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Probabilidade de ruína com fluxo de caixa e investimento governados por processos de difisão

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Material de apoio - desenvolvido por terceiros - ao curso de Ciências Atuariais

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Probabilidade de ruína com fluxo de caixa e investimento governados por processos de difisão

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16. 16.  ß·Ü ·Ó4×W×Ü½ø R ùF×on ·ÒÓ4×EØÑ­Ü9ÑgÏÜ × ß×Ü9×áÓ4Ñ£Ö× L2 (P) ÚdÛÑ ·Ò { T S ϕn(t, ω)dWt(ω)} æçÑ­ÕÜ Ô¤ÏÜ Ô Ò×õÏáØ ·ÔEÖ×ub¹ÔÏØcÝw­£×Ü L2 (P) ãFäÔ ·Ò¤Ö×ÓÔ ßÝ×Ò¤ÒÑ­ÞÕ×mÔAØÑáÒÓ4ÕÏîïÑEÖÔ ·á}Ó4×Ð­Õ4Ô ß Ö×û¡ÓedE×fÒÑ­ÞÕ× Ô£×on ·ÒÓeáØ ·ÔgÖÔÒfæçÏáîVd×Ò ϕn ÛÑdÖ×ÜÒ×Õfì ·ÒÓ4Ñ­Ò×ÜìBªÒ×áÖÔ ß ø¥X ffR ùcãXí Ò×Ð­Ï ·ÕmÔlÛÕ4×Ò4×áÓÔÜ9Ñ­Ò ÖÑ ·ÒÓ4×Ñ­Õ×Ü ÔÒãÛÕ ·Ü9× ·ÕÑ æçÑ­Õá×Ø×¦Ô ßÐ­ÏÜ ÔÒPÛÕÑ­ÛÕ ·×ÖÔÖ×Ò¤ÖÔ ·á}Ó4×Ð­Õ4Ô ß Ö×Rû¡ÓedV×RÑVÒ×Ð­ÏáÖÑ Ü9Ñ­ÒÓ4Õ4ÔgõÏ×P× ßÔgåÏÜ Ü ÔÕÓ ·áÐ}Ô ßã XíÒ¹Ö×Ü9Ñ­áÒÓ4Õ4ÔîV×Ò¹Û{ÑdÖ×ÜÒ×Õmì ·ÒÓÔÒm×ÜBªÒ×áÖÔ ß ø¥X ffR ùcã wr Â QÄGºÙ(ºÜp µs»Ô| f e g ∈ Υ(S, T)  0 ≤ S U T. ßmx¡¥ ¾ÍÌ T S f dWt = U S f dWt + T U f dWt ||1~|£yz ω ¾»¾ÍÌ T S (cf + g) dWt = c · T S f dWt + T S g dWt ||1~|gz ω ¢Wlx{zRlx7¡|x ¾»¾»¾¥Ì E[ T S f dWt] = 0 ¾í½§Ì T S f dWt  FT ° x7dº¦³§ wr Â QÄGºÙ(ºÙ µs»Ô| f(t, ω) ∈ Υ(0, T) ∀ T. ßmx¡¥ Mt(ω) = t 0 f(s, ω) dWs £d¿9|cvwyxd¶}|³Qlxvªyxl34wy | Ft § GºÙ(ºÙ( 7@îÍï Â ¡yÛÅÄä¿oÁ4å í ·á}Ó4×Ð­Õ4Ô ß Ö× û¡Óed ÛÑdÖ×1Ò×ÕÖ×©Uá ·ÖÔÛÔÕ4ÔÏÜ ÔØ ßÔÒÒ×1Ö× ·áÓ4×Ð­Õ4ÔáÖÑ­Ò f Ü Ô ·Ñ­Õ£õÏ× Υ ã ÷Ô à ×Ü9Ñ­Ò ·ÒÒÑ¦×á7æçÕ4ÔõÏ×Ø×áÖÑ1ÔÒAØÑ­áÖ ·îVd×Ò©ø··ùf× ø···ùÖÔ¦Ù×©Uá ·îïÑ R ãaXdãaXÛÔÕ4Ô¦ÔÒAÒ×Ð­Ï ·á}Ó4×Òæ ¾»¾ÍÌ4ð Î§n ·ÒÓ4×gÏÜ9ÔævÔÜ ß·ÔØÕ×ÒØ×á}Ó4×©Ö× σ þyë ßÐ­×ÞÕ4ÔÒ At; t ≥ 0 ÓÔ ·Ò¹õÏ× ÄÌ Wt å¤ÏÜ3Ü ÔÕÓ ·áÐ}Ô ß ØÑ­Ü3Õ×ÒÛ× ·Ó4Ñ Ô At × ñ Ì ft å At þyÔÖÔÛ7ÓÔÖÑ ¾»¾»¾¥ÌYð P T S f(s, ω)2 ds ∞ = 1 !CwÆÇÀy¾»ÈÉGºÙ(ºÙmÞ xjl¡| £l ΛA(S, T) |³|zg{4 f(t, ω) ∈ R ~ |vw° |Wòw×1z}| Þ »ÑCxwç¥9¬§¯d§¯ò¡wyw×5ó¢ò¡wywyw×5ó7|wy9|}§ íá}Ó4×ÒÖ×1Ö×©Uá ·Õ4Ü¦Ñ­ÒÔæçé­ÕÜÏ ßÔVÖ× û¡ÓedÚFÛÕ×Ø ·Ò4ÔÜ9Ñ­Ò©Ö×©Uá ·Õ©ÏÜIÛÕÑdØ×ÒÒÑ Ö×1û¡Óedã÷Ô à ×Ü9Ñ­Ò ·ÒÒÑ¦ÖÔRÒ×Ð­Ï ·áÓ4×£Ü Ôlá× ·Õ4Ôxæ SYh
