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

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

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  7. 7. € ¦€‚yƒ£„C… ʆƒD‡y…•ˆƒ S úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’Ñ9è£Ô’Ü Ô (2.25, 1.5) × δ = 0.1 ڐUxn‡Ô’á‡Ö‡Ñ ÎA‰£ã©ã£ãgãgãgãgãgãgãgãgãgãgã£ãgãgãqf}ñ X úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’Ñ9è£Ô’Ü Ô (2.25, 1.5) × δ = 0.1 ã¦ãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgãgã£ãgãgãqfri R úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’Ñ9è£Ô’Ü Ô (2.25, 1.5) × δ = 0.05 ã†ãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãgãgãgãqfri f úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’Ñ9è£Ô’Ü Ô (2.25, 1.5) × δ = 0.01 ã†ãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãgãgãgãqfw™ h úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦΧn7Û{Ñ­á‡×€á‡Ø ·Ô ß(2/3) × δ = 0.1 ڐUxn‡Ô’á‡Ö‡Ñ ÎA‰£ã1ãgãgã£ãgãgãgãgãgãgãgãgãgãgãŠfw„ ñ úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦΧn7Û{Ñ­á‡×€á‡Ø ·Ô ß(2/3) × δ = 0.1 ã…ãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgãgãŠfw„ i úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦΧn7Û{Ñ­á‡×€á‡Ø ·Ô ß(2/3) × δ = 0.05 ã¿ãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãˆh f ™ úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦΧn7Û{Ñ­á‡×€á‡Ø ·Ô ß(2/3) × δ = 0.01 ã¿ãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãˆh f „ úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦöEá ·æçѭ՘Ü9× (1, 2) × δ = 0.1 ڐUxn‡Ô’á‡Ö‡Ñ ÎA‰£ã¤ãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãˆhQS S f úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦöEá ·æçѭ՘Ü9× (1, 2) × δ = 0.1 ãCãgã£ãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãˆh‘X S‘S úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦöEá ·æçѭ՘Ü9× (1, 2) × δ = 0.05 ã…ãgãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãˆh‘X SYX úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Rևנ՘ϐ‰›á‚ÔV×€Ü Ó4׀Ü9Û{Ñ¡U‚á ·Ó4Ñõ‰Ï‚Ô’á‡Ö‡ÑÔ’Ò ·á‡Ö‡×€á ·›à ÔlîVd׀ÒgÓe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦöEá ·æçѭ՘Ü9× (1, 2) × δ = 0.01 ã…ãgãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãˆh R S R ÎCÕ˜Õ˜ÑˆÖ‡×”×€ÒŽÓ ·Ü Ô’îï’Ñ Û‚Ô’Õ4Ô1Ô Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×gևה՘ϐ‰›á‚Ô ×€ÜYÓ4׀Ü9ۖÑ8U‚á ·Ó4Ñ1õ‰Ï‚Ô’á‡Ö‡Ñ δ = 0.1 ã ãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgã‹h—f Sgf ÎCÕ˜Õ˜ÑˆÖ‡×”×€ÒŽÓ ·Ü Ô’îï’Ñ Û‚Ô’Õ4Ô1Ô Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×gևה՘ϐ‰›á‚Ô ×€ÜYÓ4׀Ü9ۖÑ8U‚á ·Ó4Ñ1õ‰Ï‚Ô’á‡Ö‡Ñ δ = 0.05 ãEãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgã‹h—f SYh ÎCÕ˜Õ˜ÑˆÖ‡×”×€ÒŽÓ ·Ü Ô’îï’Ñ Û‚Ô’Õ4Ô1Ô Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×gևה՘ϐ‰›á‚Ô ×€ÜYÓ4׀Ü9ۖÑ8U‚á ·Ó4Ñ1õ‰Ï‚Ô’á‡Ö‡Ñ δ = 0.01 ãEãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgã‹h‘h ì ·
  8. 8. S‘ñ úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Eևפ՘ϐ‰›á‚Ô©×€Üӕ׃Ü9Û{Ñ ·áxU‚á ·Ó4Ñ©õ‰Ï‚Ô’á‡Ö‡Ñ9Ô’Ò ·á‡Ö‡×€á ·›à ԒîV‰×€Ò¹Óe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’Ñ9è£Ô’Ü Ô (2.25, 1.5) × δ = 0.1 ã¦ãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgãgã£ãgãgãˆh‘i SYi úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Eևפ՘ϐ‰›á‚Ô©×€Üӕ׃Ü9Û{Ñ ·áxU‚á ·Ó4Ñ©õ‰Ï‚Ô’á‡Ö‡Ñ9Ô’Ò ·á‡Ö‡×€á ·›à ԒîV‰×€Ò¹Óe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦΧn7Û{Ñ­á‡×€á‡Ø ·Ô ß(2/3) × δ = 0.1 ã…ãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgãgã‹hd„ S4™ úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Eևפ՘ϐ‰›á‚Ô©×€Üӕ׃Ü9Û{Ñ ·áxU‚á ·Ó4Ñ©õ‰Ï‚Ô’á‡Ö‡Ñ9Ô’Ò ·á‡Ö‡×€á ·›à ԒîV‰×€Ò¹Óe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦΧn7Û{Ñ­á‡×€á‡Ø ·Ô ß(2/3) × δ = 0.05 ã¿ãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãYñ f S4„ úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Eևפ՘ϐ‰›á‚Ô©×€Üӕ׃Ü9Û{Ñ ·áxU‚á ·Ó4Ñ©õ‰Ï‚Ô’á‡Ö‡Ñ9Ô’Ò ·á‡Ö‡×€á ·›à ԒîV‰×€Ò¹Óe˜€Ü Ö ·Òyþ Ó4Õ ·Þ‡Ï ·îï’ѦΧn7Û{Ñ­á‡×€á‡Ø ·Ô ß(2/3) × δ = 0.01 ã¿ãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãYñ f ì ·›·
