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R úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×RÖ‡× Õ˜Ï‰›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‡×RÖ‡× Õ˜Ï‰›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‡×RÖ‡× Õ˜Ï‰›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‡×RÖ‡× Õ˜Ï‰›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‡×RÖ‡× Õ˜Ï‰›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‡×RÖ‡× Õ˜Ï‰›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‡×RÖ‡× Õ˜Ï‰›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 ·æçÑ­Õ˜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‡×RÖ‡× Õ˜Ï‰›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 ·æçÑ­Õ˜Ü9× (1, 2)
× δ = 0.1
ãCãgã£ãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãˆh‘X
S‘S úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×RÖ‡× Õ˜Ï‰›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 ·æçÑ­Õ˜Ü9× (1, 2)
× δ = 0.05
ã…ãgãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgã£ãˆh‘X
SYX úCÕ˜Ñ­Þ‚Ô’Þ ·›ß›·Ö‚Ô’Ö‡×RÖ‡× Õ˜Ï‰›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 ·æçÑ­Õ˜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×€Ü9Û–Ñ8U‚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×€Ü9Û–Ñ8U‚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×€Ü9Û–Ñ8U‚á ·Ó4Ñ1õ‰Ï‚Ô’á‡Ö‡Ñ
δ = 0.01
ãEãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgãgãgãgã£ãgãgãgãgãgãgãgã‹h‘h
ì ·

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 Ô’îV‰×€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 Ô’îV‰×€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 Ô’îV‰×€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 Ô’îV‰×€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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üE׀ҎӕԖҘ׀îï’ѤÔlÛ‡Õ4׃Ò4׃á‰Ó•Ô’Ü9Ñ­ÒXÔ ßЭχá‡Ò¤Ø€Ñ­á‡Ø€× ·Ó4Ñ­ÒXׂևשU‚á ·îV‰×€ÒXÖ‚ÔAÓ4×€Ñ­Õ ·ÔAևѥb¹ë ßØ€Ï ßÑA×€ÒŽÓ4ÑdØë’ÒŽÓ ·Ø€Ñ
×¤Ö‚Ô ·á}Ó4׀ЭÕ4Ô ß Ö‡×£û¡Óed‡ã
S R

15. G€ºÙ(€º» tE“Ày©Á  Žyȑɐ•ä¿Ä‘¾»À“Á4ŒrÚ Â Ä¼Ûwä¿Œ|oÁ4å
êd×yâ˜Ô (Ω, F, P)
×€Ò˜Û‚Ô’î€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
§
êd×¦Ï‡Ü Ô1æçχá‡î€ï’Ñ g(ω)
å Ft
þÜ9×€á‡Ò˜Ï‡Õ4ëì}× ßÚ ·ÒŽÓ4ÑVÒ ·Ð­á ·U‚ØÔ7Ú ·á}Ó4Ï ·Ó·ì lÔ’Ü9×€á}Ó4×’Ú“õ‰Ï‡×¦Ñˆì’Ô ßÑ­Õ¤Ö‡× g(ω)
Û–ÑdÖ‡×©Ò˜×€Õ–Ö‡×€Ø ·Ö ·Ö‡Ñ Û{× ßÑ­ÒAì’Ô ßѭ՘׀ÒEÖ‡× Ws(ω)
Û‚Ô’Õ4Ô s ≤ t