17. 17. !CwÆÇÀy¾»ÈÉGºÙ(º i µs»Ô| Wt da l¦wy x9±A4ËCxwv|xjV (Ω, F, P) § Ê {z ¨ Öë dxwvzwy x7wlx{|³gd {4 Xt  (Ω, F, P) z}|lc9| Xt = X0 + t 0 u(s, ω) ds + t 0 v(s, ω) dWs øÅfù lx{z v ∈ ΛA ¢¹zlc9|1~ P t 0 v(s, ω)2 ds ∞ ∀ t ≥ 0 = 1. Î |9²cY|dRwy g~ u  At °|}z}|c{¡|}z}|òlx{z At cl  @ò¡wyw×5ó×1 P t 0 |u(s, ω)| ds ∞ ∀ t ≥ 0 = 1. êd× Xt å£ÏÜ3ÛÕÑØ×ÒÒÑ Ö×£û¡ÓedÚ×á}ÓïÑ¦× ß×£ÛÑdÖ×£Ò×ÕA×ÒØÕ ·Ó4Ñ9Ö×¤æçÑ­ÕÜ ÔRÔÞÕ×ì ·ÔÖÔRØÑ­Ü9Ñ dXt = udt + vdWt. øÍhù öEÜoÖÑ­Ò Õ×ÒÏ ßÓÔÖÑ­Ò Ü Ô ·Ò ·Ü9ÛÑ­ÕÓÔáÓ4×Ò¦ÖÔ¨Ó4×Ñ­Õ ·ÔÖ× ·áÓ4×Ð­Õ4ÔîïÑ×ÒÓ4ÑdØëÒÓ ·ØÔ7Ú¹Ñ õÏÔ ß ÔÛÕ×cþ Ò×á}ÓÔÕ×Ü9Ñ­ÒÔÐÑ­Õ4Ô7ÚQå1ÔÕ×Ð­Õ4ÔØÑ­áÝ×Ø ·ÖÔØÑ­Ü9ÑVæçé­ÕÜRÏ ßÔÖ×9û¡ÓedãVûá7æçÑ­ÕÜ Ô ßÜ9×á}Ó4×ÚQ×ÒÓÔÕ×Ð­Õ4Ô åRÏÜ Ô ì­×ÕÒ4ïÑVÖÔÕ×Ð­Õ4ÔÖÔ1ØÔÖ× ·ÔÖÑØë ßØÏ ßÑÖ ·æç×Õ×áØ ·Ô ß Ø ßëÒÒ ·ØÑãúXÔÕ4ÔÏÜ Ô1ÛÕÑìÔ1æçÑ­ÕÜ Ô ß Ö×ÒÓ4×¤Ó4×Ñ­Õ×Ü Ôì}×yâÔBªÒ×áÖÔ ß ø¥X ffR ùcã wr Â QÄGºÙ(ºÙsêégsï Â ¡yÛÅÄ¡ä¿|oÁ4åyÀy¾»äy¾»QÀyg¾ÅÀ¿Ä¼ÛÍÌ µs»Ô| Xt d {4Dz ¨ Öë dxwvzwy x7wlx{|³sz}|zl dXt = udt + vdWt. µs»Ô| g(t, x) ∈ C2 ([0, ∞) × R) ò¡ws¢¶T¨z|¦4¨lxvwyx| x¨zw4xjwvº¦³ [0, ∞) × R) §²ßmx¡¥ Yt = g(t, Xt) £¡|9²c3d {4 z ¨ Öë¢W dYt = ∂g ∂t (t, Xt)dt + ∂g ∂x (t, Xt)dXt + 1 2 ∂2 g ∂x2 (t, Xt) · (dXt)2 , øñ­ù lx{z (dXt)2 = (dXt) · (dXt) 4|³d³|}z z|lz8Ý£4¡¶| dt · dt = dt · dWt = dWt · dt = 0, dWt · dWt = dt. úXÔÕ4Ô©Ô£æçé­ÕÜRÏ ßÔ¤Ö×û¡Óed©ádþÖ ·Ü9×áÒ ·Ñ­áÔ ß ì­×yâÔ©íá×n7ÑRûãí Ò×Ð­Ï ·Õ×áÏáØ ·ÔÜ9Ñ­Ò×ÛÕÑìÔÜ9Ñ­ÒÛÔÕÓ4× ÖÑ×on7×ÕØØ ·ÑfãR Ö×¥BªÒ×áÖÔ ß ø¥X ffR ùcÚÑõÏÔ ß Ò×Õ4ëÏ7Ó ·ß·à ÔÖÑgáÑØÔÛÓ4Ï ßÑñgÖ×ÒÓÔÖ ·ÒÒ×ÕÓÔîïÑã Sñ
18. 18. H Â Fôõg¾»ÈÉGºÙ(ºÙ' µs»Ô| Xt  Yt {4gz ¨ Öë1 R §ßmx¡¥ d(XtYt) = XtdYt + YtdXt + dXtdYt. Ó 4l¦l|}® êd×yâÔÜ Wt =   Xt Yt   , Ñ­ÏVÒ×yâÔ7Ú Wt å£ÏÜ3ÛÕÑdØ×ÒÒÑ Ö×£û¡Óed¦Þ ·Ö ·Ü9×áÒ ·Ñ­áÔ ß øí©f×yâÔ¦íá×on7Ñ9ûùcã êd×yâÔ Z(t, ω) = g(t, Wt) = xy ãfúX× ßÔæçé­ÕÜÏ ßÔ©Ö×£û¡Óed9ádþÖ ·Ü9×áÒ ·Ñ­áÔ ßÚ dáFöDXdÚÓ4×Ü9Ñ­ÒWõÏ× d(XtYt) = 0 + YtdXt + XtdYt + 1 2 · 1 · dYtdXt + 1 2 · dYt · dXt. ýÑ­Ð­Ñ d(XtYt) = XtdYt + YtdXt + dXtdYt. GºÙ(ºÙG H Â 5÷Qgä¿½Ä Â ¾ÅÄ¼ÈÉø¿Ä¼ä ÂYù ÁY¾»÷Ä !CwÆÇÀy¾»ÈÉGºÙ(º»r{ µs»Ô| Nt(.) : Ω → R d {414ºvw1lxvªyx­§ Þ »ÑCx{| [N]t ≡ lim ∆tk→0 tk≤t (Ntk+1 (ω) − Ntk (ω))2 (limite em probabilidade) lx{z 0 = t1 t2 ... tn = t  ∆tk = tk+1 − tk. © |9|  [N]t |1¦l|cwv|­¥1~|}zºvw4| z N xj9wyx7¡|x£y¢A [N] = {[N]t, t ≥ 0} ¤{4¦¦l|cwv|­¥9~|}zºvw|1|wv|}z9| N § ÃÜ ÔÕÓ ·áÐ}Ô ß Mt Ö×©Uá ·ÖÑ9áÑpQ×Ñ­Õ×Ü Ô R ãaXdãah©Ó4×Ü!