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  14. 14. ‡lˆš‘»Ôƒ|™¢ At œ”“ƒŒ4•š‘“•ƒx‚„§ !CŒwÆÇÀy¾»È‘ɐ•G€º»¼ºÙG§Ê ŠS‹{Œ4‡‘“••š•š‘‡ {Nt, t ≥ 0} œ|}z}|c‹{„¡|}z‰‡ |™dŠ9| ÑC³´„vŒ˜|­—™¥‰‡ {At, t ≥ 0} š‘ Nt ∈ At ∀ t. !CŒwÆÇÀy¾»È‘ɐ•G€º»¼ºÜp@Ê Š‹{Œ4‡™“••š•š‘‡T•šƒ„‡™“4º™šƒ„vw“•‡ {Mt, t ≥ 0} œ “•†‡|’Š9|}z‰‡udŠ5Š9|’Œc„vwyxd¶}|’³©“•‡lŠ Œ4•š‹‚ƒwy„‡8Ý2ÑC³´„vŒ˜|­—™¥‰‡ {At, t ≥ 0} š‘ ¾ÍÌ Mt œ At °Ž|}z}|c‹{„¡|}z‰‡‘¢ ¾»¾ÍÌ E[|Mt|] ∞ ‹‡|’Œ˜| „‡z‰‡ t  ¾»¾»¾¥Ì E[Mt+s|At] = Mt ‹‡|’Œ˜|9„‡‘z‰‡ s ≥ 0 § ٖףܠԒá‡× ·Õ4ԔԒá‚ë ßÑ­Ð}Ô7Ú7ۖÑdև×cþÒ˜×£Ö‡×©U‚á ·ÕWχÜ3Ü Ô’ÕŽÓ ·á‡Ð}Ô ß Ô”Ó4׀Ü9Û–Ñ¦Ö ·Ò˜Ø€Õ˜×ƒÓ4чã‚úXԒÕ4ÔRχܠԔևשU‚á ·îï’Ñ æçÑ­Õ˜Ü Ô ßÚ dì­×yâ˜Ô êdÝ ·Õ´­}Ԓ׃ì…ø S4„‘„­ñ}ùcã !CŒwÆÇÀy¾»È‘ɐ•G€º»¼ºÙ…mÞ |}z}|dŠ9|EÑC³´„vŒ˜|­—™¥‰‡ {At, t ≥ 0} ¢PdŠ wyx7šƒ„¡|’x‚„1|’³›4|’„ÖՒŒcw‡ Î œ dŠ „ƒŠ–‹‚‡ z‰¹‹‡|’Œ˜|}z}|ˆƒŠ3Œ•ƒ³|­—™¥‰‡9|V•šƒ„¡|2ÑC³´„vŒ˜|­—™¥‰‡‘¢Fš‘”‡ˆƒ¦’ƒx‚„‡ {T ≤ t} ∈ At ¢“‹‡|’Œ˜|9„‡z‰‡ t ≥ 0 § !CŒwÆÇÀy¾»È‘ɐ•G€º»¼ºÙ’ µs»Ôƒ| M = {Mt, 0 ≤ t ∞} dŠ ‹{Œ4‡‘“••š•š‘‡“•‡lx‚„vªyx‚‚‡­§ µs gÒ­w›šƒ„¦dŠ9|1š‘ƒ° ~€‚žƒxj“ƒwv| x{¥‰‡l°Žz‰•“ƒŒ••š“ccx‚„ {Tn}∞ n=1 z‰ „ƒŠ–‹–‡‘š z‰ ‹‡|’Œ˜|}z}| z‰ {At} ¢ „¡|’³ ~€‚ {M (n) t ≡ Mt ∧ Tn , 0 ≤ t ∞} œ©dŠYŠ9|’Œc„vwyxd¶}|’³‡‹‡|’Œ˜|…“4|}z}| n ≥ 1  P[limn→∞ Tn = ∞] = 1 ¢ ƒx‚„¡¥‰‡ z’w™ƒŠ ‡‘š©~€‚ M œgdŠ¿Š9|’Œc„vwyxd¶}|’³s³›‡‘“4|’³Q“•‡lx‚„vªyx‚‚‡­§ !CŒwÆÇÀy¾»È‘ɐ•G€º»¼º i Ê ŠÉ‹{Œ4‡‘“••š•š‘‡ N = {Nt, t ≥ 0} œEz’wy„‡–š‘ƒŒAz‰¹¦l|’Œcwv|­—™¥‰‡P³´wyŠRwy„¡|}z}|¤³›‡‘“4|’³´Š ƒx‚„ š‘ ƒ³›1œ |}z}|c‹{„¡|}z‰‡ |g•dx–—™¥‰‡ t → Nt œ z‰R¦l|’Œcwv|­—™¥d‡1³´wyŠRwy„¡|}z}| ƒŠI“4|}z}|…wyx‚„ƒŒc¦l|’³›‡ˆ³´wyŠRwy„¡|}z‰‡ z‰ R+ ~€‡|™š‘”“•ƒŒc„|’Š ƒx‚„§ !CŒwÆÇÀy¾»È‘ɐ•G€º»¼ºÙs µs»Ôƒ| X = M + V ¢g‡lx{z‰ M = {Mt, t ≥ 0} œ dŠIŠ9|’Œc„vwyxd¶}|’³C³›‡‘“4|’³¹“•‡lx‚° „vªyx‚‚‡… V = {Vt, t ≥ 0} œRdŠ ‹{Œ4‡‘“••š•š‘‡…“•‡lx‚„vªyx‚‚‡Vz‰”¦l|’Œcwv|­—™¥‰‡1³´wyŠRwy„¡|}z}|1³›‡‘“4|’³´Š ƒx‚„§Eßmx‚„¡¥‰‡ X = {Xt, t ≥ 0} œgdŠ š‘ƒŠRwyŠ |’Œc„vwyxd¶}|’³§ °²±¥à ƒ…ffxyijh–xyiq“q}”7iWijffq­”7vyijiq € vy’‚”7q“pCr‡€fx¦•–qâál”xã üE׀ҎӕԖҘ׀îï’ѤÔlۇÕ4׃Ò4׃á‰Ó•Ô’Ü9Ñ­ÒXÔ ßЭχá‡Ò¤Ø€Ñ­á‡Ø€× ·Ó4Ñ­ÒXׂևשU‚á ·îV‰×€ÒXւÔAÓ4×€Ñ­Õ ·ÔAևѥb¹ë ßØ€Ï ßÑA׀ҎÓ4ÑdØë’ÒŽÓ ·Ø€Ñ ×¤Ö‚Ô ·á}Ó4׀ЭÕ4Ô ß Ö‡×£û¡Óed‡ã S R
  15. 15. G€ºÙ(€º» tE“Ày©Á  Žyȑɐ•ä¿Ä‘¾»À“Á4ŒrÚ Â Ä¼Ûw俌|oÁ4å êd×yâ˜Ô (Ω, F, P) ׀ҘۂԒî€Ñ9ևףۇÕ4Ñ’Þ‚Ô’Þ ·›ß›·Ö‚ԒևגãAbmÑ­Ü9׀îԒܦѭÒA؀ѭÜ3Ô¦Ò˜×€Ð­Ï ·á}Ó4ףևשU‚á ·îï’ѐæ !CŒwÆÇÀy¾»È‘ɐ•G€ºÙ(€º» µs»Ôƒ| Wt dŠ Š ‡l¦‘wyŠ ƒx‚„‡±AŒ4‡—ËCx‚wv|’xj‡­§ Þ »ÑCx‚wyŠ ‡‘š Ft “•‡lŠ ‡™| σ °Žº’³¶‰²€Œ˜| ¶‰ƒŒ˜|}z}|¤‹‚ƒ³|™š£¦l|’Œcwvº’¦’ƒw›šg|’³›4|’„ÖÕlŒcwv|™š {Ws}0≤s≤t § êdצχܠÔ1æçχá‡î€ï’Ñ g(ω) å Ft þÜ9׀á‡Ò˜Ï‡Õ4ëì}× ßÚ ·ÒŽÓ4ÑVÒ ·Ð­á ·U‚ØÔ7Ú ·á}Ó4Ï ·Ó·ì lԒÜ9׀á}Ó4גړõ‰Ï‡×¦Ñˆì’Ô ßÑ­Õ¤Ö‡× g(ω) ۖÑdÖ‡×©Ò˜×€Õ–Ö‡×€Ø ·Ö ·Ö‡Ñ Û{× ßÑ­ÒAì’Ô ßѭ՘׀ÒEÖ‡× Ws(ω) ۂԒÕ4Ô s ≤ t ãB–PއҘ׀Վì­×”õ‰Ï‡× Fs ⊂ Ft ۂÔlÕ•Ô s t × õ‰Ï‡× Ft ⊂ F ∀ t ã !CŒwÆÇÀy¾»È‘ɐ•G€ºÙ(€ºÙ( µs»Ôƒ| Υ = Υ(S, T) |V“ƒ³|™š•š‘©z‰f•dx–——瑁•š f(t, ω) : [0, ∞) × Ω → R „¡|’w›š©~€‚ ¾ÍÌ (t, ω) → f(t, ω) œ B×F °Š ƒx7šƒdŒ˜º’¦’ƒ³¢X‡lx{z‰ B Œ4‹{Œ4•š‘ƒx‚„¡|g| σ °Žº’³¶‰²€Œ˜|gz‰m±P‡lŒ4ƒ³‚ƒŠ [0, ∞) § ¾»¾ÍÌkè ‹{Œ4‡‘“••š•š‘‡ f(t, ·) œ Ft °Ž|}z}|c‹{„¡|}z‰‡­§ ¾»¾»¾¥Ì E T S f(t, ω)2 dt ∞ § öEܠԔæçχá‡îï’Ñ ϕ ∈ Υ å£Ø•Ý‚Ô’Ü Ô’Ö‚ÔR× ß׀Ü9׀á}ӕԒÕWÒ˜×£× ßԔÓ4׀ܿԔæçÑ­Õ˜Ü Ô ϕ(t, ω) = j ej(ω) · I[tj,tj+1)(t), ø S™ù Ñ­á‡Ö‡× I å£Ô”æçχá‡îï’Ñ ·á‡Ö ·ØԒևѭÕ4Ô7ãA–PއҘ׀Վì­×gõ‰Ï‡×£Ø€Ñ­Ü9Ñ ϕ ∈ Υ Ú‡ØԒւԔæçχá‡îï’Ñ ej և׃ì}×£Û{׀ՎÓ4׀á‡Ø€×€ÕEÔ Ftj ã‚úXԒÕ4Ԕæçχá‡îVd׀ÒE× ß׀Ü9׀á}Ó•Ô’Õ˜×€Ò ϕ(t, ω) ڂևשU‚á ·Ü9Ñ­ÒAÔ ·á}Ó4׀ЭÕ4Ô ß Ö‡×£û¡Óed¦Ø€Ñ­Ü9Ñ T S ϕ(t, ω) dWt(ω) = j≥0 ej(ω)[Wtj+1 − Wtj ](ω). ø¥X­ù !CŒwÆÇÀy¾»È‘ɐ•G€ºÙ(€ºÙGêég7@¾»À“Á4ŒrÚ Â Ä¼Ûu俌|oÁ4å€Ì µs»Ôƒ| f ∈ Υ(S, T) § Þ »ÑCx‚wyŠ ‡‘š£|9wyx‚„¡¶’Œ˜|’³“z‰ ¨„  ë9z‰ f z‰ S | T “•‡lŠ ‡ T S f(t, ω) dWt(ω) = lim n→∞ T S ϕn(t, ω) dWt(ω), ø R ù ‡lx{z‰ {ϕn} œgdŠ9|Rš‘4~€‚žƒxj“ƒwv|1z‰‚•dx–——瑁•š”ƒ³›ƒŠ ƒx‚„¡|’Œ4•š¤„¡|’w›š”~€‚ E T S (f(t, ω) − ϕn(t, ω))2 dt → 0 quando n → ∞. Sgf