ãB–PއҘ׀Վì­×”õ‰Ï‡× Fs ⊂ Ft
Û‚ÔlÕ•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 ∞
§
öEÜ Ô”æçχá‡îï’Ñ ϕ ∈ Υ
å£Ø•Ý‚Ô’Ü Ô’Ö‚ÔR× ß×€Ü9×€á}Ó•Ô’ÕWÒ˜×£× ßÔ”Ó4׀ܿ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‡ØÔ’Ö‚Ô”æçχá‡îï’Ñ ej
և׃ì}×£Û{×€ÕŽÓ4×€á‡Ø€×€ÕEÔ
Ftj
ã‚úXÔ’Õ4Ô”æçχá‡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 Ä¼Ûuä¿Œ|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. – ß›·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
Ҙ׀õ‰Ï˜€á‡Ø ·ÔEÖ‡×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×fҘѭއ՘×
Ô£×on ·ÒŽÓe˜€á‡Ø ·ÔgÖ‚Ô’Òfæçχá‡îVd×€Ò ϕn
Û–Ñdև׀ÜҘ׀Õfì ·ÒŽÓ4Ñ­Ò‚×€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׀ЭÕ4Ô ß Ö‡×Rû¡ÓedV×RÑVҘ׀Эχá‡Ö‡Ñ
Ü9Ñ­ÒŽÓ4Õ4Ôgõ‰Ï‡×P× ßÔgå–Ï‡Ü Ü Ô’ÕŽÓ ·á‡Ð}Ô ßã Xí–ҹև׀Ü9Ñ­á‡ÒŽÓ4Õ4Ô’îV‰×€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  ¡ŽyÛÅÄŸä¿ŒoÁ4å
í ·á}Ó4׀ЭÕ4Ô ß Ö‡× û¡Óed Û–ÑdÖ‡×1Ҙ׀ՔևשU‚á ·Ö‚ÔÛ‚Ô’Õ4ÔÏ‡Ü ÔØ ßÔ’Ò˜Ò˜×1Ö‡× ·á‰Ó4׀ЭÕ4Ô’á‡Ö‡Ñ­Ò f
Ü Ô ·Ñ­Õ£õ‰Ï‡× Υ
ã
÷–Ô à ×€Ü9Ñ­Ò ·Ò˜Ò˜Ñ¦×€á7æçÕ4Ô’õ‰Ï‡×€Ø€×€á‡Ö‡Ñ1Ô’ÒA؀ѭá‡Ö ·îVd×€Ò©ø·›·ùf× ø·›·›·ù‚ւԦٖשU‚á ·îï’Ñ R ãaXdãaX”Û‚Ô’Õ4Ô¦Ô’ÒAÒ˜×€Ð­Ï ·á}Ó4×€Òæ
¾»¾ÍÌ4ð Χn ·ÒŽÓ4×gχÜ9Ô”ævԒ܉ 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
þyÔ’Ö‚Ô’Û7Ó•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‚wç—™¥‰‡9¬‰§›¯d§›¯…ò¡wyw×5ó¢–ò¡wywyw×5ó7|‰“ƒwyŠ9|}§
í–á}Ó4×€Ò”Ö‡×1ևשU‚á ·Õ4ܦѭҔԅæçé­Õ˜Ü”Ï ßÔVÖ‡× û¡Óed‡ÚFÛ‡Õ˜×€Ø ·Ò4Ô’Ü9ѭҩևשU‚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. !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‡Ü3Û‡Õ˜Ñ‰Ø€×€Ò˜Ò˜Ñ Ö‡×£û¡Óed‡Ú‚×€á}Ó•ï’Ñ¦× ß×£Û–ÑdևףҘ׀ÕA×€Ò˜Ø€Õ ·Ó4Ñ9ևפæçÑ­Õ˜Ü ÔRÔ’Þ‡Õ˜×ƒì ·Ô’Ö‚ÔR؀ѭÜ9Ñ
dXt = udt + vdWt.
øÍh’ù
öEÜoÖ‡Ñ­Ò Õ˜×€Ò˜Ï ßÓ•Ô’Ö‡Ñ­Ò Ü Ô ·Ò ·Ü9Û–Ñ­ÕŽÓ•Ô’á‰Ó4×€Ò¦Ö‚Ô¨Ó4×€Ñ­Õ ·Ô…Ö‡× ·á‰Ó4׀ЭÕ4Ô’îï’Ñ…×€ÒŽÓ4ÑdØë’ÒŽÓ ·ØÔ7Ú¹Ñ õ‰Ï‚Ô ß Ô’Û‡Õ˜×cþ
Ҙ׀á}ӕԒ՘׀Ü9Ñ­Ò”Ô’Ð’Ñ­Õ4Ô7ÚQå1ԅ՘׀ЭÕ4ԅ؀ѭá‡Ý‡×€Ø ·Ö‚ԅ؀ѭÜ9ÑVæçé­Õ˜ÜRÏ ßÔˆÖ‡×9û¡Óed‡ãVûá7æçÑ­Õ˜Ü Ô ßÜ9×€á}Ó4×’ÚQ׀Ҏӕԅ՘׀ЭÕ4Ô
åRÏ‡Ü Ô ì­×€Õ˜Ò4ï’ÑVւԈ՘׀ЭÕ4ÔˆÖ‚Ô1ØÔ’Ö‡× ·ÔˆÖ‡ÑˆØë ßØ€Ï ßÑˆÖ ·æç׀՘׀á‡Ø ·Ô ß Ø ßë’Ò˜Ò ·Ø€Ñ‡ã”úXÔ’Õ4ÔˆÏ‡Ü Ô1ۇ՘ёì’Ô1æçÑ­Õ˜Ü Ô ß
և׀ҎÓ4פÓ4×€Ñ­Õ˜×€Ü Ô”ì}×yâ˜Ô–Bª‰Ò˜×€á‡Ö‚Ô ß ø¥X f‘f‘R ùcã
wŒr  ŒQ‘Ä‘G€ºÙ(€ºÙsêég”sï  ¡ŽyÛÅÄ¡ä¿Œ|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‰ ¨„ 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)
œ”“4|’³›“ƒd³|}z‰‡ z‰”|‰“•‡lŒ˜z‰‡8Ý™š£Œ4¡¶’Œ˜|™š
dt · dt = dt · dWt = dWt · dt = 0, dWt · dWt = dt.