ìÔÕ ·ÔîïÑRõÏÔÖÕ4ëlÓ ·ØÔRÖÔÖÔ¦ÛÑ­Õ [M]t = t 0 |f(s, ω)|2 ds. PÒ9ÖÑ ·Ò¦Ó4×Ñ­Õ×Ü ÔÒ9õÏ×Ò×Ð­Ï×Ü Ò×Õ4ïÑÖ×æçÏáÖÔÜ9×á}ÓÔ ßA·Ü9Û{Ñ­ÕÓáØ ·Ô áÑØÔÛÓ4Ï ßÑñ7Ú¹áÑ õÏÔ ß Ö ·ÒØÏ7Ó ·Ü¦Ñ­ÒPÑÜ9ÑÖ× ßÑ1Ö×RÖ ·æçÏÒ4ïÑÛÔÕ4ÔÔ1Õ×Ò×ÕìlÔVÖ×RÏÜ Ô Ò4×Ð­ÏÕ4ÔÖÑ­Õ4Ô7ã ÛÕ ·Ü9× ·ÕÑ åÑ pQ×ÑÕ4×Ü Ô9ñ7ãTS£×£Ñ¦Ò×Ð­ÏáÖÑ å¤Ñ bmÑ­ÕÑ ßëÕ ·ÑfãahÖ×EbmÝÏáÐ8¨ú« ·ßß·ÔÜ¦Ò£ø S4 f ùcã wr Â QÄGºÙ(º»FÊ Å{4 W = {Wt, t ≥ 0} gd3 l¦wy x©±A4ËCxwv|xj  R  l xWggÒ­wd9|sÑC³´v|­¥ {At} ¡|³j~ W P 9|cvwyxd¶}|³³4|³sl4wyR| {At} l¿¦l|cwv|­¥9~|}zºvw4| [W] |vwv|x{z [W]t = t q.c. ∀ t. SYi
19. 19. wr Â QÄGºÙ(º»Q( µs»Ô| M = {Mt, t ≥ 0} d 9|cvwyxd¶}|³P³4|³£lxvªyxdz¨¦l|cwv|­¥ ³´wyRwy¡|}z}|9³4|³´ x§sßmx¡¥ P(Mt = M0 ∀ t ≥ 0) = 1. S4
20. 20. û Æ Ì ÀSÁ  Ì Â ü ÍVÍ Ä¤Â Ì Í Á Æ Ä{È§£SÁ Í1ÊÅÄ{ÆÇÁ ÈËÊ Ì ý ±´³ þ riWijqqdqdiq rqqdqsr7X q rvyqdiji üE×ÒÓÔ Ò×îïÑ Ö ·ÒØÏ7Ó ·Ü9Ñ­Ò Ñ Ü9ÑÖ× ßÑ Ø ßëÒÒ ·ØÑ Ö×Õ ·ÒØÑã íÒÒÏÜ ·Ü9Ñ­Ò õÏ×¨Ô×ì}Ñ ßÏîïÑ ÖÔ Õ×Ò×ÕìlÔ U = {Ut, t ≥ 0} Ö× ÏÜ ÔsØÑ­Ü9ÛÔáÝ ·ÔsÖ× Ò×Ð­ÏÕÑ­Òå ×ÜÞÔÒ4ÔÖÔT×Ü ÏÜ ×ÒÛÔîÑ Ö×ÛÕÑ­ÞÔÞ ·ß·ÖÔÖ× (Ω, F, P) ØÑ­Ü9Ñ Ò×Ð­Ï×ãÅêd×yâÔ U0 = u 0 ÔÕ×Ò×ÕìlÔ ·á ·Ø ·Ô ßã úCÕgÜ ·Ñ­Ò ØÝ×Ð}ÔÜ ØÑ­á}Ó ·ádÏÔlÜ9×áÓ4×EÔ£ÏÜ ÔPÓÔYnÔ£ØÑ­áÒÓÔáÓ4× p 0 × ·áÖ×á ·à ÔîV×ÒÒ4ïÑgÛÔÐ}ÔÒf×Ü ·áÒÓÔáÓ4×Ò Ô ß×ÔlÓ4é­Õ ·Ñ­Ò T1, T2, . . . (0 T1 T2, . . .) Ñ­áÖ×gÑ­ÒAìÔ ßÑ­Õ×ÒEÔ¦Ò×Õ×ÜYÛÔÐ­Ñ­ÒEá×ÒÒ×Ò ·áÒÓÔá}Ó4×ÒEÒ4ïÑ Ö×ÒØÕ ·Ó4Ñ­ÒFÛ{Ñ­ÕXìlÔÕ ·ëì}× ·ÒFÔ ß×ÔlÓ4é­Õ ·ÔÒQáïÑlþá×Ð}ÔlÓ ·ìlÔÒ X1, X2, . . . . )ÛÕÑdØ×ÒÒÑ£Ö×WÕ ·ÒØÑPÕ×ÒÏ ßÓÔáÓ4×Ú ÛÔÕ4Ô t 0 Úå Ut = u + pt − St, øÍiù Ñ­áÖ× St = i≥1 XiI(Ti ≤ t), × I åÔ æçÏáîïÑ ·áÖ ·ØÔÖÑ­Õ4Ô7ãsÙ×©Uá ·Ü9Ñ­Ò9Ô ìÔÕ ·ëì­× ß Ô ß×ÔlÓ4é­Õ ·Ô τ = inf{t ≥ 0 : Ut ≤ 0} ãxìÔ ßÑ­Õ£Ö× τ áÑ­ÒgÖëÑ ·áÒÓÔáÓ4×¦ÖÔÕÏáÔ7ã1Ù×ì­×Ü9Ñ­ÒgÓÔÜÞ{åÜ ØÑ­áÒ ·Ö×Õ4ÔÕVÔ Û{Ñ­ÒÒ ·Þ ·ß·ÖÔÖ×Ö×¨áïÑuÝÔì­×ÕVÕÏáÔ7ÚÑuõÏ×¨Ò ·Ð­á ·UØÔ õÏ× τ = ∞ ãÎCÒÓÔÜ9Ñ­Ò ·á}Ó4×Õ×ÒÒ4ÔÖÑ­ÒáÔ9ÛÕÑ­ÞÔÞ ·ß·ÖÔlÖ×£Ö×©ÕÏáÔ¦×Ü¿Ó4×Ü9Û{Ñ ·áxUá ·Ó4Ñ P(τ ∞) Ú{×©áÔ9ÛÕÑ­ÞÔÞ ·ß·ÖÔÖ× Ö×£ÕÏáÔRÔáÓ4×ÒAÖÑRÓ4×Ü9Û{Ñ t Ú P(τ ≤ t) ã ý ±¥à noiWqxyiijxyffqdqdvyijiqÅ¨rpsrkÿh¹gqsrp ÅÜ9ÑÖ× ßÑRØ ßëÒÒ ·ØÑ¦Ö×EbmÕ4ÔÜ9åÕAýÏáÖÞ{×ÕÐ9å£ØÔÕ4ÔØÓ4×Õ ·à ÔÖÑ¦Û{× ßÔÒWÒ×Ð­Ï ·áÓ4×ÒAÒÏÛ{Ñ­Ò ·îV×Òæ • PÒXìÔ ßÑ­Õ×ÒXÖÔÒ ·áÖ×á ·à ÔîVd×Ò X1, X2, . . . Ò4ïÑPìlÔÕ ·ëì­× ·ÒFÔ ß×ÔlÓ4é­Õ ·ÔÒ ·áÖ×Û{×áÖ×á}Ó4×Òf× ·Ö×ádþ Ó·Ø ÔÜ9×áÓ4×VÖ ·ÒÓ4Õ ·ÞÏÖÔÒRØÑ­ÜoæçÏáîïÑÖ×Ö ·ÒÓ4Õ ·ÞÏ ·îïÑ F Ú¹Ü9åÖ ·Ô µ = E(X1) ∞ × ìÔÕ ·áØ ·Ô σ2 = V ar(X1). • PÒRÓ4×Ü9Û{Ñ­Ò T1, T2, . . . áÑ­Ò¦õÏÔ ·Ò¦ÔÒ ·áÖ×á ·à ÔîVd×Ò¦ÑdØÑ­ÕÕ×Ü Ò4ïÑ ÓÔ ·ÒRõÏ×ÔÒìÔÕ ·ëì­× ·Ò Ô ß×ÔlÓ4é­Õ ·ÔÒ Zi = Ti − Ti−1, i ≥ 1 ÒïlÑ ·áÖ×Û{×áÖ×á}Ó4×Ò£× ·Ö×áÓ ·ØÔÜ9×áÓ4×©Ö ·ÒÓ4Õ ·ÞÏÖÔÒÚjØÑ­ÜaÖ ·ÒÓ4Õ ·ÞÏ ·îïÑ1×on7Û{Ñá×áØ ·Ô ß ØÑ­Ü E(Z1) = λ−1 ã • ÅáðÜ9×ÕÑ¦Ö× ·áÖ×á ·à ÔîVd×ÒAáÑ ·á}Ó4×ÕìÔ ßÑ [0, t] å£Ö×áÑÓÔÖÑ9ÛÑ­Õ Nt = sup{n ≥ 1 : Tn ≤ t}, t ≥ 0, S4
