Persamaan Diferensial Stokastik dan Beberapa Penerapannya
P❡rs❛♠❛❛♥ ❉✐❢❡r❡♥s✐❛❧ ❙t♦❦❛st✐❦ ❞❛♥ ❇❡❜❡r❛♣❛ P❡♥❡r❛♣❛♥♥②❛
❍❡rr② Pr✐❜❛✇❛♥t♦ ❙✉r②❛✇❛♥ ✸✳ ❆♣r✐❧ ✷✵✶✹
■s✐ Pr❡s❡♥t❛s✐
▼♦t✐✈❛s✐ ❉❡r❛✉ P✉t✐❤ ❞❛♥ ●❡r❛❦ ❇r♦✇♥ P❡rs❛♠❛❛♥ ❉✐❢❡r❡♥s✐❛❧ ❙t♦❦❛st✐❦ ❇❡❜❡r❛♣❛ P❡♥❡r❛♣❛♥
▼♦t✐✈❛s✐
▼♦❞❡❧ P❡rt✉♠❜✉❤❛♥ P♦♣✉❧❛s✐
▼♦❞❡❧ ▼❛❧t❤✉s ✭✶✼✾✽✮✿ dN
(t) = rN(t) N ( >
✵) = N ✵
✵
dt ❚✐❞❛❦ r❡❛❧✐st✐s✦ P❡rs❛♠❛❛♥ ▲♦❣✐st✐❦ ✴ ▼♦❞❡❧ ❱❡r❤✉❧st ✭✶✽✹✺✮✿ dN N
(t) (t) = rN(t) , N ( >
✶ − ✵) = N ✵
✵
dt K P❡♥②❡❧❡s❛✐❛♥✿
N K
✵
N (t) = −rt N
- (K − N )e
✵ ✵
❞❛♥ ♣❡r✐❧❛❦✉ ❥❛♥❣❦❛ ♣❛♥❥❛♥❣ ✭❧♦♥❣ t✐♠❡ ❜❡❤❛✈✐♦✉r✮ N (t) = K . t →∞ ❧✐♠
❇❡❜❡r❛♣❛ ❝❛r❛ ✉♥t✉❦ ♠❡♠♣❡r❜❛✐❦✐ ♠♦❞❡❧ ❧♦❣✐st✐❦✿ ✶ P❡rs❛♠❛❛♥ ❧♦❣✐st✐❦ ②❛♥❣ ❞✐♠♦❞✐✜❦❛s✐✿ dN (t) N(t) N (t)
, > = rN(t) −
✶ ✶ − ✵ < L < K, N(✵) = N ✵ ✵ ✷ dt L K ▲❛❥✉ ♣❡rt✉♠❜✉❤❛♥ t❛❦❦♦♥st❛♥✿ dN (t) N (t)
, N > = r (t)N(t) (
✶ − ✵) = N ✵ ✵ ✸ dt K
P❡rs❛♠❛❛♥ ❧♦❣✐st✐❦ st♦❦❛st✐❦ ✭♠❡♠♣❡rt✐♠❜❛♥❣❦❛♥ ❛❞❛♥②❛ ❞❡r❛✉ ✭♥♦✐s❡✮✮✿
dN t N t t t · D t = rN + αN
✶ − dt K N
= Y >
✵ ✵ t t
❑✐t❛ t✐❞❛❦ t❛❤✉ ♣❡r✐❧❛❦✉ ❡❦s❛❦ ❞❛r✐ ❞❡r❛✉ D ✱ ❤❛♥②❛ ❞✐str✐❜✉s✐ ♣❡❧✉❛♥❣ ❞❛r✐ D ②❛♥❣ ❞✐❦❡t❛❤✉✐✳
❦✉r✈❛ ❧♦❣✐st✐❦ ❞❡t❡r♠✐♥✐st✐❦ ✈s st♦❦❛st✐❦
❞❡t❡r♠✐♥✐st✐❦✿ st♦❦❛st✐❦✿
❇❡❜❡r❛♣❛ ♣❡rt❛♥②❛❛♥ ✭♠❛t❡♠❛t✐s✮ ②❛♥❣ ♠✉♥❝✉❧