úXÔ’Õ4Ô©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’Ü9Ñ­Ò‚Û‚Ô’ÕŽÓ4×
ևє×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 Ô’Ö‡Ñgá‡Ñ”ØԒې‰Ó4Ï ßÑ”ñgÖ‡×€ÒŽÓ•Ô”Ö ·Ò˜Ò˜×€ÕŽÓ•Ô’îï’чã
S‘ñ

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

 ,
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‡×on7Ñ9ûŽùcã
êd×yâ˜Ô Z(t, ω) = g(t, Wt) = xy
ãfúX× ßÔ”æçé­Õ˜Ü”Ï ßԩևףû¡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“ѭЭÑ
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×€Ñ­Õ˜×€Ü Ô’Ò9õ‰Ï‡×Ò˜×€Ð­Ï‡×€Ü Ò˜×€Õ4ï’љևׅæçχá‡Ö‚Ô’Ü9×€á}Ó•Ô ßA·Ü9Û{Ñ­ÕŽÓ˜—’á‡Ø ·Ô á‡Ñ™ØԒې‰Ó4Ï ßÑ…ñ7Ú¹á‡Ñ
õ‰Ï‚Ô ß Ö ·Ò˜Ø€Ï7Ó ·Ü¦Ñ­ÒPшÜ9Ñ‰Ö‡× ßÑ1Ö‡×RÖ ·æçχÒ4ï’шۂԒÕ4ÔˆÔ1՘׀Ҙ׀ՎìlÔVÖ‡×RÏ‡Ü Ô Ò4׃ЭχÕ4ԒևѭÕ4Ô7ブ Û‡Õ ·Ü9× ·Õ˜Ñ å”Ñ
pQ׀ђÕ4×ƒÜ Ô9ñ7ãTS£×£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. 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. 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 Ö‡×Õ ·Ò˜Ø€Ñ‡ã í–Ò˜Ò˜Ï‡Ü ·Ü9Ñ­Ò õ‰Ï‡×¨Ô™×ƒì}Ñ ßχîï’Ñ Ö‚Ô
՘׀Ҙ׀ՎìlÔ U = {Ut, t ≥ 0}
Ö‡× Ï‡Ü Ôs؀ѭÜ9Û‚Ô’á‡Ý ·ÔsÖ‡× Ò˜×€Ð­Ï‡Õ˜Ñ­Òå ׀ܔނԒÒ4Ô’Ö‚ÔT×€Ü Ï‡Ü ×€Ò˜Û‚Ô’î€Ñ
Ö‡×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 ·ádÏ‚ÔlÜ9×€á‰Ó4×EÔ£Ï‡Ü ÔPÓ•ÔYn‡Ô£Ø€Ñ­á‡ÒŽÓ•Ô’á‰Ó4× p 0
× ·á‡Ö‡×€á ·›à Ô’îV‰×€Ò‚Ò4ï’ÑgÛ‚Ô’Ð}Ô’Òf×€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, . . . .