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23. 23. ¡±¥à þ riõ¹Põ¹Pvyxyvyfq q rh§¦y êd×yâÔ ψδ(u) Ô¦ÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×¤Ö×gÕÏáÔ¦×Ü3Ó4×Ü9Û{Ñ ·áxUá ·Ó4ÑRÖ×gÏÜ Ô¦ØÑ­Ü9ÛÔáÝ ·ÔRÖ×gÒ×Ð­ÏÕÑ­Ò ØÑ­Ü3Õ×Ò×ÕìlÔ ·á ·Ø ·Ô ß u ãfÎCá}ÓïÑ ψδ(u) = P t≥0 (Uδ(t) 0) = P t≥0 (Vδ(t) 0) . ÎCÜaêdÏáÖ7ÓD¨ pQ×ÏÐ­× ßÒ¦ø S4wh­ùcÚ{Ñ­ÒPÔÏ7Ó4Ñ­Õ×ÒPÖ ·à ×ÜYõÏ×Ò× Uδ(υ) ≥ 0 ∀ υ ≤ t Új×áÓïÑ Uδ(υ) ≥ U(υ) ∀ υ ≤ t × ∀ δ ≥ 0 ã£ûÒÓ4Ñ ·Ü9Û ß·ØÔ õÏ× ψδ(υ) ≤ ψ(υ) ÚsÑ­Ï Ò×yâÔ7ÚÑÜ9ÑÖ× ßÑ1Ø ßëÒÒ ·ØÑ1ØÑ­Ü δ = 0 æçÑ­Õá×Ø×£ÏÜ ß·Ü ·Ó×PÒÏÛ{×Õ ·Ñ­ÕAÛÔÕ4Ô¦ÔRÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×PÖ×£ÕÏáÔRáÑ¦ØÔÒÑ9Ð­×Õ4Ô ß δ ≥ 0 ã úXÔÕ4Ô ÏÜ ÔÜ9× ßÝÑ­ÕØÑ­Ü9ÛÕ××áÒ4ïÑ¨ÖÔÒ¦Ö ·æç×Õ×áîÔÒæçÏáÖÔÜ9×á}ÓÔ ·ÒR×áÓ4Õ×VÑ­ÒRØÔÒÑ­Ò δ = 0 × δ 0 Úsì­×yâÔÜ9Ñ­Ò¤ÑVØÑ­Ü9Û{Ñ­ÕÓÔÜ9×á}Ó4ÑÖÑVÛÕÑØ×ÒÒÑ {Vδ(t), t ≥ 0} á×ÒÓ4×ÒgÖÑ ·ÒPØÔÒÑ­Òã¦üEÑVØÔÒÑ δ = 0 ÚÑ¦ØÑ­Ü9Û{Ñ­ÕÓÔÜ9×á}Ó4ÑRÖ× Vt = Ut = u + pt − St æçÑ · Ö×ÒØÕ ·Ó4Ñ9áÔRÒ×îïÑfãaXdã üEÑ1ØÔÒÑ δ 0 áïÑ1Ýë1ØÑ­á}ì­×ÕÐáØ ·ÔÛÔÕ4Ô +∞ Ñ­Ï −∞ ×Ò ·ÜYÛÔÕ4Ô1Ô ìlÔÕ ·ëì}× ß Ô ß×ÔlÓ4é­Õ ·Ô Uá ·ÓÔ Vδ(∞) = u + p δ − ∞ 0 e−δυ dS(υ). ø SYh­ù úXÔÕ4Ôì}×ÕAõÏ×gÔ ·á}Ó4×Ð­Õ4Ô ß áÔ¦×on7ÛÕ×ÒÒ4ïÑ ø SYh­ùmå¥Uá ·ÓÔ7ÚÞÔÒÓÔævÔ à ×Õ ∞ 0 e−δυ dS(υ) = i≥1 e−δTi Xi. í ·Ð­ÏÔ ßÖÔÖ×ÔØ ·Ü ÔAåìlë ß·ÖÔAÛ{Ñ ·ÒXÔ ·áÓ4×Ð­Õ4Ô ß ó×ÒõÏ×ÕÖÔPåmØÑ­áÒÓÔá}Ó4×m×áÓ4Õ×mÑ­Ò ·áÒÓÔá}Ó4×ÒX×ÜÃõÏ× ØÝ×Ð}ÔÜÔÒmÕ× ·áì ·Ö ·ØÔîV×ÒãAbmÑ­Ü9Ñ X × T Ò4ïÑ£ìÔÕ ·ëì­× ·ÒmÔ ß×ÔlÓ4é­Õ ·ÔÒCÛ{Ñ­Ò ·Ó·ì ÔÒf× ·áÖ×Û{×áÖ×á}Ó4×ÒÚ ÛÑdÖ×Ü9Ñ­ÒAÑ­Þ7Ó4×Õ E i≥1 e−δTi Xi = i≥1 E e−δTi Xi = i≥1 E(e−δTi )E(Xi) = µ i≥1 E(e−δ[(Ti−Ti−1)+(Ti−1−Ti−2)+···+(T1−T0)] ) = µ i≥1 [E(e−δπ )]i , π ∼ Ti − Ti−1 = µ i≥1 λ λ + δ i ∞. bmÑ­Ü9Ñ E ∞ 0 e−δυ dS(υ) ∞ Ú×á}ÓïÑ ∞ 0 e−δυ dS(υ) ∞ q.c ã Ù×ÒÓÔ1æçÑ­ÕÜ Ô7Ú{áïÑ1åé­Þ}ì ·Ñ1õÏ× ψδ(u) = 1 Ò×Ü9ÛÕ×õÏ× p ρ ã£üEÑ1×áÓÔá}Ó4ÑÚsÒ× p ≤ −δu ×á}ÓïÑ ψδ(u) = 1 ãfúFÔÕ4ÔRÜ Ô ·Ò¹Ö×ÓÔ ßÝ×ÒAÒÑ­ÞÕ×£×ÒÓ4×gÜ9ÑÖ× ßÑÚdì}×yâÔ9êdÏáÖ7ÓB¨úpQ×ÏÐ­× ßÒ©ø S4wh­ùcã XX