  16. 16. – ߛ·Ü ·Ó4×W׀ܽø R ùF×on ·ÒŽÓ4×E؀ѭÜ9ÑgÏ‡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 ·ÔEև×ub¹Ô’χØcÝw­£×€Ü L2 (P) ãFäÔ ·Ò¤Ö‡×ƒÓ•Ô ß݇׀ҤҘѭއ՘×mÔA؀ђá‡Ò˜Ó4՘χîï’ÑEÖ‚Ô ·á}Ó4׀ЭÕ4Ô ß Ö‡×‚û¡ÓedE×fÒ˜Ñ­Þ‡Õ˜× Ô£×on ·ÒŽÓe˜€á‡Ø ·ÔgւԒÒfæçχá‡îVd×€Ò ϕn ۖÑdև׀ÜҘ׀Õfì ·ÒŽÓ4ѭ҂׀Üì–Bª‰Ò˜×€á‡Ö‚Ô ß ø¥X f‘f‘R ùcãXí Ò˜×€Ð­Ï ·ÕmÔlۇÕ4׃Ò4׃á‰Ó•Ô’Ü9Ñ­Ò Ö‡Ñ ·Ò–Ó4×€Ñ­Õ˜×€Ü Ô’Ò€ã–Û‡Õ ·Ü9× ·Õ˜Ñ æçѭ՘á‡×€Ø€×¦Ô ßЭχܠԒÒPÛ‡Õ˜Ñ­Û‡Õ ·×€Ö‚Ô’Ö‡×€Ò¤Ö‚Ô ·á}Ó4׀ЭÕ4Ô ß Ö‡×Rû¡ÓedV×RÑVҘ׀Эχá‡Ö‡Ñ Ü9ѭҎÓ4Õ4Ôgõ‰Ï‡×P× ßÔgå–Ï‡Ü Ü Ô’ÕŽÓ ·á‡Ð}Ô ßã Xí–ҹև׀Ü9Ñ­á‡ÒŽÓ4Õ4ԒîV‰×€Ò¹Û{Ñdև׀ÜҘ׀Õmì ·ÒŽÓ•Ô’Òm׀܊–Bª‰Ò˜×€á‡Ö‚Ô ß ø¥X f‘f‘R ùcã wŒr  ŒQ‘Ä‘G€ºÙ(€ºÜp µs»Ôƒ|’Š f e g ∈ Υ(S, T)  0 ≤ S U T. ßmx‚„¡¥‰‡ ¾ÍÌ T S f dWt = U S f dWt + T U f dWt ‹‡|’Œ˜|1~€‡|™š‘£„y‡z‰‡ ω ¾»¾ÍÌ T S (cf + g) dWt = c · T S f dWt + T S g dWt ‹‡|’Œ˜|1~€‡|™š‘g„‡z‰‡ ω ¢W‡lx{z‰”“”œR“•‡lx7šƒ„¡|’x‚„ ¾»¾»¾¥Ì E[ T S f dWt] = 0 ¾í½§Ì T S f dWt œ FT °Š ƒx7šƒdŒ˜º’¦’ƒ³§ wŒr  ŒQ‘Ä‘G€ºÙ(€ºÙ… µs»Ôƒ| f(t, ω) ∈ Υ(0, T) ∀ T. ßmx‚„¡¥‰‡ Mt(ω) = t 0 f(s, ω) dWs œ£dŠ¿Š9|’Œc„vwyxd¶}|’³Q“•‡lx‚„vªyx‚‚‡ˆ“•‡lŠ3Œ4•š‹‚ƒwy„‡ | Ft § G€ºÙ(€ºÙ( 7@îÍï  ¡ŽyÛÅğ俌oÁ4å í ·á}Ó4׀ЭÕ4Ô ß Ö‡× û¡Óed ۖÑdև×1Ҙ׀ՔևשU‚á ·Ö‚ÔۂԒÕ4ÔχܠÔØ ßԒҘҘ×1Ö‡× ·á‰Ó4׀ЭÕ4Ԓá‡Ö‡Ñ­Ò f Ü Ô ·Ñ­Õ£õ‰Ï‡× Υ ã ÷–Ô à ׀Ü9Ñ­Ò ·Ò˜Ò˜Ñ¦×€á7æçÕ4Ԓõ‰Ï‡×€Ø€×€á‡Ö‡Ñ1ԒÒA؀ѭá‡Ö ·îVd׀ҩø·›·ùf× ø·›·›·ù‚ւԦٖשU‚á ·îï’Ñ R ãaXdãaX”Û‚Ô’Õ4ԦԒÒAÒ˜×€Ð­Ï ·á}Ó4׀Òæ ¾»¾ÍÌ4ð Χn ·ÒŽÓ4×gχÜ9ԔævԒ܉ ߐ·Ô”؀՘׀Ҙ؀׀á}Ó4×©Ö‡× σ þyë ßЭ׀އÕ4Ô’Ò At; t ≥ 0 Ó•Ô ·Ò¹õ‰Ï‡× Ä“Ì Wt å¤Ï‡Ü3Ü Ô’ÕŽÓ ·á‡Ð}Ô ß Ø€Ñ­Ü3Õ˜×€Ò˜Û–× ·Ó4Ñ Ô At × ñ Ì ft å At þyԒւԒÛ7Ó•Ô’Ö‡Ñ ¾»¾»¾¥ÌYð P T S f(s, ω)2 ds ∞ = 1 !CŒwÆÇÀy¾»È‘ɐ•G€ºÙ(€ºÙ’mÞ ƒxj‡l„¡|’Š ‡‘š£‹‚‡lŒ ΛA(S, T) |™“ƒ³|™š•š‘ˆz‰g‹{Œ4‡‘“••š•š‘‡‘š f(t, ω) ∈ R ~€‚ š|’„vw›šƒ° ƒ|‘™ƒŠWòw×1z}| Þ »ÑCx‚w痙¥‰‡9¬‰§›¯d§›¯…ò¡wyw×5ó¢–ò¡wywyw×5ó7|‰“ƒwyŠ9|}§ í–á}Ó4׀Ҕև×1ևשU‚á ·Õ4ܦѭҔԅæçé­Õ˜Ü”Ï ßÔVևנû¡Óed‡ÚFÛ‡Õ˜×€Ø ·Ò4ԒÜ9ѭҩևשU‚á ·Õ©Ï‡ÜIۇ՘ÑdØ€×€Ò˜Ò˜Ñ Ö‡×1û¡Óed‡ã÷–Ô à ׀Ü9Ñ­Ò ·Ò˜Ò˜Ñ¦Ö‚ÔRÒ˜×€Ð­Ï ·á‰Ó4ףܠÔlá‡× ·Õ4Ôxæ SYh
  17. 17. !CŒwÆÇÀy¾»È‘ɐ•G€ºÙ(€º i µs»Ôƒ| Wt dŠaŠ ‡l¦‘wyŠ ƒx‚„‡9±AŒ4‡—ËCx‚wv|’xj‡VƒŠ (Ω, F, P) § Ê Š‹{Œ•‡‘“••š•š‘‡ˆz‰ ¨„ Öë dx‚wvz’wyŠ ƒx7šƒw‡lx{|’³“œgdŠ ‹{Œ4‡‘“••š•š‘‡ Xt ƒŠ (Ω, F, P) z}|–€‡lŒcŠ9| Xt = X0 + t 0 u(s, ω) ds + t 0 v(s, ω) dWs øÅf‰ù ‡lx{z‰ v ∈ ΛA ¢¹z‰‚€‡lŒcŠ9|1~€‚ P t 0 v(s, ω)2 ds ∞ ∀ t ≥ 0 = 1. Î |’Š9²cœƒŠY|™š•šƒdŠRwyŠ ‡‘šg~€‚ u œ At °Ž|}z}|c‹{„¡|}z}|òŽ‡lx{z‰ At œ”“c‡lŠ ‡ ƒŠ@ò¡wyw×5ó×1 P t 0 |u(s, ω)| ds ∞ ∀ t ≥ 0 = 1. êd× Xt å£Ï‡Ü3ۇ՘щ؀׀ҘҘѠևףû¡Óed‡Ú‚×€á}ӕï’Ñ¦× ßףۖÑdևףҘ׀ÕA×€Ò˜Ø€Õ ·Ó4Ñ9ևפæçѭ՘ܠÔRÔ’Þ‡Õ˜×ƒì ·Ô’Ö‚ÔR؀ѭÜ9Ñ dXt = udt + vdWt. øÍh’ù öEÜoÖ‡Ñ­Ò Õ˜×€Ò˜Ï ßÓ•Ô’Ö‡Ñ­Ò Ü Ô ·Ò ·Ü9ۖѭՎӕԒá‰Ó4׀ҦւԨÓ4×€Ñ­Õ ·Ô…Ö‡× ·á‰Ó4׀ЭÕ4Ԓîï’Ñ…×€ÒŽÓ4ÑdØë’ÒŽÓ ·ØÔ7Ú¹Ñ õ‰Ï‚Ô ß Ô’Û‡Õ˜×cþ Ҙ׀á}ӕԒ՘׀Ü9ѭҔԒВѭÕ4Ô7ÚQå1ԅ՘׀ЭÕ4ԅ؀ѭá‡Ý‡×€Ø ·Ö‚ԅ؀ѭÜ9ÑVæçé­Õ˜ÜRÏ ßԈև×9û¡Óed‡ãVûá7æçÑ­Õ˜Ü Ô ßÜ9׀á}Ó4גÚQ׀Ҏӕԅ՘׀ЭÕ4Ô åRχܠԠì­×€Õ˜Ò4ï’ÑVւԈ՘׀ЭÕ4ԈւÔ1ØÔ’Ö‡× ·ÔˆÖ‡ÑˆØë ßØ€Ï ßÑˆÖ ·æç׀՘׀á‡Ø ·Ô ß Ø ßë’Ò˜Ò ·Ø€Ñ‡ã”úXԒÕ4ԈχܠÔ1ۇ՘ёì’Ô1æçÑ­Õ˜Ü Ô ß Ö‡×€ÒŽÓ4פÓ4׀ѭ՘׀ܠԔì}×yâ˜Ô–Bª‰Ò˜×€á‡Ö‚Ô ß ø¥X f‘f‘R ùcã wŒr  ŒQ‘Ä‘G€ºÙ(€ºÙsêég”sï  ¡ŽyÛÅġ俌|oÁ4冎yÀy¾»äy¾»‘ŒQÀyg¾Å“À¿Ä¼ÛÍÌ µs»Ôƒ| Xt dŠ ‹{Œ4‡‘“••š•š‘‡Dz‰ ¨„ Öë dx‚wvz’wyŠ ƒx7šƒw‡lx{|’³sz}|‰z‰‡–‹–‡lŒ dXt = udt + vdWt. µs»Ôƒ| g(t, x) ∈ C2 ([0, ∞) × R) ò¡w›šƒ„‡sœ•¢”¶Tœ¨z’‡|™š¦’4™•š¨“•‡lx‚„vwyx‚‡|’Š ƒx‚„¨z’w€ƒŒ4ƒxj“ƒwvº’¦’ƒ³–ƒŠ [0, ∞) × R) §²ßmx‚„¡¥‰‡ Yt = g(t, Xt) œ£„¡|’Š9²cœƒŠ3dŠ ‹{Œ4‡‘“••š•š‘‡ z‰ ¨„ Ö둢W 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) œ”“4|’³›“ƒd³|}z‰‡ z‰”|‰“•‡lŒ˜z‰‡8ݙš£Œ4¡¶’Œ˜|™š dt · dt = dt · dWt = dWt · dt = 0, dWt · dWt = dt. úXԒÕ4Ô©Ô£æçé­Õ˜ÜRÏ ßԤևזû¡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’Ü9ѭ҂זۇ՘ёì’Ô’Ü9ѭ҂ۂԒՎÓ4× Ö‡Ñ”×on7׀՘؉Ø ·Ñf‡ãR ևץ–Bª‰Ò˜×€á‡Ö‚Ô ß ø¥X f‘f‘R ùcډєõ‰Ï‚Ô ß Ò˜×€Õ4ë”Ï7Ó ·›ß›·à ԒևÑgá‡Ñ”ØԒې‰Ó4Ï ßєñgÖ‡×€ÒŽÓ•Ô”Ö ·Ò˜Ò˜×€ÕŽÓ•Ô’îï’чã S‘ñ
  18. 18. H  Fôõ“g¾»È‘ɐ†G€ºÙ(€ºÙ' µs»Ôƒ| Xt  Yt ‹{Œ4‡‘“••š•š‘‡‘šgz‰ ¨„ Öë1ƒŠ R §•ßmx‚„¡¥‰‡ d(XtYt) = XtdYt + YtdXt + dXtdYt. Ó Œ4‡l¦l|}® êd×yâ˜Ô’Ü Wt =   Xt Yt   , Ñ­ÏVҘ×yâ˜Ô7Ú Wt å£Ï‡Ü3ۇ՘Ñd؀׀ҘҘѠևףû¡Óed¦Þ ·Ö ·Ü9׀á‡Ò ·Ñ­á‚Ô ß øí©f×yâ˜Ô¦í–á‡×on7Ñ9ûŽùcã êd×yâ˜Ô Z(t, ω) = g(t, Wt) = xy ãfúX× ßԔæçé­Õ˜Ü”Ï ßԩևףû¡Óed9ádþÖ ·Ü9׀á‡Ò ·Ñ­á‚Ô ßÚ dáFöDXdڇÓ4׀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  5÷‘ŒQg†ä¿Œ½Ä  ¾Åļȑɐ†ø€Ž¿Ä¼ä ÂYù ÁY¾»÷‘Ä !CŒwÆÇÀy¾»È‘ɐ•G€ºÙ(€º»r{ µs»Ôƒ| Nt(.) : Ω → R dŠ ‹{Œ4‡‘“••š•š‘‡1•šƒ„‡‘“4º™šƒ„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. © †‡|’Š9|’Š ‡‘š [N]t |1¦l|’Œcwv|­—™¥‰‡1~€‡|}z’Œ˜º’„vw“4| z‰ N xj‡9wyx7šƒ„¡|’x‚„£„y¢A [N] = {[N]t, t ≥ 0} ‡¤‹{Œ4‡‘“••š•š‘‡¦¦l|’Œcwv|­—™¥‰‡9~€‡|}z’Œ˜º’„vw“•|1|™š•š‘‡‘“ƒwv|}z‰‡9| N § –ÃÜ Ô’ÕŽÓ ·á‡Ð}Ô ß Mt ևשU‚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׀ѭ՘׀ܠԒÒ9õ‰Ï‡×Ò˜×€Ð­Ï‡×€Ü Ò˜×€Õ4ï’љևׅæçχá‡Ö‚Ô’Ü9׀á}Ó•Ô ßA·Ü9Û{ѭՎӘ—’á‡Ø ·Ô á‡Ñ™ØԒې‰Ó4Ï ßхñ7Ú¹á‡Ñ õ‰Ï‚Ô ß Ö ·Ò˜Ø€Ï7Ó ·Ü¦Ñ­ÒPшÜ9Ñ‰Ö‡× ßÑ1և×RÖ ·æçχÒ4ï’шۂԒÕ4ԈÔ1՘׀Ҙ׀ՎìlÔVև×RχܠԠÒ4׃ЭχÕ4ԒևѭÕ4Ô7ブ Û‡Õ ·Ü9× ·Õ˜Ñ å”Ñ pQ׀ђÕ4׃ܠÔ9ñ7ãTS£×£Ñ¦Ò˜×€Ð­Ï‡á‡Ö‡Ñ å¤Ñ bmÑ­Õ˜Ñ ßë’Õ ·Ñƒf‡ãah”Ö‡×Ebm݉χá‡Ð8¨ú« ·›ß›ß·Ô’ܦңø S4„‘„ f ùcã wŒr  ŒQ‘Ä‘G€ºÙ(€º»F‹Ê ŠÅ‹{Œ4‡‘“••š•š‘‡ W = {Wt, t ≥ 0} œgdŠ3Š ‡l¦‘wyŠ ƒx‚„‡©±AŒ4‡—ËCx‚wv|’xj‡ ƒŠ R š‘” š‘‡lŠ ƒx‚„Wš‘ggÒ­w›šƒ„–dŠ9|sÑC³´„vŒ˜|­—™¥‰‡ {At} „¡|’³j~€‚ W œP‰Š Š9|’Œc„vwyxd¶}|’³–³›‡‘“4|’³s“•‡lŠŒ4•š‹‚ƒwy„‡R| {At} ”“•‡lŠ¿¦l|’Œcwv|­—™¥‰‡9~€‡|}z’Œ˜º’„vw“4| [W] š|’„vw›švƒ|‘™ƒx{z‰‡ [W]t = t q.c. ∀ t. SYi