–)Û‡Õ˜Ñd؀׀ҘҘѣև×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 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‡Ñ­ÒgÖ‚ë…Ñ ·á‡ÒŽÓ•Ô’á‰Ó4צւԅ՘ϐ‰›á‚Ô7ã1ٖ׃ì­×€Ü9Ñ­ÒgÓ•Ô’Ü”Þ{å€Ü
؀ѭá‡Ò ·Ö‡×€Õ4Ô’ÕVÔ Û{Ñ­Ò˜Ò ·Þ ·›ß›·Ö‚Ô’Ö‡×ևרá‚ï’ÑuÝ‚Ôì­×€ÕV՘ϐ‰›á‚Ô7Ú–Ñuõ‰Ï‡×¨Ò ·Ð­á ·U‚ØÔ õ‰Ï‡× τ = ∞
ãÎCÒŽÓ•Ô’Ü9Ñ­Ò
·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‚ÔRÔ’á‰Ó4×€ÒAÖ‡ÑRÓ4×€Ü9Û{Ñ t
Ú P(τ ≤ t)
ã
ý ±¥à noiW•–q“xyi†ijxyffqdqdvyiji•–qѨr‡€‚†psr‰k—ÿ”h–’–•–¹gqsr‡p
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•
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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).
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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 ·ÒŽÓ4Õ ·Þ‡Ï‰›Ö‚Ô’Ò€Új؀ѭÜaÖ ·ÒŽÓ4Õ ·Þ‡Ï ·îï’Ñ1×on7Û{Ñ’á‡×€á‡Ø ·Ô ß Ø€Ñ­Ü
E(Z1) = λ−1 ã
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å£Ö‡×€á‡Ñ’Ó•Ô’Ö‡Ñ9Û–Ñ­Õ
Nt = sup{n ≥ 1 : Tn ≤ t}, t ≥ 0,
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21. Ñ­á‡Ö‡× sup{∅} = 0.
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í–ÒEҘ׀õ‰Ï˜€á‡Ø ·Ô’Ò X1, X2, . . .
× T1, T2, . . .
Ò4ï’Ñ ·á‡Ö‡×€Û–×€á‡Ö‡×€á‰Ó4×€Ò€ã
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ågχÜ3Û‡Õ˜ÑdØ€×€Ò˜Ò˜Ñ Ö‡×©úQÑ ·Ò˜Ò˜Ñ­á…؀ѭÜ!Ó•ÔYn‡Ô λ.
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Ú‚å
Ut = u + pt − St
ø»™}ù
Ñ­á‡Ö‡× St =
Nt
i=1
Xi
ã‚úQÑdևוþÒ4×£Ü9Ñ­ÒŽÓ4Õ4Ô’Õgøí©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{×€Õ4Ô’Ö‡ÑTÖ‡× ·á‡Ö‡×ƒá ·›à Ô’îVd×€ÒVÛ{Ñ­ÕˆÏ‡á ·Ö‚Ô’Ö‡× Ö‡× Ó4×€Ü9Û{чãöEÒ4Ô’á‡Ö‡Ñ
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Ú
×€á}Ó•ï’Ñ Ut → ∞
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×€á‡õ‰Ï‚Ô’á}Ó4ÑRÒ˜× p ρ
Úd×€á}Ó•ï’Ñ Ut → −∞
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ã
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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
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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
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bmÑ­Ü9Ñ Ô’á}Ó4×€Õ ·Ñ­Õ˜Ü9×€á}Ó4×’Ú‚Ò˜×yâ˜Ô Xi
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Ti
× Nt
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ã¢í–Ò˜Ò˜Ï‡Ü ·Ü9Ñ­Ò
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×9õ‰Ï‡× Ñ­ÒgìlÔ ßѭ՘׀ÒgÖ‚Ô’Ò