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25. 25. öEÜ ÔÕ× ßÔîïÑ¦Õ×ØÏÕÒ ·ìlÔ¦Ü Ô ·ÒWÐ­×Õ4Ô ß Û{ÑÖ×£Ò×ÕEÖ×ÖÏ à·ÖÔRØÑ­Ü9Ñ Uv = e∆v−∆t Ut − v t e−(∆τ −∆t) dBτ , t ≤ v. ø S4}ù º»¼ºÙ( x5ÈQä¿ Â »À¿Ä êd×yâÔ (Ω, F, P) ÏÜS×ÒÛÔîÑÖ×¦ÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×Ú A = {At, t ≥ 0} ÏÜ Ô U ßÓ4Õ4ÔîïÑÚÑ­áÖ× At ⊂ F Õ×ÛÕ×Ò×á}ÓÔ¨Ô ·á7æçÑ­ÕÜ ÔlîïÑÖ ·ÒÛ{Ñ­áì­× ß ÔlÓ4å ÑÓ4×Ü9ÛÑ t ã3Õ ·ÒØÑÖ×Ò×Ð­ÏÕÑ å1Ü9ÑÖ× ßÔÖÑ ævÔ à ×áÖÑ {Bt, t ≥ 0} ÏÜIÛÕÑdØ×ÒÒÑ ×ÒÓ4ÑdØëÒÓ ·ØÑ A þÒ×Ü ·Ü9ÔÕÓ ·áÐ}Ô ß ØÑ­ÜË×Ò4Û×Õ4ÔáîÔ|Uá ·ÓÔ7ÚXÑ­Ï Ò×yâÔ7Ú {Bt, t ≥ 0} å A þyÔÖÔÛ7ÓÔÖÑ×VØÑ­Ü Ó4Õ4Ôlâ×Ó4é­Õ ·ÔlÒ¦ØÑ­á}ÓeáÏÔÒ9óÖ ·Õ× ·ÓÔØÑ­Ü ß·Ü ·Ó×Ò¦Û× ßÔ ×ÒõÏ×ÕÖÔ7ã ÙPÔ Ü9×ÒÜ ÔæçÑ­ÕÜ Ô Ñ Õ·Ò ØÑUáÔáØ× ·ÕÑuå ·á}Ó4ÕÑdÖÏ à·ÖÑævÔ à ×áÖÑ {∆t, t ≥ 0} ÏÜ ÛÕÑØ×ÒÒÑ ×ÒÓ4ÑdØëÒÓ ·ØÑ A þÒ×Ü ·Ü ÔÕÓ ·áÐ}Ô ßã íÕÏáÔVÖÔVÒ×Ð­ÏÕ4ÔÖÑ­Õ4Ô7ÚQÖ×©Uá ·ÖÔVØÑ­Ü9ÑÑ×ì­×á}Ó4Ñ×Ü õÏ× ÔVÕ×Ò×ÕìÔVÓ4Ñ­ÕáÔþÒ×¦Ü9×áÑ­Õ£Ñ­Ï ·Ð­ÏÔ ß Ô à ×ÕÑÚ{åÑ1ÔÒÒÏá}Ó4ÑÛÕ ·áØ ·ÛÔ ß Ö×ÒÓÔ1Ö ·ÒÒ×ÕÓÔîïÑãÎCÒÓÔÜ9Ñ­Ò ·á}Ó4×Õ×ÒÒ4ÔÖÑ­ÒP×ÜY×ÒÓ4ÏÖÔÕPÔ ÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×PÖ×£ÕÏáÔR×Ü!Ó4×Ü9ÛÑUá ·Ó4ÑÚ ψ(u, v) = P inf s∈[0,v] Us ≤ 0|U0 = u ø S4}ù = P sup s∈[0,v] s 0 e−∆τ dBτ ≥ u|U0 = u , ø¥X f ù ×£Ô¦ÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×PÖ×£ÕÏáÔR×Ü!Ó4×Ü9ÛÑ ·áxUá ·Ó4ÑÚ ψ(u) = P inf s≥0 Us ≤ 0|U0 = u ø¥XQSù = P sup s≥0 s 0 e−∆τ dBτ ≥ u|U0 = u . ø¥XX­ù ²±¥à noiWqxyiq¢vhq¢fi PCrrqqdqsr7X bmÑ­Ü9Ñ1Ø ·ÓÔÖÑÔá}Ó4×Õ ·Ñ­ÕÜ9×áÓ4×Ú{ØÑ­áÒ ·Ö×Õ4ÔáÖÑÏÜ Ô ØÔÕÓ4× ·Õ4Ô Ö×Ò×Ð­ÏÕÑ­ÒPõÏ×Rå©æç× ·ÓÔ Ö×ÏÜ Ð­Õ4ÔáÖ×áðÜ9×ÕÑ Ö×Õ ·ÒØÑ­Ò ·áÖ ·ì ·ÖÏÔ ·Ò ·áÖ×Û×áÖ×áÓ4×ÒÚØÑ­áÒ ·Ö×Õ4ÔáÖÑ õÏ×á×áÝdÏÜÖ× ß×Ò1å Ð­Õ4ÔáÖ×mÑ¤ÒÏxUØ ·×á}Ó4×WÛÔÕ4ÔPÔlæç×ÓÔÕXÑPìlÔ ßÑ­ÕQÓ4ÑÓÔ ß ÖÔÒ ·áÖ×á ·à ÔîVd×ÒCÔÐ­Õ×Ð}ÔlÖÔÒFÒ ·Ð­á ·UØÔlÓ ·ìlÔÜ9×á}Ó4×Ú ÔÛÕÑgn ·Ü ÔÜ¦Ñ­ÒWÔæçÏáîïÑ Bt Û{Ñ­ÕWÏÜ ÔRÖ ·æçÏÒ4ïÑÚ ·ÒÓ4Ñ¦åÚ Bt = −βt − σbWb,t, ø¥X R ù Ñ­áÖ× β × σb Ò4ïÑØÑ­áÒÓÔá}Ó4×Ò©× Wb,t = {Wb,t, t ≥ 0} å9ÏÜ Ü9Ñì ·Ü9×á}Ó4ÑômÕÑ4`Aá ·ÔáÑÛÔÖÕ4ïÑã íWõÏ · β Õ×ÛÕ×Ò×á}ÓÔÑVÐ}ÔáÝÑ×ÒÛ{×Õ4ÔÖÑÛÑ­Õ£Ïá ·ÖÔÖ×¦Ö×RÓ4×Ü9Û{ÑÚ¤Ö×ì ·ÖÑÔÏÜ ØÑ×©UØ ·×áÓ4×9Ö× Ò×Ð­ÏÕ4ÔáîÔ θ 0 áÑPÛÕgÜ ·Ñ× σb å¹ÑPÖ×Òì ·Ñ¤ÛÔÖÕ4ïÑPÖÑPÓ4ÑÓÔ ß Ö×¹ÛÔÐ}ÔÜ9×á}Ó4Ñ­ÒXÖÔÒ ·áÖ×á ·à ÔîVd×Ò Xf