  19. 19. wŒr  ŒQ‘Ä‘G€ºÙ(€º»Q( µs»Ôƒ| M = {Mt, t ≥ 0} dŠ Š9|’Œc„vwyxd¶}|’³P³›‡‘“4|’³£“•‡lx‚„vªyx‚‚‡d™z‰¨¦l|’Œcwv|­—™¥‰‡ ³´wyŠRwy„¡|}z}|9³›‡‘“4|’³´Š ƒx‚„§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{ȧ£SÁ Í1ÊÅÄ{ÆÇÁ ÈËÊ Ì ý ±´³ þ r‡iWijq“qdqdi†•–q r‡q“qdqsr7‰X€ •–q r‡vyqdiji üE×€ÒŽÓ•Ô Ò˜×€îï’Ñ Ö ·Ò˜Ø€Ï7Ó ·Ü9Ñ­Ò Ñ Ü9Ñ‰Ö‡× ßÑ Ø ßë’Ò˜Ò ·Ø€Ñ և×Õ ·Ò˜Ø€Ñ‡ã í–Ò˜Ò˜Ï‡Ü ·Ü9Ñ­Ò õ‰Ï‡×¨Ô™×ƒì}Ñ ßχîï’Ñ Ö‚Ô Õ˜×€Ò˜×€ÕŽìlÔ U = {Ut, t ≥ 0} Ö‡× Ï‡Ü Ôs؀ѭÜ9ۂԒá‡Ý ·ÔsÖ‡× Ò˜×€Ð­Ï‡Õ˜Ñ­Òå ׀ܔނԒÒ4ԒւÔT×€Ü Ï‡Ü ×€Ò˜Û‚Ô’î€Ñ և×Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡× (Ω, F, P) ؀ѭÜ9Ñ Ò˜×€Ð­Ï‡×’ãÅêd×yâ˜Ô U0 = u 0 ԙ՘׀Ҙ׀ՎìlÔ ·á ·Ø ·Ô ßã úCÕg˜€Ü ·Ñ­Ò ؕ݇׀Ð}Ô’Ü Ø€Ñ­á}Ó ·ádςÔlÜ9׀á‰Ó4×EԣχܠÔPӕÔYn‡Ô£Ø€Ñ­á‡ÒŽÓ•Ô’á‰Ó4× p 0 × ·á‡Ö‡×€á ·›à ԒîV‰×€Ò‚Ò4ï’ÑgۂԒÐ}ԒÒf×€Ü ·á‡ÒŽÓ•Ô’á‰Ó4×€Ò Ô ß×ÔlÓ4é­Õ ·Ñ­Ò T1, T2, . . . (0 T1 T2, . . .) Ñ­á‡Ö‡×gÑ­ÒAì’Ô ßѭ՘׀ÒEԦҘ׀՘׀ÜYۂԒЭѭÒEá‡×€Ò˜Ò˜×€Ò ·á‡ÒŽÓ•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, . . . . –)ۇ՘Ñd؀׀ҘҘѣև×WÕ ·Ò˜Ø€ÑPÕ˜×€Ò˜Ï ßӕԒá‰Ó4×’Ú Û‚Ô’Õ4Ô t 0 ڇå Ut = u + pt − St, øÍi’ù Ñ­á‡Ö‡× St = i≥1 XiI(Ti ≤ t), × I åÔ æçχá‡îï’Ñ ·á‡Ö ·ØԒևѭÕ4Ô7ãsٖשU‚á ·Ü9Ñ­Ò9Ô ì’Ô’Õ ·ëì­× ß Ô ß×ÔlÓ4é­Õ ·Ô τ = inf{t ≥ 0 : Ut ≤ 0} ãx–ì’Ô ßÑ­Õ£Ö‡× τ á‡Ñ­Ògւë…Ñ ·á‡ÒŽÓ•Ô’á‰Ó4צւԅ՘ϐ‰›á‚Ô7ã1ٖ׃ì­×€Ü9Ñ­ÒgӕԒܔÞ{å€Ü ؀ѭá‡Ò ·Ö‡×€Õ4ԒÕVÔ Û{Ñ­Ò˜Ò ·Þ ·›ß›·Ö‚Ô’Ö‡×ևרá‚ï’Ñu݂Ôì­×€ÕV՘ϐ‰›á‚Ô7ږÑuõ‰Ï‡×¨Ò ·Ð­á ·U‚ØÔ õ‰Ï‡× τ = ∞ ãÎCҎӕԒÜ9Ñ­Ò ·á}Ó4׀՘׀ҘÒ4ԒևѭҖá‚Ô9Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ôlևףևש՘ϐ‰›á‚Ԧ׀ܿÓ4׀Ü9Û{Ñ ·áxU‚á ·Ó4Ñ P(τ ∞) Ú{שá‚Ô9Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡× ևף՘ϐ‰›á‚ÔRԒá‰Ó4׀ÒAևÑRÓ4׀Ü9Û{Ñ t Ú P(τ ≤ t) ã ý ±¥à noiW•–q“xyi†ijxyffqdqdvyiji•–qѨr‡€‚†psr‰k—ÿ”h–’–•–¹gqsr‡p –ÅÜ9Ñ‰Ö‡× ßÑRØ ßë’Ò˜Ò ·Ø€Ñ¦Ö‡×EbmÕ4ԒÜ9å€ÕAý“χá‡Ö‡Þ{׀՘Ð9å£ØԒÕ4Ԓ؃Ó4×€Õ ·›à ԒևѦÛ{× ßԒÒWÒ˜×€Ð­Ï ·á‰Ó4׀ÒAҘχÛ{Ñ­Ò ·îV‰×€Òæ • –PÒXì’Ô ßѭ՘׀ÒXÖ‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîVd×€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 ·ÒŽÓ4Õ ·Þ‡Ï‰›Ö‚Ô’ÒR؀ѭÜoæçχá‡îï’Ñ™Ö‡×…Ö ·ÒŽÓ4Õ ·Þ‡Ï ·îï’Ñ F Ú¹Ü9åƒÖ ·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 ԒîVd׃ҦÑdØ€Ñ­Õ˜Õ˜×€Ü Ò4ï’Ñ Ó•Ô ·ÒRõ‰Ï‡×ԒҔì’Ô’Õ ·ëì­× ·Ò Ô ß×ÔlÓ4é­Õ ·Ô’Ò Zi = Ti − Ti−1, i ≥ 1 ҕïlÑ ·á‡Ö‡×€Û{׀á‡Ö‡×€á}Ó4×€Ò£× ·Ö‡×€á‰Ó ·Ø€Ô’Ü9׀á‰Ó4×©Ö ·ÒŽÓ4Õ ·Þ‡Ï‰›Ö‚Ô’Ò€Új؀ѭÜaÖ ·ÒŽÓ4Õ ·Þ‡Ï ·îï’Ñ1×on7Û{ђá‡×€á‡Ø ·Ô ß Ø€Ñ­Ü E(Z1) = λ−1 ã • –Åá‰ð‡Ü9×€Õ˜Ñ¦Ö‡× ·á‡Ö‡×€á ·›à ԒîVd׀ÒAá‡Ñ ·á}Ó4׀Վì’Ô ßÑ [0, t] å£Ö‡×€á‡Ñ’Ó•Ô’Ö‡Ñ9Û–Ñ­Õ Nt = sup{n ≥ 1 : Tn ≤ t}, t ≥ 0, S4„
  21. 21. Ñ­á‡Ö‡× sup{∅} = 0. • í–ÒEҘ׀õ‰Ï˜€á‡Ø ·Ô’Ò X1, X2, . . . × T1, T2, . . . Ò4ï’Ñ ·á‡Ö‡×€Û–×€á‡Ö‡×€á‰Ó4׀Ҁã öEÜ ÔR؀ѭá‡Ò˜×€õ‰Ï˜€á‡Ø ·Ô9ւԦևשU‚á ·îï’Ñ Ô’Ø ·Ü Ô”ågõ‰Ï‡× Nt ågχÜ3ۇ՘Ñd؀׀ҘҘѠևשúQÑ ·Ò˜Ò˜Ñ­á…؀ѭÜ!ӕÔYn‡Ô λ. – Û‡Õ˜Ñ‰Ø€×€Ò˜Ò˜Ñ Ö‡×£Õ ·Ò˜Ø€Ñ¦Õ˜×€Ò˜Ï ßӕԒá‰Ó4גڇۂԒÕ4Ô t 0 ڂå Ut = u + pt − St ø»™}ù Ñ­á‡Ö‡× St = Nt i=1 Xi ã‚úQÑdևוþÒ4×£Ü9ѭҎÓ4Õ4ԒÕgøí©fאâ˜Ô¦í–á‡×on7Ñ9û˜ûŽù¹õ‰Ï‡× E(St) = E Nt i=1 Xi = λµt ø»„}ù × V ar(St) = V ar Nt i=1 Xi = λt(σ2 + µ2 ). ø S f ù êd×yâ˜Ô ρ = λµ ÑuìlÔ ßÑ­Õ1Ó4Ñ’Ó•Ô ß ×€Ò˜Û{׀Õ4ԒևÑTÖ‡× ·á‡Ö‡×ƒá ·›à ԒîVd׀ÒVÛ{Ñ­ÕˆÏ‡á ·Ö‚Ô’Ö‡× Ö‡× Ó4׀Ü9Û{чãöEÒ4Ԓá‡Ö‡Ñ Õ˜×€Ò˜Ï ßӕԒևѭÒgÒ˜Ñ­Þ‡Õ˜× Û‚Ô’Ò˜Ò˜× ·Ñ­Ò”Ô ß×ÔlÓ4é­Õ ·Ñ­Ò€ÚQå Û–Ñ­Ò˜Òg‰ì}× ß Û‡Õ˜Ñ‘ì’Ô’Õ1øyì}×yâ˜Ôêdχá‡Ö7ӑÚsSg„‘„ R ù¤õ‰Ï‡× Ò˜× p ρ Ú ×€á}ӕï’Ñ Ut → ∞ õ–ãؒãCõ‰Ï‚Ô’á‡Ö‡Ñ t ↑ ∞ ׀á‡õ‰Ï‚Ô’á}Ó4ÑRÒ˜× p ρ Úd׀á}ӕï’Ñ Ut → −∞ õ–ãؒãfõ‰Ï‚Ô’á‡Ö‡Ñ t ↑ ∞ ã úQ× ßхҘ׀Эχá‡Ö‡Ñ…Õ4׃Ò4Ï ßÓcÔlÖ‡Ñ‡Ú ψ(u) = 1 ∀ u ã ÎCÒŽÓ•Ô…Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚ԒևהӕԒܔÞ{å€Üaå ·Ð­Ï‚Ô ß Ô¡S¦á‡Ñ…ØÔ’Ò˜Ñ p = ρ Ú{ԒÛ{׀Ò4ԒÕEև×gá‚ï’Ñ9݂Ôì­×€Õ–Õ˜×€Ò˜Ï ßӕԒևÑ9և×g؀ѭá}ì­×€Õ˜Ð‘˜€á‡Ø ·Ô7ãAêd׀á‡Ö‡Ñ1Ô’Ò˜Ò ·Ü…ڂχܠԦ؀ѭá‡Ö ·î€ï’Ñ9é­Þ‰ì ·Ô Ö‡×£Ò˜Ñ ß쑘€á‡Ø ·ÔRå£Ò4ÔlÓ ·ÒŽæç× ·Ó•Ô”Ò˜× p = (1 + θ)λµ, θ 0. ø S‘S™ù íÅ؀ѭá‡ÒŽÓ•Ô’á}Ó4× θ ågØc݂ԒܠÔlÖ‚Ô “•‡‘»Ñm“ƒwƒx‚„”z‰£š‘¡¶’dŒ˜|’x–—™| ã¹úFԒÕ4Ô9Ü9Ô ·ÒAÖ‡×ƒÓ•Ô ß݇׀ÒAì}×yâ˜Ô ÎCܔއ՘׀ØcÝ}Ó4Ò–× Ñ­Ï7Ó4՘ѭÒgø¥X f‘f‘R ùcãCÎCÜYê ·›ßìlԃ¨Åí¹Ó4χá‡ØԒÕgø¥X f‘f f‰ùcڇѭÒAԒÏ7Ó4ѭ՘׀ÒWÜ9ѭҎÓ4Õ4ԒÜ!Õ˜×€Ò˜Ï ßӕԒևѭÒWÖ‡×¤Ò ·ÜRÏ ßÔlîVd×€Ò Ö‡×€ÒŽÓ4×£Ü9Ñ‰Ö‡× ßÑRۂԒÕ4ԔѦØë ßØ€Ï ßÑRÖ‚Ô”Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ԓևזևף՘ϐ‰›á‚Ô”×€Ü Ó4׀Ü9Û{ÑU‚á ·Ó4ÑRף؀ѭܦۂԒÕ4ԒÜ!Ø€Ñ­Ü Ñ­ÒWÕ˜×€Ò˜Ï ßӕԒևѭÒAÑ­Þ7Ó ·Ö‡Ñ­ÒWÛ{× ßÔRԒۇ՘Ñ4n ·Ü Ô’îï’Ñ”Û{Ñ­ÕAÖ ·æçχÒ4ï’Ñ7ã X f