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Ú’Ñ­á‡Ö‡× E(X) = µ
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êd×yâ˜Ô
St =

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Nt
i=1
Xi
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0
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ø SYX­ù
Ñ ì’Ô ßÑ’Õ1Ó4Ñ’Ó•Ô ß Ö‚Ô’Ò ·á‡Ö‡×€á ·›à Ô’îVd׀҅ԒЭ՘׀ЭԒւԒÒVá‡Ñ ·á‰Ó4×€ÕŽìlÔ ßÑ Ö‡× Ó4×€Ü9Û{Ñ (0, t]
ã –ËÛ‡Õg˜€Ü ·Ñ õ‰Ï‡×
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ã í
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ã–êd×yâ˜Ô Uδ(t)
Ñ
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õ‰Ï‡×
Uδ(t) = ueδt
+ p
t
0
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dυ −
t
0
eδ(t−υ)
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Ú
t
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t
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eδ(t−υ)
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Ñ­á‡Ö‡× β
× σb
Ò4ï’Ñ؀ѭá‡ÒŽÓ•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’чã
íWõ‰Ï · β
՘׀ۇ՘׀Ҙ׀á}Ó•ÔÑVÐ}Ô’á‡Ý‡Ñ…×€Ò˜Û{×€Õ4Ô’Ö‡ÑÛ–Ñ­Õ£Ï‡á ·Ö‚Ô’Ö‡×¦Ö‡×RÓ4×€Ü9Û{Ñ‡Ú¤Ö‡×ƒì ·Ö‡ÑÔˆÏ‡Ü Ø€Ñ‰×©U‚Ø ·×€á‰Ó4×9Ö‡×
Ҙ׀ЭχÕ4Ô’á‡îÔ θ 0
á‡ÑPÛ‡Õg˜€Ü ·Ñ–× σb
å¹ÑPÖ‡×€ÒŽì ·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. Û–Ñ­ÕRÏ‡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×€Ò˜Ü Ô©æçÑ­Õ˜Ü Ô7Ú‡Ô’Ò˜Ò4Ï‡Ü ·Ü9Ñ­ÒWõ‰Ï‡×£Ñ¦ævÔ™Ó•Ñ’ÕWևףև׀Ҙ؀ѭá‰Ó4Ñ ∆t
å£Ï‡Ü ÔRÖ ·æçχÒ4ï’чڇѭÏVÒ˜×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‰Ó4ÑVômÕ˜ÑY`Aá ·Ô’á‡ÑVÛ‚Ô’Ö‡Õ4ï’чã
êd× σd
å©Û{Ñ­Ò ·Ó·ì ­Ñ‡Ú–×€ÒŽÓ•Ô’Ü9Ñ­ÒPÔ’Ö ·Ø ·Ñ­á‚Ô’á‡Ö‡Ñ9Ï‡Ü Ô9Û{×€ÕŽÓ4χ՘ނԒîï’Ñ1×€ÒŽÓ4ÑdØë’ÒŽÓ ·ØÔ1Ô Ï‡Ü Ô¦Ó•ÔYn‡Ô Ö‡×AâŽÏ‡Õ˜Ñ­Ò
؀ѭá‡ÒŽÓ•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× ßÔEٖשU‚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×ƒÑ­Õ˜×€Ü Ô R ãaXdã™7Ú Zt e Yt
Ó•Ô’Ü”Þ{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׀ܦѭÒWÓ•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 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×€Õ˜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. =
1
Utσ2
d + σ2
b
(U2
t σ2
d + σ2
b )dt = dt.
ý“Ñ­Ð­Ñ‡Ú ·á‰Ó4׀ЭÕ4Ô’á‡Ö‡Ñ¦Ô’ÜRÞ–Ñ­ÒWÑ­Ò ßԒևѭÒWև׀ҎӕԦ׀õ‰Ï‚Ô’îï’Ñ9á‡Ñ ·á}Ó4×€ÕŽì’Ô ßÑ [0, t]
Ú‚Ñ­Þ7Ó4×€Ü9Ñ­Ò
d[W]t = dt ⇒ [W]t = t.