26. 26. ÛÑ­ÕRÏá ·ÖÔÖ×Ö×1Ó4×Ü9Û{Ñã ÎCÜ áÑ­ÒÒÑ Ó4Õ4ÔÞÔ ßÝÑÚCÏ7Ó ·ß·à ÔÕ×Ü¦Ñ­ÒÔÒ¦Ö ·æçÏÒgV×Ò¦ÛÔlÕÔ ÔÛÕÑgn ·Ü ÔÕ©Ñ Ü9ÑÖ× ßÑRØ ßëÒÒ ·ØÑ¦ØÑ­Ü ·á}ì}×ÒÓ ·Ü9×á}Ó4Ñãfêd×áÖÑ1ÔÒÒ ·ÜÚ7Õ×ÛÕ×Ò×áÓÔÜ9ÑÒ β × σb Û{Ñ­Õ β = p − λµ, e ø¥Xfù σb = λ(σ2 + µ2). ø¥Xh­ù ÙPÔ¦Ü9×ÒÜ Ô©æçÑ­ÕÜ Ô7ÚÔÒÒ4ÏÜ ·Ü9Ñ­ÒWõÏ×£Ñ¦ævÔÓÑÕWÖ×£Ö×ÒØÑ­áÓ4Ñ ∆t å£ÏÜ ÔRÖ ·æçÏÒ4ïÑÚÑ­ÏVÒ×yâÔ7Ú ∆t = δt + σdWd,t, ø¥Xñ}ù Ñ­áÖ× δ × σd Ò4ïÑØÑ­áÒÓÔá}Ó4×Ò©× Wd,t = {Wd,t, t ≥ 0} å¦ÏÜ Ü9Ñì ·Ü¦×áÓ4ÑVômÕÑY`Aá ·ÔáÑVÛÔÖÕ4ïÑã êd× σd å©Û{Ñ­Ò ·Ó·ì ­ÑÚ×ÒÓÔÜ9Ñ­ÒPÔÖ ·Ø ·Ñ­áÔáÖÑ9ÏÜ Ô9Û{×ÕÓ4ÏÕÞÔîïÑ1×ÒÓ4ÑdØëÒÓ ·ØÔ1Ô ÏÜ Ô¦ÓÔYnÔ Ö×AâÏÕÑ­Ò ØÑ­áÒÓÔá}Ó4× δ ã íÒÒÏÜ ·Ü9Ñ­ÒsÓÔÜÞ{åÜ¢õÏ× Wb,t × Wd,t Ò4ïÑ ·áÖ×Û×áÖ×á}Ó4×ÒãCúQ× ßÔEÙ×©Uá ·îïÑ R ãaXdãaidÚÓ4×Ü9Ñ­Ò¤õÏ× Bt × ∆t Ò4ïÑ ÛÕÑØ×ÒÒÑ­ÒPÖ×gû¡Óed ·áÖ×Û{×áÖ×á}Ó4×ÒãEêd×yâÔÜ Zt = e∆t × Yt = U0 − t 0 e−∆τ dBτ Ú Ñ­ÏVÒ×yâÔ7Ú Ut = ZtYt ãfúX× ßÑpQ×Ñ­Õ×Ü Ô R ãaXdã7Ú Zt e Yt ÓÔÜÞ{åÜ!Ò4ïÑ¦ÛÕÑdØ×ÒÒÑ­ÒEÖ×£û¡Óed9×Ü R+ ã úQ× ßÔ¦úCÕÑ­Û{Ñ­Ò ·îïÑ R ãaXdãR×£Û× ßÑpQ×Ñ­Õ×Ü Ô R ãaXdã7Ú7Ó4×Ü9Ñ­ÒWõÏ×dæ d(Ut) = d(ZtYt) = −dBt + Ut e∆t d(e∆t ) − d(e∆t )e−∆t dBt. pQ×Ü¦Ñ­ÒWÓÔÜRÞåÜ3õÏ×dæ dBt = −βdt − σbdWb,t × d(e∆t ) = δe∆t dt + σde∆t dWd,t + 1 2 σ2 de∆t dt = e∆t δ + 1 2 σ2 d dt + σde∆t dWd,t. Ù×ÛÑ ·ÒAÖ×gÔ ßÐ­ÏáÒWØë ßØÏ ßÑ­ÒWØÑ­áØ ßÏÜ9Ñ­Ò¹õÏ×dæ dUt = Ut δ + 1 2 σ2 d + β dt + UtσddWd,t + σbdWb,t. ø¥Xi­ù íÐ­ÕÏÛÔáÖÑ¦Ñ­ÒWÓ4×ÕÜ9Ñ­ÒWÖÔ¦Ö ·æçÏÒ4ïÑRáÔ¦×on7ÛÕ×ÒÒ4ïÑ ø¥Xi­ùcÚÖ×©Uá ·Ü9Ñ­Ò dWt = UtσddWd,t + σbdWb,t (U2 t σ2 d + σ2 b ) 1 2 . êd×áÖÑ ÔÒÒ ·ÜÚ Wt = t 0 (U2 s σ2 d + σ2 b )− 1 2 (UsσddWd,s + σbdWb,s). pQ×Ü¦Ñ­ÒAõÏ× d[W]t = (dWt)2 = 1 Utσ2 d + σ2 b (UtσddWd,t + σbdWb,t)2 = Xh
27. 27. = 1 Utσ2 d + σ2 b (U2 t σ2 d + σ2 b )dt = dt. ýÑ­Ð­ÑÚ ·áÓ4×Ð­Õ4ÔáÖÑ¦ÔÜRÞÑ­ÒWÑ­Ò ßÔÖÑ­ÒWÖ×ÒÓÔ¦×õÏÔîïÑ9áÑ ·á}Ó4×ÕìÔ ßÑ [0, t] ÚÑ­Þ7Ó4×Ü9Ñ­Ò d[W]t = dt ⇒ [W]t = t. Ù×ÒÓÔ)æçÑ­ÕÜ Ô7Ú©Ó4×Ü9Ñ­Ò õÏ× W = {Wt, t ≥ 0} åuÏÜ%Ü ÔÕÓ ·áÐ}Ô ß ØÑ­á}ÓeáÏÑÉØÑ­ÜDìÔÕ ·ÔîïÑ õÏÔÖÕ4ëlÓ ·ØÔ [W]t = t ãúQ× ßÑypQ×Ñ­Õ×Ü Ô R ãaXdãTSYXdÚXÓ4×Ü9Ñ­ÒõÏ× W å1ÏÜÜ9Ñì ·Ü9×á}Ó4ÑômÕ4Ñ4`Aá ·ÔáÑ ÛÔÖÕ4ïÑã¢ýÑ­Ð­ÑÚWÛÑdÖ×Ü9Ñ­Ò9Ó4Õ4ÔlÓÔÕ ø¥Xi­ùRØÑ­Ü9Ñ ÏÜ2ØÔÒÑ ×ÒÛ{×Ø ·Ô ß ÖÔ×õÏÔîïÑ Ö ·æç×Õ×áØ ·Ô ß ÖÔ æçÑ­ÕÜ Ô dUt = µ(t, Ut)dt + σ(t, Ut)dWt, ø¥Xd}ù Ñ­áÖ× W åfÏÜ Ü9Ñì ·Ü9×á}Ó4ÑEômÕÑ4`Aá ·ÔáÑAÛÔÖÕ4ïÑE×fÛ{ÑdÖ×Ü9Ñ­Ò¤ØÑ­áÒ ·Ö×Õ4ÔÕ At = σ{Ws; 0 ≤ s ≤ t} ã ¤ ·Ü9ÛÑ­ÕÓÔáÓ4×Ö ·à ×Õ¦õÏ×Ú¹ØÑ­Ü9Ñ Ut åÏÜ Ô¨Ö ·æçÏÒ4ïÑÚ×áÓïÑ× ß×ÓÔÜRÞåÜ åÏÜoÛÕÑdØ×ÒÒÑ äÔÕgªÑì ·ÔáÑã íÃÛÕ ·áØÛ ·Ñ×ÒÓÔÕ×Ü9Ñ­Ò ·á}Ó4×Õ×ÒÒ4ÔÖÑ­ÒEáÔRÒ×Ð­Ï ·á}Ó4×£ÛÔÕ4ÔÜ9×Ó4Õ ·à ÔîïÑæ µ(t, u) = u(δ(t, u) + 1 2 σ2 d(t, u)) + β(t, u) ø¥Xd}ù σ2 (t, u) = u2 σ2 