  22. 22.   Æ Ì ÀSÁ € Ì Â Ì Æ Ä{ȧ£SÁ Í1ÊÅÄ{ÆÇÁÉÈËÊÌ ¡•±´³ µ q“qdi{r‡v ·£¢fi•–i †iW•–q“xyi bmÑ­Ü9ѠԒá}Ó4×€Õ ·Ñ­Õ˜Ü9׀á}Ó4גڂҘ×yâ˜Ô Xi ѦìlÔ ßÑ­ÕWÖ‚Ô ·þå€Ò ·Ü Ô ·á‡Ö‡×€á ·›à Ԓîï’Ѧõ‰Ï‡×©Ô’؀ѭá‰Ó4׀؀×gá‡Ñ ·á‡ÒŽÓ•Ô’á}Ó4× Ti × Nt Ñ á‰ð‡Ü9×€Õ˜Ñ Ö‡× ·á‡Ö‡×€á ·›à ԒîVd׀ҠÑdØ€Ñ­Õ˜Õ ·Ö‚Ô’Ò9á‡Ñ ·á}Ó4׀ՎìlÔ ßљև×Ó4׀Ü9Û{Ñ (0, t] ã¢í–Ò˜Ò˜Ï‡Ü ·Ü9Ñ­Ò õ‰Ï‡× {Nt, t ≥ 0} å Ï‡ÜIÛ‡Õ˜Ñ‰Ø€×€Ò˜Ò˜Ñ Ö‡×1úQÑ ·Ò˜Ò˜Ñ­á™Ý‡Ñ­Ü9ѭБ˜€á‡×€Ñ؀ѭÜSӕÔYn‡Ô λ ×9õ‰Ï‡× Ñ­ÒgìlÔ ßѭ՘׀ÒgÖ‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîV‰×€Ò“Ò4ï’ÑAÜRÏ7Ó4ςԒÜ9׀á}Ó4× ·á‡Ö‡×€Û{׀á‡Ö‡×€á}Ó4׀ÒQ× ·Ö‡×€á}Ó ·ØԒÜ9׀á}Ó4×FÖ ·ÒŽÓ4Õ ·Þ‡Ï‰›Ö‡Ñ­ÒsØ€Ñ­Ü Ö ·ÒŽÓ4Õ ·Þ‡Ï ·îï’Ñ F ڒѭá‡Ö‡× E(X) = µ ãFêd׀á‡Ö‡Ñ¤Ô’Ò˜Ò ·Ü…ÚlÑEìlÔ ßѭդ׀ҘÛ{׀Õ4ԒևÑPÖ‡× ·á‡Ö‡×€á ·›à ԒîVd׀ÒQÛ{Ñ­ÕQÏ‡á ·Ö‚Ô’Ö‡×‚Ö‡×fÓ4׀Ü9Û{ÑEå Ö‚Ô’Ö‡Ñ”Û–Ñ­Õ ρ = λµ ãõpXԒܔÞ{å€Ü Ô’Ò˜Ò˜Ï‡Ü ·Ü9ѭ҂õ‰Ï‡×Pѭ҂ì’Ô ßѭ՘׀ÒmÖ‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîVd׀ÒmÒ4ï’Ñ ·á‡Ö‡×€Û{׀á‡Ö‡×€á}Ó4×€Ò Ö‡Ñ¦Û‡Õ˜Ñd؀׀ҘҘѠևѦá‰ð‡Ü9×€Õ˜Ñ¦Ö‡× ·á‡Ö‡×ƒá ·›à ԒîVd׀Ҁã êd×yâ˜Ô St =    Nt i=1 Xi Ò˜× Nt 0, 0 Ò˜× Nt = 0. ø SYX­ù Ñ ì’Ô ßђÕ1Ó4Ñ’Ó•Ô ß Ö‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîVd׀҅ԒЭ՘׀ЭԒւԒÒVá‡Ñ ·á‰Ó4׀ՎìlÔ ßÑ Ö‡× Ó4׀Ü9Û{Ñ (0, t] 㠖ËۇÕg˜€Ü ·Ñ õ‰Ï‡× ԙ؀ѭÜ9ۂԒá‡Ý ·Ô…Ö‡× Ҙ׀Эχ՘ѭ҈՘׀؀׃Þ{× å Û‚Ô’Ð­Ñ Ø€Ñ­á}Ó ·á‰Ï‚Ô’Ü9׀á‰Ó4×؀ѭÜ2χܠԅӕÔYn‡Ô ؀ѭá‡ÒŽÓ•Ô’á}Ó4× p ã í ؀ѭÜ9ۂԒá‡Ý ·ÔRӕԒܔÞ{å€Ü¿Õ˜×€Ø€×€Þ{×EâŽÏ‡Õ˜Ñ­ÒEևשҘςԠ՘׀Ҙ׀Վì’Ԡ؀ѭÜYχܠԦӕÔYn‡Ô9؀ѭá‡ÒŽÓ•Ô’á‰Ó4× δ ã–êd×yâ˜Ô Uδ(t) Ñ ìlÔ ßÑ­ÕQւԖ՘׀Ҙ׀Վì’ÔPá‡Ñ ·á‡ÒŽÓ•Ô’á}Ó4ׂÓãõbmÑ­ÜÅԒÒFÒ˜Ï‡Û–Ñ­Ò ·îVd׀ÒCÔ’Ø ·Ü Ô7Ú­êdχá‡Ö7ÓA¨êpQ×€Ï‡Ð­× ßÒAø S4„‘„wh­ù“Ü9ѭҎÓ4Õ4Ô’Ü õ‰Ï‡× Uδ(t) = ueδt + p t 0 eδυ dυ − t 0 eδ(t−υ) dSυ, ø S R ù Ø€Ñ­Ü u = U(0) ãPü–Ô9×on7ۇ՘׀Ҙҕï’ÑVÔ’Ø ·Ü Ô7Ú ueδt å”Ñ9ì’Ô ßѭՖÔlÓ4Ï‚Ô ß Ö‚Ô9՘׀Ҙ׀Վì’Ô ·á ·Ø ·Ô ß á‡Ñ ·á‡ÒŽÓ•Ô’á}Ó4× t Ú t 0 peδυ dυ å¤ÑRì’Ô ßÑ­ÕWÔlÓ4Ï‚Ô ß Ö‡ÑRÓ4Ñ’Ó•Ô ß Ö‡×£Û‡Õg˜€Ü ·Ñ­Ò¹Õ˜×€Ø€×€Þ ·Ö‡Ñ­ÒAÛ–× ßÔRҘ׀ЭχÕ4ԒևѭÕ4Ô¦ÔlÓ4å¤Ñ ·á‡ÒŽÓ•Ô’á}Ó4× t × t 0 eδ(t−υ) dSυ å¦ÑˆìlÔ ßÑ­Õ£ÔlÓ4Ï‚Ô ß Ö‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîVd׀ҤۂԒÐ}ԒҤÛ{× ßԈҘ׀ЭχÕ4ԒևѭÕ4ÔVÔlÓ4åRÑ ·á‡ÒŽÓcÔlá‰Ó4× t 㥤 ·Ü9ۖѭՎӕԒá‰Ó4×mѭއҘ׀ՎìlԒÕfõ‰Ï‡×EÔ¤ð ßÓ·Ü 9Ô ·á}Ó4׀ЭÕ4Ô ß á‚Ô¤×on7ۇ՘׀ҘÒ4ï’Ñ1ø S R ùQåA؀ѭá‡ÒŽÓ•Ô’á}Ó4×A׀á}Ó4՘×WÑ­Ò ·á‡Ò˜Ó•Ô’á}Ó4×€Ò Ö‚Ô’ÒAؕ݇׀Ð}ԒւԒÒAÖ‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîVd׀Ҁã bmÑ­á‡Ò ·Ö‡×€Õ˜×¤Ó•Ô’Ü”Þ{å€Ü!Ñ ¦l|’³›‡lŒgz‰•š‘“•‡lx‚„¡|}z‰‡ Ö‡× Ut á‡ÑRÓ4׀Ü9Û{Ñ à ×€Õ˜Ñ‡Ú ·ÒŽÓ4Ѧå’Ú Vδ(t) = e−δt Uδ(t) = u + p a (δ) t − t 0 e−δυ dSυ ø Sgf‰ù Ø€Ñ­Ü a (δ) t =    t Ò˜× δ = 0, ou 1−e−δt δ Ò˜× δ 0. –PÜ ·Ó·Õ ˜×€Ü¦Ñ­Ò¹Ñ‰›á‡Ö ·Ø€× δ õ‰Ï‚Ô’á‡Ö‡Ñ Ô”Ó•ÔYn‡ÔRև׹âŽÏ‡Õ˜Ñ­ÒWæçÑ­Õ à ׀՘чã XQS
  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‚Ԓևפև×g՘ϐ‰›á‚Ԧ׀Ü3Ó4׀Ü9Û{Ñ ·áxU‚á ·Ó4ÑRև×gχܠԦ؀ѭÜ9ۂԒá‡Ý ·ÔRև×gÒ˜×€Ð­Ï‡Õ˜Ñ­Ò Ø€Ñ­Ü3՘׀Ҙ׀ՎìlÔ ·á ·Ø ·Ô ß u ãfÎCá}ӕï’Ñ ψδ(u) = P t≥0 (Uδ(t) 0) = P t≥0 (Vδ(t) 0) . ÎCÜaêdχá‡Ö7ÓD¨ pQ×€Ï‡Ð­× ßÒ¦ø S4„‘„wh­ùcÚ{Ñ­ÒPԒÏ7Ó4ѭ՘׀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ÒŽÓ4Ñ ·Ü9Û ß·ØÔ õ‰Ï‡× ψδ(υ) ≤ ψ(υ) ÚsÑ­Ï Ò˜×yâ˜Ô7ړшÜ9Ñ‰Ö‡× ßÑ1Ø ßë’Ò˜Ò ·Ø€Ñ1Ø€Ñ­Ü δ = 0 æçѭ՘á‡×€Ø€×£Ï‡Ü ߛ·Ü ·Ó˜×PҘχÛ{×€Õ ·Ñ­ÕAۂԒÕ4Ô¦ÔRÛ‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Pևף՘ϐ‰›á‚ÔRá‡Ñ¦ØԒҘÑ9Э׀Õ4Ô ß δ ≥ 0 ã úXԒÕ4Ô Ï‡Ü ÔÜ9× ß݇ѭՔ؀ѭÜ9ۇ՘׀׀á‡Ò4ï’Ñ¨Ö‚Ô’Ò¦Ö ·æç׀՘׀á‡îԒҔæçχá‡Ö‚Ô’Ü9׀á}Ó•Ô ·ÒR׃á‰Ó4՘×VÑ­ÒRØÔ’Ò˜Ñ­Ò δ = 0 × δ 0 Úsì­×yâ˜Ô’Ü9Ñ­Ò¤ÑV؀ѭÜ9Û{ѭՎӕԒÜ9׀á}Ó4шևÑVÛ‡Õ˜Ñ‰Ø€×€Ò˜Ò˜Ñ {Vδ(t), t ≥ 0} á‡×€ÒŽÓ4׀ÒgÖ‡Ñ ·ÒPØԒҘѭҀã¦üEÑVØÔ’Ò˜Ñ δ = 0 ڂѦ؀ѭÜ9Û{ѭՎӕԒÜ9׀á}Ó4ÑRÖ‡× Vt = Ut = u + pt − St æçÑ · Ö‡×€Ò˜Ø€Õ ·Ó4Ñ9á‚ÔRҘ׀îï’Ñf‡ãaXdã üEÑ1ØÔ’Ò˜Ñ δ 0 á‚ï’Ñ1݂ë1؀ѭá}ì­×€Õ˜Ð‘˜€á‡Ø ·ÔˆÛ‚Ô’Õ4Ô +∞ Ñ­Ï −∞ ×”Ò ·ÜYۂԒÕ4Ô1Ô ìlÔ’Õ ·ëì}× ß Ô ß×ÔlÓ4é­Õ ·Ô U‚á ·Ó•Ô Vδ(∞) = u + p δ − ∞ 0 e−δυ dS(υ). ø SYh­ù úXԒÕ4Ԕì}׀ÕAõ‰Ï‡×gÔ ·á}Ó4׀ЭÕ4Ô ß á‚Ô¦×on7ۇ՘׀ҘÒ4ï’Ñ ø SYh­ùmå¥U‚á ·Ó•Ô7ڇނԒҎӕԔævÔ à ×€Õ ∞ 0 e−δυ dS(υ) = i≥1 e−δTi Xi. í ·Ð­Ï‚Ô ßÖ‚Ô’Ö‡×‚Ô’Ø ·Ü ÔAå‚ìlë ߛ·Ö‚ÔAÛ{Ñ ·ÒXÔ ·á‰Ó4׀ЭÕ4Ô ß ó–×€Ò˜õ‰Ï‡×€Õ˜Ö‚ÔPåm؀ѭá‡ÒŽÓ•Ô’á}Ó4×m׀á‰Ó4՘×mÑ­Ò ·á‡ÒŽÓ•Ô’á}Ó4׀ÒX׀ÜÃõ‰Ï‡× ؕ݇׀Ð}ԒÜԒÒmÕ˜× ·á‰ì ·Ö ·ØԒîV‰×€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–Ñdև׀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) ∞ ڂ׀á}ӕï’Ñ ∞ 0 e−δυ dS(υ) ∞ q.c ã ٖ׀ҎӕÔ1æçѭ՘ܠÔ7Ú{á‚ï’Ñ1å”é­Þ}ì ·Ñ1õ‰Ï‡× ψδ(u) = 1 Ҙ׀Ü9ۇ՘הõ‰Ï‡× p ρ ã£üEÑ1׀á‰Ó•Ô’á}Ó4чÚsÒ˜× p ≤ −δu ׀á}ӕï’Ñ ψδ(u) = 1 ãfúFԒÕ4ÔRÜ Ô ·Ò¹Ö‡×ƒÓ•Ô ß݇׀ÒAҘѭއ՘ף׀ҎÓ4×gÜ9Ñ‰Ö‡× ßчÚdì}×yâ˜Ô9êdχá‡Ö7ÓB¨úpQ×€Ï‡Ð­× ßÒ©ø S4„‘„wh­ùcã X‘X