Ù–×€ÒŽÓ•Ô)æçÑ­Õ˜Ü Ô7Ú©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×€Ü9Ñ­Ò”õ‰Ï‡× W
å1χÜÜ9Ñ‘ì ·Ü9×€á}Ó4ÑômÕ4Ñ4`Aá ·Ô’á‡Ñ
Û‚Ô’Ö‡Õ4ï’чã¢ý“ѭЭчÚWÛ–Ñdև׀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}Ó4ÑEômÕ˜Ñ4`Aá ·Ô’á‡ÑAÛ‚Ô’Ö‡Õ4ï’ÑE×fÛ{Ñdև׀Ü9ѭҤ؀ѭá‡Ò ·Ö‡×€Õ4Ô’Õ At = σ{Ws; 0 ≤ s ≤ t}
ã
¤ ·Ü9Û–Ñ­ÕŽÓ•Ô’á‰Ó4×ˆÖ ·›à ×€Õ¦õ‰Ï‡×’ڹ؀ѭÜ9Ñ Ut
å…Ï‡Ü Ô¨Ö ·æçχÒ4ï’чڂ׀á‰Ó•ï’Ñ…× ß׈ӕԒÜRÞ–å€Ü å…χÜoÛ‡Õ˜ÑdØ€×€Ò˜Ò˜Ñ
äÔ’Õgª’Ñ‘ì ·Ô’á‡Ñ‡ã
íÃÛ‡Õ ·á‡Ø‰›Û ·Ñ”׀ҎӕԒ՘׀Ü9Ñ­Ò ·á}Ó4׀՘׀ҘÒ4ԒևѭÒEá‚ÔRÒ˜×€Ð­Ï ·á}Ó4×£Û‚Ô’Õ4Ô’Ü9׃Ó4Õ ·›à Ô’îï’ѐæ
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‡×ˆÑ­ÒRØ€Ñd×oU‚Ø ·×€á‰Ó4×€Ò β, σb, δ, e σd
և׀Û{×€á‡Ö‚Ô’Ü Ö‡ÑÓ4×€Ü9Û{Ñ ×1Ö‚Ô՘׀Ҙ׀Վì’Ô
ÔlÓ4Ï‚Ô ßã
í–Ò˜Ò˜Ï‡Ü ·Ü9Ñ­ÒFõ‰Ï‡×AÑ­ÒCØ€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”×VÛ‚Ô’Õ4ԨԒҘҘ׀ЭχÕ4Ô’Õ Ô¨×on ·ÒŽÓe˜€á‡Ø ·Ô¨×V؀ѭá}Ó ·ádÏ ·Ö‚ÔlÖ‡×Vև׈Ó4ÑdÖ‚Ô’Ò9Ô’Ò
Ö‡×€Õ ·ìlÔ’Ö‚Ô’ÒAõ‰Ï‡×gԒۇ՘׀Ҙ׀á‰Ó•Ô’՘׀Ü9Ñ­ÒAá‚ÔRÛ‡Õ˜é4n ·Ü ԔҘ׀îï’чã
²±»° 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 ·ØԒևѭÕ4Ô©Ö‡ÑR׃ì}×€á}Ó4Ñ A
ãwbmÑ­á‡Ò ·Ö‡×€Õ˜×€Ü9Ñ­ÒAÔ”Ö ·ÒŽÓ4Õ ·Þ‡Ï ·îï’Ñ”Ö‡×¤Û‡Õ˜Ñ­Þ‚Ô’Þ ·›ß›·þ
Ö‚Ô’Ö‡×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×€ÕŽìlÔ ßєև×PÓ4×€Ü9Û{Ñ
U‚á ·Ó4чÚ
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. 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}
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‚ԒևשäÔ’Õgª’Ñ‘ì ·Ô’á‚ÔRÖ‡Ñ9Û‡Õ˜Ñd؃׀ҘÒ4Ñ9ւԦ՘׀Ҙ׀ՎìlÔ7ã
í–Û ß›·ØÔ’á‡Ö‡Ñ…ÔVæçé­Õ˜Ü”Ï ßÔˆÖ‡×9û¡Óed…Э׀á‡×€Õ4Ô ß›·›à Ô’Ö‚Ôbm݉χá‡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
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}Ó•×V×VχÒ4Ô’á‡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 ·Ñ­ÒFÛ‚Ô’Õ4ÔEÑ­Þ7Ó4×€á‡îï’ÑPև׀ҎӕÔP×€õ‰Ï‚Ô’îï’Ñ–ì’ï’Ñ¤Ô ßå€Ü Ö‡ÑP׀Ҙ؀ѭÛ{ÑPև׀ҎӕÔPÖ ·Ò˜Ò˜×€ÕŽÓ•Ô’îï’чã
bmÑ­Ü9Ñ£Ô–æçχá‡îï’Ñ It(1−P(t, Ut, v, y))
åW؀ѭá}Óe‰›ádÏ‚ÔP×ƒÜ t