d(t, u) + σ2 b (t, u), ø Rf ù áÔõÏÔ ß Û×ÕÜ ·Ó·Ü ¦Ñ­Ò©õÏ×Ñ­ÒRØÑd×oUØ ·×áÓ4×Ò β, σb, δ, e σd Ö×Û{×áÖÔÜ ÖÑÓ4×Ü9Û{Ñ ×1ÖÔÕ×Ò×ÕìÔ ÔlÓ4ÏÔ ßã íÒÒÏÜ ·Ü9Ñ­ÒFõÏ×AÑ­ÒCØÑd×©UØ ·×á}Ó4×Ò µ(t, u) × σ(t, u) Ò4ïÑ¤Ò4ÏxUØ ·×áÓ4×Ü9×á}Ó4×AÒÏÔì­×ÒÛÔlÕÔ¤ÔÖÜ ·Ó·Õ ÏÜ Ô ðá ·ØÔ ÒÑ ßÏîïÑÛÔÕ4Ôsø¥Xd}ù×VÛÔÕ4Ô¨ÔÒÒ×Ð­ÏÕ4ÔÕ Ô¨×on ·ÒÓeáØ ·Ô¨×VØÑ­á}Ó ·ádÏ ·ÖÔlÖ×VÖ×Ó4ÑdÖÔÒ9ÔÒ Ö×Õ ·ìlÔÖÔÒAõÏ×gÔÛÕ×Ò×áÓÔÕ×Ü9Ñ­ÒAáÔRÛÕé4n ·Ü ÔÒ×îïÑã ²±»° þ riõ¹Põ¹Pvyxyvyfq q rh§¦y ºÙGº» H Â  ñ Ä ñ ¾»Û»¾»ä¿Ä¼ä¿ä¿ Â »À¿Ä¡Q Á4QôõÆÇÀy¾íÁ4 êd×yâÔ I[A] Ô©æçÏáîïÑ ·áÖ ·ØÔÖÑ­Õ4Ô©ÖÑR×ì}×á}Ó4Ñ A ãwbmÑ­áÒ ·Ö×Õ×Ü9Ñ­ÒAÔÖ ·ÒÓ4Õ ·ÞÏ ·îïÑÖ×¤ÛÕÑ­ÞÔÞ ·ß·þ ÖÔÖ×PØÑ­áÖ ·Ø ·Ñ­áÔ ß ÖÑRÜá ·Ü¦ÑÔlÓ ·áÐ ·ÖÑ©Û{× ßÑRÛÕÑØ×ÒÒÑ9ÖÔÕ×Ò×ÕìÔR×Ü!ÏÜ ·áÓ4×ÕìlÔ ßÑÖ×PÓ4×Ü9Û{Ñ Uá ·Ó4ÑÚ P(t, u, v, y) = P inf s∈[t,v] Us ≤ y|Ut = u , 0 ≤ t ≤ v; u, y ∈ R. ø R Sù Xñ
28. 28. úXÔÕ4Ô v × y Uxn7Ñ­ÒÚÒ×yâÔ Mt = P inf s∈[0,v] Us y|At = It(1 − P(t, Ut, v, y)), ø R X­ù Ñ­áÖ× It = 1 inf s∈[0,t] Us y . ÎCá}ÓïÑ Mt = {Mt, t ≥ 0} ågÏÜ¿Ü9ÔÕÓ ·áÐ}Ô ß øí©f×yâÔ¦íá×on7Ñ ûûûùcãíÅÒ×Ð­ÏáÖÔ ·Ð­ÏÔ ßÖÔÖ×¤×ÜËø R X­ù å¤Ö×ì ·ÖÔ9óRÛÕÑ­ÛÕ ·×ÖÔÖ×©äÔÕgªÑì ·ÔáÔRÖÑ9ÛÕÑdØ×ÒÒ4Ñ9ÖÔ¦Õ×Ò×ÕìlÔ7ã íÛ ß·ØÔáÖÑÔVæçé­ÕÜÏ ßÔÖ×9û¡ÓedÐ­×á×Õ4Ô ß·à ÔÖÔbmÝÏáÐ|¨ì« ·ßß·ÔÜ¦Ò ø S4 f Ú¿pQ×Ñ­Õ×Ü Ôx7ãaX­ù£ÔÑ ÛÕÑØ×ÒÒÑ Mt ÚÑ­ÞÒ×ÕìÔáÖÑõÏ×Ô ÛÔÕÓ4×VØÑ­á}ÓeáÏÔ ÖÔ æçÏáîïÑ It åVØÑ­áÒÓÔá}Ó×V×VÏÒ4ÔáÖÑdø¥Xd}ù Ñ­Þ7Ó4×Ü9Ñ­ÒWÑ¦Ò×Ð­Ï ·á}Ó4×£Õ×ÒÏ ßÓÔÖÑæ dMt = − It ∂ ∂t P(t, Ut, v, y)dt − It ∂ ∂x P(t, Ut, v, y)(µ(t, Ut)dt + σ(t, Ut)dWt) − It 1 2 ∂2 ∂x2 P(t, Ut, v, y)σ2 (t, Ut)dt + It(1 − P(t, Ut, v, y)) ø RR ù − It−(1 − P(t−, Ut− , v, y)). PÒXØë ßØÏ ßÑ­ÒQá×Ø×ÒÒ4ëÕ ·Ñ­ÒFÛÔÕ4ÔEÑ­Þ7Ó4×áîïÑPÖ×ÒÓÔP×õÏÔîïÑìïÑ¤Ô ßåÜ ÖÑP×ÒØÑ­Û{ÑPÖ×ÒÓÔPÖ ·ÒÒ×ÕÓÔîïÑã bmÑ­Ü9Ñ£ÔæçÏáîïÑ It(1−P(t, Ut, v, y)) åWØÑ­á}ÓeádÏÔP×Ü t õÏÔÒ×AØ×ÕÓÔÜ9×á}Ó4×ÚÔ¤ÛÔÕÓ4×¹ÖÑ£Ò4Ô ßÓ4ÑÚ ØÑ­ÕÕ×ÒÛÑ­áÖ×áÓ4×¤ÔÑð ßÓ·Ü ¦Ñ£Ó4×ÕÜ9ÑÚdå à ×ÕÑõÏÔÒ×¤Ø×ÕÓcÔlÜ9×áÓ4×ãw Ó4×Ñ­Õ×Ü9ÔÔgÒ×Ð­Ï ·ÕÚ7Ñ­Þ7Ó ·ÖÑ©×Ü üEÑ­ÕÞ×ÕÐø S4}ùcÚdæçÑ­Õá×Ø×gÏÜ Ô×õÏÔîïÑ9Ö ·æç×Õ×áØ ·Ô ß ÛÔÕ4Ô¦Ô¦ÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×P×ÜIø R Sùcã wr Â QÄºÙGº» u {4­²|}²wy³´wvz}|}z©Wò¬«o×|vwv||V4~|­¥ zw4xjwv|³ ∂ ∂t P(t, u, v, y) + ∂ ∂x P(t, u, v, y)µ(t, u) + 1 2 ∂2 ∂x2 P(t, u, v, y)σ2 (t, u) = 0, ø R fù ©Ôwy¡| ÝRlx{zwçç P(v, u, v, y) =    1  u ≤ y, 0  u y, P(t, y, v, y) = 1, 0 ≤ t v, P(t, ∞, v, y) = 0, 0 ≤ t v. Ó 4l¦l|}® ÎCÜ¿ÛÕ ·Ü9× ·ÕÑ ßÏÐ}ÔÕÚÔÒÒ4ÏÜ ·Ü9Ñ­ÒAõÏ× ∂ ∂x P(t, Ut, v, y)σ(t, Ut)dt åDUá ·ÓÔ¦õãØããWÎCÜø RR ùcÚÓ4ÕÑlþ ØÔÜ9Ñ­Ò©ÑÓ4×ÕÜ9Ñ ØÑ­ÜËÑævÔlÓ4Ñ­Õ dWt ÛÔÕ4ÔÑ ßÔÖÑ×ÒõÏ×ÕÖÑÖÔ×õÏÔîïÑãbmÑ­Ü9Ñ W × M Ò4ïÑ Ü ÔÕÓ ·áÐ}Ô ·ÒÚdÑõÏ×P×ÒÓëÖÑ ßÔÖÑ©Ö ·Õ× ·Ó4ÑgÖÔ×õÏÔîïÑÓÔÜRÞåÜÖ×ì}×PÒ×Õ£øçÑ ·áØÕ×Ü9×á}Ó4ÑÖ×ùÏÜ Xi