  24. 24. ¨ Æ Ì ÀSÁ € Ì ÀSÁ ÀSÄ©všËÍ8½Ì üE׀ҎÓ4×gØԒې‰Ó4Ï ßѦԒۇ՘׀Ҙ׀á}ӕԒÜ9Ñ­ÒEԦԒۇ՘Ñ4n ·Ü9Ԓîï’ÑRÛ{Ñ­ÕWÖ ·æçχÒ4ï’Ѧۇ՘ѭÛ{ѭҎӕÔRÛ{Ñ­ÕAüEѭ՘Þ{׀՘Ðø S4„‘„‘„}ù ۂԒÕ4Ô¹Ü9ÑdÖ‡× ßԒÕsÔW՘׀Ҙ׀ՎìlÔAև×fχܠԹҘ׀ЭχÕ4ԒևѭÕ4ÔW×fԹӕÔYn‡ÔWԒ؀χÜRÏ ßԒւÔmև×jâŽÏ‡Õ˜Ñ­Ò¤óWõ‰Ï‚Ô ß ÔW՘׀Ҙ׀Վì’ÔA׀Ҏӕë ҘÏlâŽ× ·Ó•Ô7ã–bm݂ԒܠԒÜ9Ñ­ÒAÑ Õ ·Ò˜Ø€Ñ Õ˜× ßÔ’Ø ·Ñ­á‚Ô’Ö‡Ñ9ó9Õ4׃Ò4׃՘ìlÔ1Ö‡× Œcw›š‘“•‡ˆz‰£š‘¡¶’dŒ•‡ ×©Ñ Õ ·Ò˜Ø€Ñ Õ˜× ßÔ’Ø ·Ñ­á‚Ô’Ö‡Ñ Ô’Ñ ·á‰ì­×€ÒŽÓ ·Ü9׀á}Ó4ÑÖ‡× Œcw›š‘“•‡ÑCx{|’xj“•ƒwyŒ4‡ ã…ê7Ô ß›·×€á‰Ó•Ô’ܦѭңõ‰Ï‡× ×€ÒŽÓ•ÔԒۇ՘Ñgn ·Ü Ô’îï’Ñ…å9ìlë ߛ·Ö‚Ô…Ô’Û–×€á‚Ô’Ò ۂԒÕ4ԈÑVØԒҘх׀ÜSõ‰Ï‡× ÔˆÖ ·ÒŽÓ4Õ ·Þ‡Ï ·îï’ÑˆÖ‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîVd׀ңå¦Ö‡×¦ØÔ’Ï‡Ö‚Ô ß×cì}גڤÑVõ‰Ï‡×9Ò ·Ð­á ·U‚ØÔ1õ‰Ï‡× á‡×€á‡Ý‰Ï‡Ü Ô£Ö‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîVd×€Ò ·Ò˜Ñ ßԒւԒÜ9׃á‰Ó4×WåAЭÕ4Ԓá‡Ö‡×AÑgҘÏxU‚Ø ·×€á}Ó4×EۂԒÕ4ÔgÔlæç׃ӕԒÕCÑ£ìlÔ ßÑ­ÕFÓ4Ñ’Ó•Ô ß Ö‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîV‰×€ÒEԒЭ՘׀Ð}ԒւԒÒWÒ ·Ð­á ·U‚ØÔlÓ ·ì’Ԓܦ׀á}Ó4ג㠲±´³ ƒ…if’–ijq“v¡”7ifq•¹Pffqdvyijifq ’€º»¼º» §ô  5÷‘ŒQg†ä¿Ä  ŒQŒ  ½Ä bmÑ­á‡Ò ·Ö‡×€Õ˜×”χܠÔ9؀ѭÜ9ۂԒá‡Ý ·Ô9ևשҘ׀Эχ՘ѭÒP؀ѭÜ9׀îԒá‡Ö‡Ñ1á‡Ñ Ó4׀Ü9Û–Ñ f Ú{הҘ×yâ˜Ô Bt Ñ9ì’Ô ßÑ­ÕAÓ4Ñ’Ó•Ô ß Ö‡× ·á‡Ö‡×€á ·›à ԒîVd׀ңۂԒÐ}ԒÒgÜ9׀á‡Ñ­Ò£Û‡Õg˜€Ü ·Ñ­Ò¤Õ˜×€Ø€×€Þ ·Ö‡Ñ­Ògá‡Ñ ·á}Ó4׀Վì’Ô ßÑVևצÓ4׀ܦÛ{Ñ [0, t] ã1Ù–× Ô’Ø€Ñ­Õ˜Ö‡Ñ Ø€Ñ­Ü3á‡Ñ­Ò˜Ò4Ô¦á‡Ñ’Ó•Ô’îï’Ñ‡Ú Bt = St − pt, ø S‘ñ}ù Ø€Ñ­Ü St ևשU‚á ·Ö‡Ñ ؀ѭÜ9Ñá‡Ñ ØԒې‰Ó4Ï ßÑ Ô’á}Ó4×€Õ ·Ñ­Õ€ã…êdχÛ{Ñ­á‡Ý‚Ôõ‰Ï‡×…Ô՘׀Ҙ׀Վì’Ô Ö‡×€ÒŽÓ•Ô Ø€Ñ­Ü9ۂԒá‡Ý ·Ôå ·á}ì­×€ÒŽÓ ·Ö‚ԩ؀ѭá}Ó ·ádςԒÜ9׀á}Ó4זá‡Ñ©Ó4׀Ü9Û{ÑVøçÑ­Ï Ó4ѭܠԒւÔg׀Ü9ۇ՘׀ҎӕԒւÔ}ù‚×€Ü Ü9׀՘ØԒևѭÒuU‚á‚Ôlá‡Ø€× ·Õ˜Ñ­Ò¹õ‰Ï‡× Û‡Õ˜Ñ‰Ö‡Ï à ׀ÜÉâŽÏ‡Õ˜Ñ­Ò€ãfêd×¹×on ·ÒŽÓ4׹χܠÔEӕÔYn‡ÔPև×QâŽÏ‡Õ˜Ñ­Ò δt ×€Ü ØÔ’Ö‚Ô ·á‡ÒŽÓ•Ô’á‰Ó4× t Ú}׀á}ӕï’ÑPÑPì’Ô ßÑ­ÕXá‡ÑPÓ4׀Ü9Û{Ñ t և×WÏ‡Ü Ô¤Ï‡á ·Ö‚Ô’Ö‡×WÜ9Ñ­á‡×ƒÓ•ë’Õ ·Ô ·á}ì}×€ÒŽÓ ·Ö‚ÔPá‡Ñ¤Ó4׀Ü9Û{Ñ f å e∆t øyì’Ô ßÑ­ÕXæçÏ7Ó4χ՘щùQØ€Ñ­Ü ∆t = t 0 δτ dτ ã ÙPԅÜ9׀ҘܠԈæçѭ՘ܠÔ7ڤхì’Ô ßÑ­Õ£á‡Ñ…Ó4׀Ü9Û{Ñ f ևנχÜ9Ô…Ï‡á ·Ö‚Ô’Ö‡×9Ü9Ñ­á‡×ƒÓ•ë’Õ ·ÔVԅҘ׀ÕgۂԒÐ}ԅá‡Ñ…Ó4׀Ü9Û{Ñ t å e−∆t øyì’Ô ßѭՖۇ՘׀Ҙ׀á}Ó4בùPגÚs׀ܽЭ׀Õ4Ô ßÚ jÑ1ì’Ô ßÑ­Õ–Ö‡×”Ï‡Ü Ô Ï‡á ·Ö‚Ô’Ö‡×”Ü9Ñ­á‡×ƒÓ•ë’Õ ·Ô9á‡Ñ1Ó4׀Ü9Û{Ñ t ×€Ü Õ˜× ßԒîï’ÑRԒѩÓ4׀Ü9Û{Ñ s å e∆t−∆s Ú7Ï‡Ü ævÔlÓ4Ñ­Õmևפև׀Ҙ؀ѭá}Ó4Ñ¦Ò˜× s ≥ t פχÜævÔlÓ4Ñ­Õ¹Ö‡×¤Ô’Ø€Ï‡Ü”Ï ßԒîï’Ñ”Ò˜× s ≤ t ã êd×yâ˜Ô U0 ԅ՘׀Ҙ׀Վì’Ô ·á ·Ø ·Ô ß Ö‚Ô™Ò˜×€Ð­Ï‡Õ4ԒևѭÕ4ԙá‡Ñ ·á‡ÒŽÓ•Ô’á‰Ó4× f × Ut ԅ՘׀Ҙ׀Վì’Ô ւԅÜ9׀Ҙܠԅá‡Ñ ·á‡ÒŽÓ•Ô’á}Ó4× t ãwbmѭܿԒÒAÒ˜Ï‡Û–Ñ­Ò ·îVd×€Ò–Ô’Ø ·Ü Ô7Ú7üEѭ՘Þ{׀՘Ðø S4„‘„‘„}ù‚Ü9ѭҎÓ4Õ4ÔRõ‰Ï‡× Ut = e∆t U0 − t 0 e−∆τ dBτ . ø SYi­ù ísÕ˜× ßԒîï’ÑPÔ’Ø ·Ü ÔAÜ9ѭҎÓ4Õ4ÔAіævÔlÓ4іև×mõ‰Ï‡×¹Ô–՘׀Ҙ׀ՎìlÔPá‡Ñ ·á‡ÒŽÓ•Ô’á}Ó4× t å‚æç× ·Ó•ÔEևіØÔ’Û ·Ó•Ô ߉·á ·Ø ·Ô ß Ü9׀á‡Ñ­Ò Ñ ßχ؀՘хøçۇ՘×yâŽÏ‰ à щù ߉›õ‰Ï ·Ö‡Ñ”ÔlÓ4åPÑ©Ó4׀Ü9Û{Ñ t ÚdÓ4ÑdևєÜ9Ñ­á‰Ó•Ô’á}Ó4×PÔ’Ø€Ï‡Ü”Ï ßÔ’Ö‡Ñ”Ø€Ñ­Ü âŽÏ‡Õ˜Ñ­Ò¹Ø€Ñ­Ü9Û{ѭҎÓ4ѭҀã X R
  25. 25. öEÜ Ô”Õ˜× ßԒîï’Ñ¦Õ˜×€Ø€Ï‡Õ˜Ò ·ìlÔ¦Ü Ô ·ÒWЭ׀Õ4Ô ß Û{щևףҘ׀ÕEÖ‡×€Ö‡Ï à€·Ö‚ÔR؀ѭÜ9Ñ Uv = e∆v−∆t Ut − v t e−(∆τ −∆t) dBτ , t ≤ v. ø S4™}ù ’€º»¼ºÙ( x5ȀŒQä¿Œ  Ž»À¿Ä êd×yâ˜Ô (Ω, F, P) χÜS׀ҘۂԒî€Ñ…Ö‡×¦Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×’Ú A = {At, t ≥ 0} Ï‡Ü Ô U ßÓ4Õ4Ԓîï’чړѭá‡Ö‡× At ⊂ F ՘׀ۇ՘׀Ҙ׀á}Ó•Ô¨Ô ·á7æçѭ՘ܠÔlîï’ÑÖ ·Ò˜Û{ѭᐉì­× ß ÔlÓ4å ÑÓ4׀Ü9Û–Ñ t 㟖3Õ ·Ò˜Ø€ÑÖ‡×ˆÒ˜×€Ð­Ï‡Õ˜Ñ å1Ü9Ñ‰Ö‡× ßÔ’Ö‡Ñ ævÔ à ׀á‡Ö‡Ñ {Bt, t ≥ 0} χÜIۇ՘ÑdØ€×€Ò˜Ò˜Ñ ×€ÒŽÓ4ÑdØë’ÒŽÓ ·Ø€Ñ A þÒ˜×€Ü ·Ü9Ô’ÕŽÓ ·á‡Ð}Ô ß Ø€Ñ­ÜË׃Ò4ۖ׀Õ4Ԓá‡îÔ|U‚á ·Ó•Ô7ÚXÑ­Ï Ò˜×yâ˜Ô7Ú {Bt, t ≥ 0} å A þyԒւԒÛ7ӕԒևх×VØ€Ñ­Ü Ó4Õ4ÔlâŽ×ƒÓ4é­Õ ·ÔlҦ؀ѭá}Óe‰›á‰Ï‚Ô’Ò9ó…Ö ·Õ˜× ·Ó•ÔØ€Ñ­Ü ß›·Ü ·Ó˜×€Ò¦Û–× ßÔ ×€Ò˜õ‰Ï‡×€Õ˜Ö‚Ô7ã ÙPÔ Ü9׀ҘܠԙæçÑ­Õ˜Ü Ô Ñ Õ·Ò ˜Ø€Ñ†U‚á‚Ô’á‡Ø€× ·Õ˜Ñuå ·á}Ó4՘ÑdÖ‡Ï à€·Ö‡Ñ™ævÔ à ׀á‡Ö‡Ñ {∆t, t ≥ 0} Ï‡Ü Û‡Õ˜Ñ‰Ø€×€Ò˜Ò˜Ñ ×€ÒŽÓ4ÑdØë’ÒŽÓ ·Ø€Ñ A þÒ˜×€Ü ·Ü Ô’ÕŽÓ ·á‡Ð}Ô ßã í՘ϐ‰›á‚ÔVւÔVҘ׀ЭχÕ4ԒևѭÕ4Ô7ÚQևשU‚á ·Ö‚ÔV؀ѭÜ9хх׃ì­×€á}Ó4Ñ×€Ü õ‰Ï‡× ÔV՘׀Ҙ׀Վì’ÔVÓ4ѭ՘á‚Ô™þÒ˜×¦Ü9׀á‡Ñ­Õ£Ñ­Ï ·Ð­Ï‚Ô ß Ô à ׀՘чÚ{å”Ñ1ԒҘҘχá}Ó4ÑˆÛ‡Õ ·á‡Ø ·Û‚Ô ß Ö‡×€ÒŽÓ•Ô1Ö ·Ò˜Ò˜×€ÕŽÓ•Ô’îï’чã–ÎCҎӕԒÜ9Ñ­Ò ·á}Ó4׀՘׀ҘÒ4ԒևѭÒP׀ÜY׀ҎÓ4χւԒÕPÔ Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×Pևף՘ϐ‰›á‚ÔR׀Ü!