΛ࣋ͭϚϧί ϑաఔʹ͓͚Δ࠷ద੓ࡦ αn∗ (s) ͷ୯ௐੑΛߟ͕͑ͨɼ͜ͷύϥϝʔλΛ࣋ͭ֬཰ม਺ྻ. {T (s)|s ∈ (0, ∞)} ʹରͯ͠ఆٛ͞Εͨ֬཰తತੑ͸ɼShaked and Shanthikumar [12] ΍ Kijima and Ohnishi [4] ͳͲʹ༻͍ΒΕ͍ͯΔɼತॱং (convex order) ͱ͸ҟͳΔ΋ͷͰ ͋Δɻ ஫ 1 ঢ়ଶۭؒΛ (0, ∞) ͱ͠ɼঢ়ଶΛද͢஋ s ͕େ͖͘ͳΕ͹ঢ়ଶ͕ѱ͘ͳΔͱ͢Δɻ͜ ͷͱ͖ͭ͗ͷΑ͏ͳ໰୊Λߟ͑Δɻঢ়ଶ͕ s ͷͱ͖ɼܾఆ α ΛऔΕ͹ɼ৽͍͠ঢ়ଶΛ αs ͱ Ͱ͖ (0 < α ≤ 1)ɼͦͷඅ༻Λ C(α) ͱ͠ɼu(s) Λ࠷‫ޙ‬ͷ‫ظ‬ͷঢ়ଶ͕ s ͷͱ͖ͷऴ୺རಘͱ ͢Δɻ‫ܭ‬ը‫ ͕ؒظ‬n Ͱঢ়ଶ͕ s ͷͱ͖ɼ࠷దʹৼΔ෣ͬͨͱ͖ͷ૯‫ظ‬଴རಘΛ vn (s) ͱ͢ Ε͹ɼঢ়ଶ͕Ϛϧίϑաఔʹ͕ͨͬͯ͠ਪҠ͢Δ͔Βɼ࠷దํఔࣜ͸ͭ͗ͷΑ͏ʹͳΔɻ. vn (s) =. max {−C(α) + E[vn−1 (T (αs))]},. 0<α≤1. (19). ͜͜Ͱɼ. v1 (s) =. max {−C(α) + E[u(T (αs))]}. 0<α≤1. ͱ͢Δɻͨͩ͠ɼu(s) ͸ s ͷ૿Ճؔ਺ͱ͠ɼC(α) ͸ α ͷ‫ݮ‬গؔ਺ͱ͢Δɻ͜ͷ৔߹ʹ΋ ಉ༷ͷੑ࣭Λ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻ·ͨɼ࠷େԽ໰୊ͱͯ͠΋ಉ༷Ͱ͋Δɻ( தҪ [9]) ஫ 2 4 અͰ͸ɼ3 અͷެ‫ڞ‬౤ࢿϞσϧʹ‫ͮ͘ج‬ஞܾ࣍ఆϞσϧΛෆ‫׬‬උ৘ใͷϚϧίϑա ఔͱͯ͠ϞσϧԽ͠ɼ࠷ద੓ࡦʹ͕ͨͬͨ͠ͱ͖ͷ૯‫ظ‬଴རಘͷ୯ௐੑΛٞ࿦ͨ͠ɻಉ͡ Α͏ʹɼ5 અͷϝϯςφϯεΛߟྀͨ͠ଟஈܾఆ໰୊ʹ͍ͭͯ΋ಉ༷ʹɼෆ‫׬‬උ৘ใͷϚ ϧίϑաఔͱͯ͠ϞσϧԽ͠ɼ࠷ద੓ࡦʹ͕ͨͬͨ͠ͱ͖ͷ૯‫ظ‬଴རಘͷ୯ௐੑΛ‫ٻ‬ΊΔ ͜ͱ͕ग़དྷΔ (Nakai[8])ɻ. ࢀߟจ‫ݙ‬ [1] Albright, S. C., Structural results for partially observable Markov decision processes, Oper. Res. 27, 1041–1053, 1979. [2] Grosfeld-Nir, A., A two-state partially observable Markov decision process with uniformly distributed observations, Oper. Res. 44, 458–463, 1996. .

(32) . [3] Itoh, H. and Nakamura, K., Partially observable Markov decision processes with imprecise parameters, Artificial Intelligence 171 , 453–490, 2007. [4] Kijima, M. and Ohnishi, M., Stochastic orders and their applications in financial optimization, Math. Methods of Oper. Res., 50, 351–372, 1999. [5] Monahan, G. E., Optimal selection with alternative information, Naval Res. Logist. Quart. 33, 293–307, 1986. [6] Nakai,T., A generalization of multivariate total positivity of order two with an application to Bayesian learning procedure, Journal of Information & Optimization Sciences 23, 163–176, 2002. [7] Nakai,T., A Sequential decision problem based on the rate depending on a Markov process, Recent Advances in Stochastic Operations Research 2 (Eds. T. Dohi, S. Osaki and K. Sawaki), World Scientific Publishing, 11–30, 2009. [8] Nakai,T., Sequential decision problem with maintenance on a partially observable Markov process, Scientiae Mathematicae Japonicae 72, 11–20, 2010. [9] தҪɹୡ, Partial Maintenance Λߟྀͨ͠ϚϧίϑաఔͰͷଟஈܾఆ໰୊ʹ͍ͭͯ, ‫౎ژ‬େֶ਺ཧղੳ‫ڀߨॴڀݚ‬࿥ʮෆ࣮֬ੑԼʹ͓͚Δҙࢥܾఆ໰୊ʯ 1734, 220–227, 2011. [10] Nakai,T., Monotonic properties for a sequential decision problem with maintenance on a Markov process, Scientiae Mathematicae Japonicae 74, 275–283, 2011. [11] Ohnishi, M., Kawai, H. and Mine, H., An optimal inspection and replacement policy under incomplete state information, European J. Oper. Res. 27, 117–128, 1986. [12] Shaked, M. and Shanthikumar, J. G., Stochastic orders and their applications (Probability and mathematical statistics : a series of monographs and textbooks), Academic Press, Boston, Massachusetts, 1994. [13] White, D. J., Structural properties for contracting state partially observable Markov decision processes, J. Math. Anal. Appl. 186, 486–503, 1994.. .

(33)