29. 29. Ü ÔÕÓ ·áÐ}Ô ßã CpQ×Ü9Ñ­Ò©õÏ×1×ÒÓ4×1Ü ÔÕÓ ·áÐ}Ô ß å1ÔÞÒÑ ßÏ7ÓÔÜ9×áÓ4×9ØÑáÓeáÏÑ× Û{Ñ­ÕÓÔá}Ó4ÑÖ× ìlÔÕ ·ÔîïÑ ß·Ü ·ÓÔÖÔ7ãFýÑ­Ð­ÑÚÛ{× ßÑpQ×Ñ­Õ×Ü Ô R ãaXdãTS R × ß×¤Ö×ì­×gÒ×ÕAØÑ­áÒÓÔáÓ4×ÚÑ­ÏÒ×yâÔ7Ú ∂ ∂t P(t, u, v, y)dt + ∂ ∂x P(t, u, v, y)µ(t, u)dt + 1 2 ∂2 ∂x2 P(t, u, v, y)σ2 (t, u)dt = K, Ñ­áÖ×ØÅå£ØÑ­áÒÓÔá}Ó4×ãuÅÕ×ÒÏ ßÓÔÖÑ¦Ò×Ð­Ï× ·Ü9×Ö ·ÔlÓÔÜ¦×á}Ó×lã ¤ ·Ü9ÛÑ­ÕÓÔáÓ4×gÑ­ÞÒ×ÕìlÔÕ¤õÏ×RÔ ÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×gÖ×ÕÏáÔ ×ÜYÓ4×Ü9Û{Ñ8Uá ·Ó4Ñ ø S4}ùAåÏÜ½ØÔÒÑ ×ÒÛ×Ø ·Ô ß ÖÔ¦ÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×P×ÜIø R SùcÚÑ­ÏÒ×yâÔ7Ú ψ(t, u, v) = P(t, u, v, 0). ºÙGºÙ( tEÄ¼!¿FÚ\$#QÀ¿rÀ¿Á4Qôõ üEÑØÔÒÑÝÑ­Ü9Ñ­Ðá×ÑáÑVÓ4×Ü9ÛÑÚ¤õÏÔáÖÑÑ­Ò£ØÑ×©UØ ·×á}Ó×Ò µ × σ Ò4ïÑ ·áÖ×Û{×áÖ×á}Ó4×Ò©Ö× t Ú ÔVæçÏáîïÑuø R Sù¤Ö×Û×áÖ×Ö× t × v ÒÑ­Ü9×á}Ó4×1ÔlÓ4Õ4Ôì}åÒ©Ö× v − t ãÙ×ÒÓÔæçÑ­ÕÜ Ô7Ú¤Ó4Ñ­ÕáÔþÒ×9ð7Ó ·ß Ô ·á}Ó4ÕÑdÖÏîïÑ£ÖÔ¤æçÏáîïlÑ©Ö×WÓ4ÕgÒÛÔÕeÜ9×Ó4ÕÑ­Ò P(u, v, y) = P(0, u, v, y) Ú×EØÑ­ÕÕ×ÒÛÑ­áÖ×áÓ4×Ü9×á}Ó4× ØÑ ßÑØÔÜ9Ñ­Ò ψ(u, v) = P inf s∈[0,v] Us ≤ 0|U0 = u . ø R h­ù í Ò×Ð­Ï ·ÕEÔÛÕ×Ò×á}ÓÔÜ9Ñ­ÒAÖÑ ·ÒWØÑ­ÕÑ ßëÕ ·Ñ­Ò¹ÖÑ¦Ó4×Ñ­Õ×Ü ÔRÔáÓ4×Õ ·Ñ­Õã tE Â Û ùÂ ¾ÅºÙGºÙ( uD{4}²|}²wy³´wvz}|}zÃzÃcdªyx{|§ò¬%o×!|  v |vwv||¿4~|­¥ zw4xjwv|³ − ∂ ∂v ψ(u, v) + ∂ ∂u ψ(u, v)µ(u) + 1 2 ∂2 ∂u2 ψ(u, v)σ2 (u) = 0, ø R ñ}ù ©Ôwy¡| ÝRlx{zwç ψ(u, 0) =    1  u ≤ 0, 0  u 0, ø R i­ù ψ(0, v) = 1, v 0, ø R }ù ψ(∞, v) = 0, v 0. ø R }ù êd×£Ñ­ÞÒ×ÕìlÔÕÜ9Ñ­ÒAõÏ× ψ(u, v) = P(T − v, u, T, 0) ÚÔRÛÕ4ÑìlÔ¦Ö×ÒÓ4×gØÑ­ÕÑ ßëÕ ·ÑRå ·Ü9×Ö ·ÔlÓÔ7ã Ù×ì­×Ü9Ñ­ÒÑ­ÞÒ×ÕìÔÕEõÏ×©×ÒÓÔÒÒ4ïÑ ØÑ­áÖ ·îV×Ò ·á ·Ø ·Ô ·ÒA×gÖ×£æçÕÑ­á}Ó4× ·Õ4Ô7ãmíÅ×õÏÔîïÑ¨ø R i­ù¹áÑ­Ò Ö ·à õÏ×ÚWÛÔÕ4Ô v = 0 ÚWÔÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×Ö×ÕÏáÔ¨å ·Ð­ÏÔ ß Ô à ×ÕÑØÑ­ÜoÏÜ ÔÕ×Ò×ÕìlÔ ·á ·Ø ·Ô ß ÛÑ­Ò ·Ó·ì Ô£× ·Ð­ÏÔ ß ÔSØÑ­Ü ÏÜ ÔgÕ×Ò×ÕìlÔ ·á ·Ø ·Ô ß áïÑ©ÛÑ­Ò ·Ó·ì Ô7ãXí ×õÏÔîïÑVø R }ùCáÑ­ÒmÖ ·à õÏ×ØÑ­Ü ÏÜ ØÔÛ ·ÓÔ ßf·á ·Ø ·Ô ßC·Ð­ÏÔ ß Ô à ×ÕÑÚCÔÛÕÑ­ÞÔÞ ·ß·ÖÔÖ×¦Ö× ÕÏáÔVå ·Ð­ÏÔ ß ÔS ÛÔÕ4ÔVÓ4ÑdÖÑ ·áÒÓÔá}Ó4× Xd