õ‰Ï‚Ô’Ò˜×AØ€×€ÕŽÓ•Ô’Ü9×€á}Ó4×’Ú‰Ô¤Û‚Ô’ÕŽÓ4׹ևѣÒ4Ô ßÓ4чÚ
؀ѭ՘՘׀Ҙۖѭá‡Ö‡×€á‰Ó4פԒєð ßÓ·Ü ¦Ñ£Ó4×€Õ˜Ü9чÚdå à ׀՘єõ‰Ï‚ԒҘפ؃׀ՎÓcÔlÜ9×€á‰Ó4×’ãw– Ó4׀ѭ՘׀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׀՘׀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¬‰«o×”š|’„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„¡| Ý™šR“•‡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}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’чã†bmÑ­Ü9Ñ W
× M
Ò4ï’Ñ
Ü Ô’ÕŽÓ ·á‡Ð}Ô ·Ò€ÚdÑ”õ‰Ï‡×P×€ÒŽÓ•ë”Ö‡Ñ ßÔ’Ö‡Ñ©Ö ·Õ˜× ·Ó4ÑgÖ‚Ô”×€õ‰Ï‚Ô’î€ï’єӕԒÜRÞ–å€Üև׃ì}×PҘ׀գøçÑ ·á‡Ø€Õ˜×€Ü9×€á}Ó4єևבù‚χÜ
X‘i

29. Ü Ô’ÕŽÓ ·á‡Ð}Ô ßã CpQ×€Ü9Ñ­Ò©õ‰Ï‡×1×€ÒŽÓ4×1Ü Ô’ÕŽÓ ·á‡Ð}Ô ß å1Ô’Þ‡Ò˜Ñ ßÏ7Ó•Ô’Ü9×€á‰Ó4×9؀ђá‰Óe‰›á‰Ï‡Ñ× Û{Ñ­ÕŽÓ•Ô’á}Ó4ÑÖ‡× ìlÔ’Õ ·Ô’îï’Ñ
ß›·Ü ·Ó•Ô’Ö‚Ô7ãFý“ѭЭчڇÛ{× ßсpQ×€Ñ­Õ˜×€Ü Ô R ãaXdãTS R × ßפև׃ì­×gҘ׀ÕA؀ѭá‡ÒŽÓ•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‡ÒŽÓ•Ô’á}Ó4×’ãu–ÅÕ˜×€Ò˜Ï ßÓ•Ô’Ö‡Ñ¦Ò˜×€Ð­Ï‡× ·Ü9×ƒÖ ·ÔlӕԒܦ׀á}Ó•×lã
¤ ·Ü9Û–Ñ­ÕŽÓ•Ô’á‰Ó4×gѭއҘ׀ՎìlÔ’Õ¤õ‰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
üEÑ…ØԒҘх݇ѭÜ9ѭБ˜€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
ãÙ–×€ÒŽÓ•ÔæçÑ­Õ˜Ü Ô7Ú¤Ó4Ñ­Õ˜á‚Ô™þÒ˜×9ð7Ó ·›ß Ô
·á}Ó4Õ˜Ñdևχîï’Ñ£Ö‚Ô¤æçχá‡îïlÑ©Ö‡×WÓ4Õg˜€Ò‚Û‚Ô’Õe—’Ü9׃Ó4Õ˜Ñ­Ò P(u, v, y) = P(0, u, v, y)
Ú‰×E؀ѭ՘՘׀Ҙۖѭá‡Ö‡×€á‰Ó4×€Ü9×€á}Ó4×
Ø€Ñ ßщØÔ’Ü9Ñ­Ò
ψ(u, v) = P inf
s∈[0,v]
Us ≤ 0|U0 = u .
ø R h­ù
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z’w€ƒŒ4ƒxj“ƒwv|’³
−
∂
∂v
ψ(u, v) +
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∂u
ψ(u, v)µ(u) +
1
2
∂2
∂u2
ψ(u, v)σ2
(u) = 0,
ø R ñ}ù
šƒ©Ô€ƒwy„¡| Ý™šR“•‡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׀ՎìlÔ’Õ˜Ü9Ñ­ÒAõ‰Ï‡× ψ(u, v) = P(T − v, u, T, 0)
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Ö ·›à õ‰Ï‡×’ÚWÛ‚Ô’Õ4Ô v = 0
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