Ó4׀Ü9ۖуU‚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‡×Pևף՘ϐ‰›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’ևшԒá}Ó4×€Õ ·Ñ­Õ˜Ü9׀á‰Ó4גÚ{؀ѭá‡Ò ·Ö‡×€Õ4Ԓá‡Ö‡ÑˆÏ‡Ü Ô ØԒՎÓ4× ·Õ4ԠևהҘ׀Эχ՘ѭÒPõ‰Ï‡×Rå©æç× ·Ó•Ô Ö‡×”Ï‡Ü ЭÕ4Ԓá‡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׀Ҁږ؀ѭá‡Ò ·Ö‡×€Õ4Ԓá‡Ö‡Ñ õ‰Ï‡×á‡×€á‡ÝdχÜÖ‡× ß׀Ò1å ЭÕ4Ԓá‡Ö‡×mѤҘÏxU‚Ø ·×€á}Ó4×WۂԒÕ4ÔPÔlæç׃ӕԒÕXÑPìlÔ ßÑ­ÕQÓ4Ñ’Ó•Ô ß Ö‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîVd׀ÒCԒЭ՘׀Ð}ÔlւԒÒFÒ ·Ð­á ·U‚ØÔlÓ ·ìlԒÜ9׀á}Ó4×’Ú Ô’Û‡Õ˜Ñgn ·Ü Ô’ܦѭÒWԔæçχá‡îï’Ñ Bt Û{Ñ­ÕWχܠÔRÖ ·æçχÒ4ï’Ñ‡Ú ·ÒŽÓ4Ѧå’Ú Bt = −βt − σbWb,t, ø¥X R ù Ñ­á‡Ö‡× β × σb Ò4ï’Ñ؀ѭá‡ÒŽÓ•Ô’á}Ó4×€Ò©× Wb,t = {Wb,t, t ≥ 0} å9Ï‡Ü Ü9Ñ‘ì ·Ü9׀á}Ó4хôm՘Ñ4`Aá ·Ô’á‡Ñ…Û‚Ô’Ö‡Õ4ï’чã íWõ‰Ï · β ՘׀ۇ՘׀Ҙ׀á}ӕÔÑVÐ}Ԓá‡Ý‡Ñ…×€Ò˜Û{׀Õ4ԒևÑÛ–Ñ­Õ£Ï‡á ·Ö‚Ô’Ö‡×¦Ö‡×RÓ4׀Ü9Û{Ñ‡Ú¤Ö‡×ƒì ·Ö‡ÑÔˆÏ‡Ü Ø€Ñ‰×©U‚Ø ·×€á‰Ó4×9Ö‡× Ò˜×€Ð­Ï‡Õ4Ԓá‡îÔ θ 0 á‡ÑPۇÕg˜€Ü ·Ñ–× σb å¹ÑPÖ‡×€ÒŽì ·Ñ¤Û‚Ô’Ö‡Õ4ï’ÑPևÑPÓ4Ñ’Ó•Ô ß Ö‡×¹Û‚Ô’Ð}ԒÜ9׀á}Ó4Ñ­ÒXÖ‚Ô’Ò ·á‡Ö‡×€á ·›à ԒîVd×€Ò X—f
  26. 26. ۖѭÕRÏ‡á ·Ö‚Ô’Ö‡×ˆÖ‡×1Ó4׀Ü9Û{чã ÎCÜ á‡Ñ­Ò˜Ò˜Ñ Ó4Õ4Ô’Þ‚Ô ß݇чÚCÏ7Ó ·›ß›·›à Ô’Õ˜×€Ü¦Ñ­Ò”Ô’Ò¦Ö ·æçχÒgV‰×€Ò¦Û‚ÔlÕ•Ô Ô’Û‡Õ˜Ñgn ·Ü Ô’Õ©Ñ Ü9Ñ‰Ö‡× ßÑRØ ßë’Ò˜Ò ·Ø€Ñ¦Ø€Ñ­Ü ·á}ì}×€ÒŽÓ ·Ü9׀á}Ó4чãfêd׀á‡Ö‡Ñ1Ô’Ò˜Ò ·Ü…Ú7՘׀ۇ՘׀Ҙ׀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׀Ҙܠԩæçѭ՘ܠÔ7ڇԒҘÒ4Ï‡Ü ·Ü9Ñ­ÒWõ‰Ï‡×£Ñ¦ævԙӕђÕWևףև׀Ҙ؀ѭá‰Ó4Ñ ∆t å£Ï‡Ü ÔRÖ ·æçχÒ4ï’чڇѭÏVҘ×yâ˜Ô7Ú ∆t = δt + σdWd,t, ø¥X’ñ}ù Ñ­á‡Ö‡× δ × σd Ò4ï’х؀ѭá‡ÒŽÓ•Ô’á}Ó4×€Ò©× Wd,t = {Wd,t, t ≥ 0} å¦Ï‡Ü Ü9Ñ™ì ·Ü¦×€á‰Ó4ÑVôm՘ÑY`Aá ·Ô’á‡ÑVۂԒևÕ4ï’чã êd× σd å©Û{Ñ­Ò ·Ó·ì ­Ñ‡Ú–×€ÒŽÓ•Ô’Ü9Ñ­ÒPÔ’Ö ·Ø ·Ñ­á‚Ô’á‡Ö‡Ñ9χܠÔ9Û{׀ՎÓ4χ՘ނԒîï’Ñ1׀ҎÓ4ÑdØë’ÒŽÓ ·ØÔ1ԠχܠԦӕÔYn‡Ô Ö‡×AâŽÏ‡Õ˜Ñ­Ò ؀ѭá‡ÒŽÓ•Ô’á}Ó4× δ ã í–Ò˜Ò˜Ï‡Ü ·Ü9Ñ­ÒsӕԒܔÞ{å€Ü¢õ‰Ï‡× Wb,t × Wd,t Ò4ï’Ñ ·á‡Ö‡×€Û–×€á‡Ö‡×€á}Ó4׀ÒãCúQ× ßÔEٖשU‚á ·îï’Ñ R ãaXdãaidڑÓ4׀Ü9Ñ­Ò¤õ‰Ï‡× Bt × ∆t Ò4ï’Ѡۇ՘щ؀׀ҘҘѭÒPև×gû¡Óed ·á‡Ö‡×€Û{׀á‡Ö‡×€á}Ó4׀ҀãEêd×yâ˜Ô’Ü Zt = e∆t × Yt = U0 − t 0 e−∆τ dBτ Ú Ñ­ÏVҘ×yâ˜Ô7Ú Ut = ZtYt ãfúX× ßÑpQ×ƒÑ­Õ˜×€Ü Ô R ãaXdã™7Ú Zt e Yt ӕԒܔÞ{å€Ü!Ò4ï’Ѧۇ՘Ñd؀׀ҘҘѭÒEևףû¡Óed9×€Ü R+ ã úQ× ßÔ¦úC՘ѭÛ{Ñ­Ò ·îï’Ñ R ãaXdã„R×£Û–× ßсpQ×€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׀ܦѭÒWӕԒÜRޖå€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 ·ÒAև×gÔ ßЭχá‡ÒWØë ßØ€Ï ßÑ­ÒW؀ѭá‡Ø ßϐ‰›Ü9Ñ­Ò¹õ‰Ï‡×dæ dUt = Ut δ + 1 2 σ2 d + β dt + UtσddWd,t + σbdWb,t. ø¥X‘i­ù í–Э՘χۂԒá‡Ö‡Ñ¦Ñ­ÒWÓ4׀՘Ü9Ñ­ÒWÖ‚Ô¦Ö ·æçχÒ4ï’ÑRá‚Ô¦×on7ۇ՘׀ҘÒ4ï’Ñ ø¥X‘i­ùcڇևשU‚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׀ܦѭÒAõ‰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׀ЭÕ4Ԓá‡Ö‡Ñ¦Ô’ÜRޖѭÒWÑ­Ò ßԒևѭÒWև׀ҎӕԦ׀õ‰Ï‚Ô’îï’Ñ9á‡Ñ ·á}Ó4׀Վì’Ô ßÑ [0, t] ڂѭÞ7Ó4׀Ü9Ñ­Ò d[W]t = dt ⇒ [W]t = t. ٖ׀ҎӕÔ)æçѭ՘ܠÔ7Ú©Ó4׀Ü9Ñ­Ò õ‰Ï‡× W = {Wt, t ≥ 0} åuχÜ%Ü Ô’ÕŽÓ ·á‡Ð}Ô ß Ø€Ñ­á}Óe‰›á‰Ï‡ÑÉ؀ѭÜDì’Ô’Õ ·Ô’îï’Ñ õ‰Ï‚Ô’Ö‡Õ4ëlÓ ·ØÔ [W]t = t ã…úQ× ßÑypQ×€Ñ­Õ˜×€Ü Ô R ãaXdãTSYXdÚXÓ4׀Ü9ѭҔõ‰Ï‡× W å1χÜÜ9Ñ‘ì ·Ü9׀á}Ó4ÑômÕ4Ñ4`Aá ·Ô’á‡Ñ ۂԒևÕ4ï’чã¢ý“ѭЭчÚWۖÑdև׀Ü9Ñ­Ò9Ó4Õ4ÔlÓ•Ô’Õ ø¥X‘i­ùR؀ѭÜ9Ñ Ï‡Ü2ØÔ’Ò˜Ñ ×€Ò˜Û{×€Ø ·Ô ß Ö‚Ô™×€õ‰Ï‚Ô’îï’Ñ Ö ·æç׀՘׀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}Ó4ÑEôm՘Ñ4`Aá ·Ô’á‡ÑAۂԒևÕ4ï’ÑE×fÛ{Ñdև׀Ü9ѭҤ؀ѭá‡Ò ·Ö‡×€Õ4Ô’Õ At = σ{Ws; 0 ≤ s ≤ t} ã ¤ ·Ü9ۖѭՎӕԒá‰Ó4×ˆÖ ·›à ׀զõ‰Ï‡×’ڹ؀ѭÜ9Ñ Ut å…Ï‡Ü Ô¨Ö ·æçχÒ4ï’чڂ׀á‰Ó•ï’Ñ…× ß׈ӕԒÜRޖå€Ü å…χÜoۇ՘ÑdØ€×€Ò˜Ò˜Ñ äԒÕgª’Ñ‘ì ·Ô’á‡Ñ‡ã íÃÛ‡Õ ·á‡Ø‰›Û ·Ñ”׀ҎӕԒ՘׀Ü9Ñ­Ò ·á}Ó4׀՘׀ҘÒ4ԒևѭÒEá‚ÔRÒ˜×€Ð­Ï ·á}Ó4ףۂԒÕ4ԒÜ9׃Ó4Õ ·›à Ԓîï’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‡×ˆÑ­ÒR؀Ñd×oU‚Ø ·×€á‰Ó4×€Ò β, σb, δ, e σd և׀Û{׀á‡Ö‚Ô’Ü ևÑÓ4׀Ü9Û{Ñ ×1ւÔ՘׀Ҙ׀Վì’Ô ÔlÓ4Ï‚Ô ßã í–Ò˜Ò˜Ï‡Ü ·Ü9Ñ­ÒFõ‰Ï‡×AÑ­ÒC؀ÑdשU‚Ø ·×€á}Ó4×€Ò µ(t, u) × σ(t, u) Ò4ï’ѤÒ4ÏxU‚Ø ·×€á‰Ó4׀Ü9׀á}Ó4×AҘςԑì­×€Ò‚Û‚ÔlÕ•Ô¤Ô’Ö‡Ü ·Ó·Õ Ï‡Ü Ô ð‡á ·ØÔ Ò˜Ñ ßχîï’Ñ…Û‚Ô’Õ4Ôsø¥Xd™}ù”×VۂԒÕ4ԨԒҘҘ׀ЭχÕ4ԒՠԨ×on ·ÒŽÓe˜€á‡Ø ·Ô¨×V؀ѭá}Ó ·ádÏ ·Ö‚Ôlև×Vև׈Ó4ÑdւԒÒ9Ô’Ò Ö‡×€Õ ·ìlԒւԒÒAõ‰Ï‡×gԒۇ՘׀Ҙ׀á‰Ó•Ô’՘׀Ü9Ñ­ÒAá‚ÔRۇ՘é4n ·Ü Ô”Ҙ׀îï’ч㠲±»° þ r‡iõ¹P€õ¹Pvyxyvy•–€f•–q •–q r‡h§¦y’–€ ’€ºÙG€º» H   ñ Ä ñ ¾»Û»¾»ä¿Ä¼ä¿Œä¿Œ  Ž»À¿Ä¡ŒQ Á4ŒQ‘ôõŸÆÇÀy¾íÁ4 êd×yâ˜Ô I[A] Ô©æçχá‡îï’Ñ ·á‡Ö ·ØԒևѭÕ4ԩևÑR׃ì}׀á}Ó4Ñ A ãwbmÑ­á‡Ò ·Ö‡×€Õ˜×€Ü9Ñ­ÒAÔ”Ö ·ÒŽÓ4Õ ·Þ‡Ï ·îï’Ñ”Ö‡×¤Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·þ ւԒև×P؀ѭá‡Ö ·Ø ·Ñ­á‚Ô ß Ö‡ÑR܉›á ·Ü¦Ñ”ÔlÓ ·á‡Ð ·Ö‡Ñ©Û{× ßÑRۇ՘щ؀׀ҘҘÑ9ւԔ՘׀Ҙ׀Վì’ÔR׀Ü!