❆♣❛ ❛rt✐♥②❛ ❞❛♥ ❢♦r♠✉❧❛s✐ ♠❛t❡♠❛t✐❦❛ ❞❛r✐✿ t ❑✉❛♥t✐t❛s ❛❝❛❦ N ✉♥t✉❦ s❡t✐❛♣ ✇❛❦t✉ t ❂❃ ♣❡✉❜❛❤ ❛❝❛❦ ✭r❛♥❞♦♠ ✈❛r✐❛❜❧❡✮ t
) ❑❡❧✉❛r❣❛ ❦✉❛♥t✐t❛s ❛❝❛❦ (N t ≥ ②❛♥❣ ❞✐✐♥❞❡❦s ♦❧❡❤ ✇❛❦t✉ t ❂❃ ♣r♦s❡s
✵ st♦❦❛st✐❦ ✭st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✮ t
❉❡r❛✉ D ❂❃ ❞❡r❛✉ ♣✉t✐❤ ●❛✉ss✐❛♥ ✭●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡✮ ✭t✉r✉♥❛♥ ❞❛r✐ ❣❡r❛❦ ❇r♦✇♥✮ ■♥t❡❣r❛❧ st♦❦❛st✐❦ Z T t t
N · D dt
✵
❂❃ ✐♥t❡❣r❛❧ ■t♦ ❛t❛✉ ✐♥t❡❣r❛❧ ❙tr❛t♦♥♦✈✐❝❤ P❡rs❛♠❛❛♥ ❞✐❢❡r❡♥s✐❛❧ st♦❦❛st✐❦ t t dN N t t t
= rN + αN · D ✶ − dt K
❂❃ ♣❡rs❛♠❛❛♥ ✐♥t❡❣r❛❧ st♦❦❛st✐❦
❉❡r❛✉ P✉t✐❤ ❞❛♥ ●❡r❛❦ ❇r♦✇♥
❚♦♥❣❣❛❦ s❡❥❛r❛❤ ❣❡r❛❦ ❇r♦✇♥ ❞❛♥ ❞❡r❛✉ ♣✉t✐❤ ✶ ✷ ❘✳ ❇r♦✇♥ ✭✶✽✷✼✮✿ ♣❡r❝♦❜❛❛♥ s❡r❜✉❦ s❛r✐ t✉♠❜✉❤❛♥ ♣❛❞❛ ❧❛r✉t❛♥ ✸ ▲✳ ❇❛❝❤❡❧✐❡r ✭✶✾✵✵✮✿ ♣❡♠♦❞❡❧❛♥ ❜✉rs❛ s❛❤❛♠ P❛r✐s ❞❡♥❣❛♥ ❣❡r❛❦ ❇r♦✇♥
❆✳ ❊✐♥st❡✐♥ ✭✶✾✵✺✮✿ t❡♦r✐ ♣❡rt❛♠❛ ❣❡r❛❦ ❇r♦✇♥ t❡r❦❛✐t ❞❡♥❣❛♥ ♣❡rs❛♠❛❛♥ ✹ ♣❛♥❛s✴❞✐❢✉s✐ ✺ ◆✳ ❲✐❡♥❡r ✭✶✾✷✸✮✿ ❢♦♥❞❛s✐ ♠❛t❡♠❛t✐❦❛ ②❛♥❣ r✐❣♦r ✉♥t✉❦ ❣❡r❛❦ ❇r♦✇♥ ✻ ❑✳ ■t♦ ✭✶✾✹✷✮✿ ♣❡♥❡♠✉❛♥ ❦❛❧❦✉❧✉s st♦❦❛st✐❦ ✭✐♥t❡❣r❛❧ t❡r❤❛❞❛♣ ❣❡r❛❦ ❇r♦✇♥✮ ❋✳ ❇❧❛❝❦ ❞❛♥ ▼✳ ❙❝❤♦❧❡s ✭✶✾✼✸✮✿ ❘✉♠✉s ❇❧❛❝❦✲❙❝❤♦❧❡s ✉♥t✉❦ ❤❛r❣❛ ♦♣s✐ ✼ t✐♣❡ ❊r♦♣❛ ❞❛❧❛♠ ❦❡✉❛♥❣❛♥ ❚✳ ❍✐❞❛ ✭✶✾✼✻✮✿ ❢♦♥❞❛s✐ ♠❛t❡♠❛t✐❦❛ ②❛♥❣ r✐❣♦r ✉♥t✉❦ ❞❡r❛✉ ♣✉t✐❤ ✭✇❤✐t❡ ✽ ♥♦✐s❡ ❛♥❛❧②s✐s✮ ▲✳ ❙tr❡✐t ✭✶✾✽✸✮✿ P❡♠❡❝❛❤❛♥ ♠❛s❛❧❛❤ ✐♥t❡❣r❛❧ ❋❡②♥♠❛♥ ❞✐ ❞❛❧❛♠ ♠❡❦❛♥✐❦❛ ✾ ❦✉❛♥t✉♠ ❞❡♥❣❛♥ ❛♥❛❧✐s✐s ❞❡r❛✉ ♣✉t✐❤ ▼✳ ❙❝❤♦❧❡s ❞❛♥ ❘✳ ▼❡rt♦♥ ✭✶✾✾✼✮✿ ◆♦❜❡❧ ❊❦♦♥♦♠✐ ✉♥t✉❦ r✉♠✉s ✶✵ ❇❧❛❝❦✲❙❝❤♦❧❡s ❲✳ ❲❡r♥❡r ✭✷✵✵✻✮✿ ▼❡❞❛❧✐ ❋✐❡❧❞ ✉♥t✉❦ ♠❛s❛❧❛❤ s❡❧❢✲✐♥t❡rs❡❝t✐♦♥ ❣❡r❛❦ ❇r♦✇♥ ❞✐♠❡♥s✐ t✐♥❣❣✐
❉❡r❛✉ P✉t✐❤ ✭❲❤✐t❡ ◆♦✐s❡✮
❉❡r❛✉✿ t❛❦♣❡r✐♦❞✐❦✱ ❦♦♠♣❧❡❦s✱ t✐❞❛❦ ♠❡♥②❡♥❛♥❣❦❛♥✱ s✉❛r❛ ❛t❛✉ s✐♥②❛❧ ②❛♥❣ r✉s❛❦ ✭❝♦rr✉♣t❡❞✮ ❉❡r❛✉ ♣✉t✐❤✿ ❞❡r❛✉ ❛❦✉st✐❦ ❛t❛✉ ❡❧❡❦tr✐❦ ②❛♥❣ ♠❡♠✉❛t s❡♠✉❛ ❢r❡❦✉❡♥s✐ ②❛♥❣ ❞❛♣❛t ❞✐❞❡♥❣❛r ❞❡♥❣❛♥ ✐♥t❡♥s✐t❛s ②❛♥❣ s❛♠❛ P✉t✐❤ ❜❡r❛rt✐ ❞❡r❛✉ t❡rs❡❜✉t t❡rs✉s✉♥ ❞❛r✐ s❡♠✉❛ ❢r❡❦✉❡♥s✐ ♣❛❞❛ s♣❡❦tr✉♠ ②❛♥❣ ❞❛♣❛t ❞✐❞❡♥❣❛r✱ t❡r❞✐str✐❜✉s✐ s❡❝❛r❛ ❛❝❛❦✳ ❍❛❧ ✐♥✐ ❛♥❛❧♦❣ ❞❡♥❣❛♥ ❝❛❤❛②❛ ♣✉t✐❤ ②❛♥❣ t❡rs✉s✉♥ ❞❛r✐ s❡♠✉❛ ✇❛r♥❛ ♣❛❞❛ s♣❡❦tr✉♠ ✈✐s✉❛❧✳ ❉✐ ❞❛❧❛♠ ♣❡♥❡r❛♣❛♥✱ ❞❡r❛✉ ♣✉t✐❤ ❞✐❣✉♥❛❦❛♥ s❡❜❛❣❛✐ s❡❜✉❛❤ ✐❞❡❛❧✐s❛s✐ ♠❛t❡♠❛t✐s ❞❛r✐ ❢❡♥♦♠❡♥❛✲❢❡♥♦♠❡♥❛ ②❛♥❣ ♠❡♠✉❛t ✢✉❦t✉❛s✐ ②❛♥❣ ♠❡♥❞❛❞❛❦ ❞❛♥ s❛♥❣❛t ❜❡s❛r✳ ❉❡r❛✉ ♣✉t✐❤ ●❛✉ss✐❛♥ ✭●❛✉ss✐❛♥ ✇❤✐t❡ ♥♦✐s❡✮✿ t❡r❦❛✐t ❞❡♥❣❛♥ t❡♦r✐ ❜❛❤✇❛ ❞❡r❛✉ ♣✉t✐❤ ❛❞❛❧❛❤ t✉r✉♥❛♥ ✭t❡r❤❛❞❛♣ ✇❛❦t✉✮ ❞❛r✐ ❣❡r❛❦ ❇r♦✇♥✳
❉❡r❛✉ P✉t✐❤✿
●❡r❛❦ ❇r♦✇♥ ❛❞❛❧❛❤ ♣r♦s❡s st♦❦❛st✐❦ B = (B ✵ ②❛♥❣ t❡r❞❡✜♥✐s✐ ♣❛❞❛ s❡❜✉❛❤ r✉❛♥❣ ♣❡❧✉❛♥❣ (Ω, F, P) s❡❤✐♥❣❣❛✿ ✶
B = ✷
✵ ✵ P✲❤❛♠♣✐r ♣❛st✐