Ï‡Ü ·á‰Ó4׀ՎìlÔ ßєև×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. úXԒÕ4Ô v × y Uxn7ѭҀڂҘ×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}ӕï’Ñ Mt = {Mt, t ≥ 0} ågχܿÜ9Ô’ÕŽÓ ·á‡Ð}Ô ß øí©f×yâ˜Ô¦í–á‡×on7Ñ û˜ûŽûŽùcã–íÅҘ׀Эχá‡Ö‚Ô ·Ð­Ï‚Ô ßւԒևפ׃ÜËø R X­ù å¤Ö‡×ƒì ·Ö‚Ô9óRÛ‡Õ˜Ñ­Û‡Õ ·×€Ö‚ԒևשäԒÕgª’Ñ‘ì ·Ô’á‚ÔRևÑ9ۇ՘Ñd؃׀ҘÒ4Ñ9ւԦ՘׀Ҙ׀ՎìlÔ7ã í–Û ߛ·ØԒá‡Ö‡Ñ…ÔVæçé­Õ˜Ü”Ï ßԈև×9û¡Óed…Э׀á‡×€Õ4Ô ß›·›à ԒւÔbm݉χá‡Ð|¨ì« ·›ß›ß·Ô’ܦҠø S4„‘„ f Ú¿pQ׀ѭ՘׀ܠÔx„7ãaX­ù£Ô’Ñ Û‡Õ˜Ñ‰Ø€×€Ò˜Ò˜Ñ Mt ڂѭއҘ׀Վì’Ô’á‡Ö‡Ñ…õ‰Ï‡×Ô Û‚Ô’ÕŽÓ4×V؀ѭá}Óe‰›á‰Ï‚Ô Ö‚Ô æçχá‡îï’Ñ It åV؀ѭá‡ÒŽÓ•Ô’á}ӕ×V×VχÒ4Ԓá‡Ö‡Ñdø¥Xd™}ù Ñ­Þ7Ó4׀Ü9Ñ­ÒWÑ¦Ò˜×€Ð­Ï ·á}Ó4×£Õ˜×€Ò˜Ï ßӕԒևѐæ 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 ·Ñ­ÒFۂԒÕ4ÔEÑ­Þ7Ó4׀á‡îï’ÑPև׀ҎӕÔP׀õ‰Ï‚Ô’îï’Ñ–ì’ï’Ñ¤Ô ßå€Ü ևÑP׀Ҙ؀ѭÛ{ÑPև׀ҎӕÔPÖ ·Ò˜Ò˜×€ÕŽÓ•Ô’îï’чã bmÑ­Ü9ѣԖæçχá‡îï’Ñ It(1−P(t, Ut, v, y)) åW؀ѭá}Óe‰›ádςÔP×ƒÜ t õ‰Ï‚Ô’Ò˜×A؀׀ՎӕԒÜ9׀á}Ó4גډԤۂԒՎÓ4׹ևѣÒ4Ô ßÓ4Ñ‡Ú Ø€Ñ­Õ˜Õ˜×€Ò˜Û–Ñ­á‡Ö‡×€á‰Ó4פԒєð ßÓ·Ü ¦Ñ£Ó4׀՘Ü9чÚdå à ׀՘єõ‰Ï‚ԒҘפ؃׀ՎÓcÔlÜ9׀á‰Ó4גãw– Ó4׀ѭ՘׀Ü9ԔÔgÒ˜×€Ð­Ï ·Õ€Ú7Ñ­Þ7Ó ·Ö‡Ñ©×€Ü üEѭ՘ޖ׀՘Ðø S4„‘„‘„}ùcÚdæçѭ՘á‡×€Ø€×gχܠԔ׀õ‰Ï‚Ô’îï’Ñ9Ö ·æç׀՘׀á‡Ø ·Ô ß Û‚Ô’Õ4Ô¦Ô¦Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×P׀ÜIø R S™ùcã wŒr  ŒQ‘Ä‘’€ºÙG€º» u ‹{Œ4‡­²•|}²€wy³´wvz}|}z‰©ƒŠWò¬‰«oהš|’„vw›švƒ|‘”|V4~€‡|­—™¥‰‡ z’w€ƒŒ4ƒxj“ƒwv|’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 šƒ©Ô€ƒwy„¡| ݙšR“•‡lx{z’w痗瑁•š P(v, u, v, y) =    1 š‘ u ≤ y, 0 š‘ u y, P(t, y, v, y) = 1, 0 ≤ t v, P(t, ∞, v, y) = 0, 0 ≤ t v. Ó Œ4‡l¦l|}® ÎCÜ¿Û‡Õ ·Ü9× ·Õ˜Ñ ßχÐ}ԒՀڂԒҘÒ4Ï‡Ü ·Ü9Ñ­ÒAõ‰Ï‡× ∂ ∂x P(t, Ut, v, y)σ(t, Ut)dt åDU‚á ·Ó•Ô¦õ–ãؒããWÎCÜø R‘R ùcڇÓ4՘Ñlþ ØԒÜ9Ñ­Ò©ÑÓ4׀՘Ü9Ñ Ø€Ñ­ÜËÑævÔlÓ4Ñ­Õ dWt ۂԒÕ4ÔÑ ßԒևÑ׀Ҙõ‰Ï‡×€Õ˜Ö‡Ñ…Ö‚Ô׀õ‰Ï‚Ô’îï’чã†bmÑ­Ü9Ñ W × M Ò4ï’Ñ Ü Ô’ÕŽÓ ·á‡Ð}Ô ·Ò€Údєõ‰Ï‡×P׀Ҏӕë”Ö‡Ñ ßÔ’Ö‡Ñ©Ö ·Õ˜× ·Ó4ÑgւԔ׀õ‰Ï‚Ô’î€ï’єӕԒÜRޖå€Üև׃ì}×PҘ׀գøçÑ ·á‡Ø€Õ˜×€Ü9׀á}Ó4єևבù‚Ï‡Ü X‘i
  29. 29. Ü Ô’ÕŽÓ ·á‡Ð}Ô ßã CpQ׀Ü9Ñ­Ò©õ‰Ï‡×1׀ҎÓ4×1Ü Ô’ÕŽÓ ·á‡Ð}Ô ß å1Ô’Þ‡Ò˜Ñ ßÏ7ӕԒÜ9׀á‰Ó4×9؀ђá‰Óe‰›á‰Ï‡Ñ× Û{ѭՎӕԒá}Ó4ÑևנìlÔ’Õ ·Ô’îï’Ñ ߛ·Ü ·Ó•Ô’Ö‚Ô7ãFý“ѭЭчڇÛ{× ßсpQ×€Ñ­Õ˜×€Ü Ô R ãaXdãTS R × ßפև׃ì­×gҘ׀ÕA؀ѭá‡ÒŽÓ•Ô’á‰Ó4גڂѭυҘ×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‡ÒŽÓ•Ô’á}Ó4גãu–ÅÕ˜×€Ò˜Ï ßÓ•Ô’Ö‡Ñ¦Ò˜×€Ð­Ï‡× ·Ü9×ƒÖ ·ÔlӕԒܦ׀á}ӕ×lã ¤ ·Ü9ۖѭՎӕԒá‰Ó4×gѭއҘ׀ՎìlԒդõ‰Ï‡×RÔ Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×gևה՘ϐ‰›á‚Ô ×€ÜYÓ4׀Ü9Û{Ñ8U‚á ·Ó4Ñ ø S4„}ùAå”χܽØÔ’Ò˜Ñ ×€Ò˜Û–×€Ø ·Ô ß Ö‚Ô¦Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×P׀ÜIø R S™ùcڇѭυҘ×yâ˜Ô7Ú ψ(t, u, v) = P(t, u, v, 0). ’€ºÙG€ºÙ( tEļ!¿“‘FÚ$#QÀ¿Œr•À¿ŸÁ4ŒQ‘ôõ üEхØԒҘх݇ѭÜ9ѭБ˜€á‡×€Ñ…á‡ÑVÓ4׀Ü9ۖчڤõ‰Ï‚Ô’á‡Ö‡Ñѭң؀щשU‚Ø ·×€á}Ó•×ƒÒ µ × σ Ò4ï’Ñ ·á‡Ö‡×€Û{׀á‡Ö‡×€á}Ó4×€Ò©Ö‡× t Ú ÔVæçχá‡îï’Ñuø R S™ù¤Ö‡×€Û–×€á‡Ö‡×ˆÖ‡× t × v ҘѭÜ9׀á}Ó4×1ÔlÓ4Õ4Ôì}å€Ò©Ö‡× v − t ãٖ׀ҎӕÔæçѭ՘ܠÔ7Ú¤Ó4ѭ՘á‚Ô™þÒ˜×9ð7Ó ·›ß Ô ·á}Ó4՘Ñdևχîï’Ñ£Ö‚Ô¤æçχá‡îïlѩև×WÓ4Õg˜€Ò‚Û‚Ô’Õe—’Ü9׃Ó4Õ˜Ñ­Ò P(u, v, y) = P(0, u, v, y) ډ×E؀ѭ՘՘׀Ҙۖѭá‡Ö‡×€á‰Ó4׀Ü9׀á}Ó4× Ø€Ñ ßщØԒÜ9Ñ­Ò ψ(u, v) = P inf s∈[0,v] Us ≤ 0|U0 = u . ø R h­ù í Ò˜×€Ð­Ï ·ÕEԒۇ՘׀Ҙ׀á}ӕԒÜ9Ñ­ÒAÖ‡Ñ ·ÒWØ€Ñ­Õ˜Ñ ßë’Õ ·Ñ­Ò¹Ö‡Ñ¦Ó4׀ѭ՘׀ܠÔRԒá‰Ó4×€Õ ·Ñ­Õ€ã tE  “Û ùÂ ¾Å‘’€ºÙG€ºÙ( uD‹{Œ4‡}²•|}²€wy³´wvz}|}z‰Ãz‰ÃŒcdªyx{|§ò¬%o×!|’„œ ‡„ƒŠ–‹‚‡ v š|’„vw›švƒ|‘†|¿4~€‡|­—™¥‰‡ z’w€ƒŒ4ƒxj“ƒwv|’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 šƒ©Ô€ƒwy„¡| ݙšR“•‡lx{z’w“˜ç‘•š ψ(u, 0) =    1 š‘ u ≤ 0, 0 š‘ u 0, ø R i­ù ψ(0, v) = 1, v 0, ø R ™}ù ψ(∞, v) = 0, v 0. ø R „}ù êdףѭއҘ׀ՎìlԒ՘Ü9Ñ­ÒAõ‰Ï‡× ψ(u, v) = P(T − v, u, T, 0) ږÔRۇÕ4ёìlԦև׀ҎÓ4×gØ€Ñ­Õ˜Ñ ßë’Õ ·ÑRå ·Ü9×€Ö ·ÔlӕÔ7ã ٖ׃ì­×€Ü9ѭҖѭއҘ׀Վì’Ô’ÕEõ‰Ï‡×©×€ÒŽÓ•Ô’Ò–Ò4ï’Ѡ؀ѭá‡Ö ·îV‰×€Ò ·á ·Ø ·Ô ·ÒA×gևףæç՘ѭá}Ó4× ·Õ4Ô7ãmíÅ׀õ‰Ï‚Ô’îï’Ѩø R i­ù¹á‡Ñ­Ò Ö ·›à õ‰Ï‡×’ÚWۂԒÕ4Ô v = 0 ÚWÔ…Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ԓև׈և×՘ϐ‰›á‚Ô¨å ·Ð­Ï‚Ô ß Ô à ׀՘љ؀ѭÜoχܠԅ՘׀Ҙ׀ՎìlÔ ·á ·Ø ·Ô ß Û–Ñ­Ò ·Ó·ì ’Ô£× ·Ð­Ï‚Ô ß ÔS–Ø€Ñ­Ü χܠÔg՘׀Ҙ׀ՎìlÔ ·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 ׀՘чÚCÔ…Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ԓևצևנ՘ϐ‰›á‚ÔVå ·Ð­Ï‚Ô ß ÔŸS Û‚Ô’Õ4ÔVÓ4ÑdÖ‡Ñ ·á‡ÒŽÓ•Ô’á}Ó4× Xd™

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