B ✸ t s ♠❡♠✐❧✐❦✐ ❦❡♥❛✐❦❛♥ ②❛♥❣ ❜❡❜❛s ✭✐♥❞❡♣❡♥❞❡♥t ✐♥❝r❡♠❡♥ts✮ B − B ∼ N ( ✹ ✵, t − s) ✭♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞✮ t
(ω) P✲❤❛♠♣✐r ♣❛st✐ t 7→ B ❦♦♥t✐♥✉
P❛rt✐❦❡❧ ❇r♦✇♥✐❛♥ t✐❞❛❦ ♠❡♠✐❧✐❦✐ ❧❛❥✉✿ B t t dB t B t t +ε ✶ +ε − B − B
∼ N ( ) =⇒ =
✵, ❧✐♠ t✐❞❛❦ ❛❞❛✦ ε→ ε ε dt ✵ ε
❋❛❦t❛✿ ❉❡♥❣❛♥ ♣❡❧✉❛♥❣ s❛t✉✱ tr❛②❡❦t♦r✐ ✭❧✐♥t❛s❛♥ s❛♠♣❡❧✮ ❣❡r❛❦ ❇r♦✇♥✐❛♥ ❜❡rs✐❢❛t ❦♦♥t✐♥✉ ❞✐♠❛♥❛✲♠❛♥❛ t❛♣✐ t✐❞❛❦ t❡r❞✐❢❡r❡♥s✐❛❧ ❞✐♠❛♥❛✲♠❛♥❛✳ ❂❃ ✐♥t❡❣r❛❧
❘✐❡♠❛♥♥✲❙t✐❡❧t❥❡s t✐❞❛❦ ❜✐s❛ ❞✐❣✉♥❛❦❛♥
●❡r❛❦ ❇r♦✇♥ ❜❡rs✐❢❛t s❡r✉♣❛ ❞✐r✐ ✭s❡❧❢✲s✐♠✐❧❛r✮ ❂❃ t❡r❦❛✐t ❞❡♥❣❛♥ ❢r❛❦t❛❧ ●❡r❛❦ ❇r♦✇♥ ❛❞❛❧❛❤ ♣r♦s❡s ▼❛r❦♦✈ ❂❃ t✐❞❛❦ ♣✉♥②❛ ♠❡♠♦r✐ ●❡r❛❦ ❇r♦✇♥ ❛❞❛❧❛❤ ♣r♦s❡s ●❛✉ss✐❛♥ ❂❃ ❑❛❥✐❛♥ ♣r♦❜❛❜✐❧✐t✐❦ ❞❛♥ ❛♥❛❧✐t✐❦♥②❛
r❡❧❛t✐❢ ♠✉❞❛❤ Pr♦s❡s st❛t✐♦♥❡r ❧❡♠❛❤ ❛❞❛❧❛❤ ♣r♦s❡s st♦❦❛st✐❦ (X ❞❡♥❣❛♥ s✐❢❛t✿ ✶ t
✵
) = m ✷ E(X t u
− m)(X − m)) = F (t) E((X +u
✷ ✷
F u − m) = σ
♣♦s✐t✐❢ ❞❡✜♥✐t ❞❛♥ F (✵) = E(X ✳ ❉❡♥❣❛♥ ❛s✉♠s✐ F ❦♦♥t✐♥✉✱ t❡♦r❡♠❛ ❇♦❝❤♥❡r ♠❡♠❜❡r✐❦❛♥ Z itx F (t) = e f (x) dx. R ❉❡r❛✉ ♣✉t✐❤ ❛❞❛❧❛❤ s❡❜❛❣❛✐ ♣r♦s❡s ●❛✉ss✐❛♥ st❛t✐♦♥❡r ❧❡♠❛❤ s❡❤✐♥❣❣❛ ❢✉♥❣s✐
✷
= ∞ ❦❡♣❛❞❛t❛♥ s♣❡❦tr❛❧♥②❛ f ❦♦♥st❛♥✳ ❆❦✐❜❛t♥②❛✱ σ ✳ t ❉❡r❛✉ ♣✉t✐❤ ❛❞❛❧❛❤ ♣r♦s❡s ●❛✉ss✐❛♥ D ②❛♥❣ s❛❧✐♥❣ ❜❡❜❛s ♣❛❞❛ ✇❛❦t✉ ②❛♥❣ ❜❡r❜❡❞❛ ❞❛♥ ♠❡♠✐❧✐❦✐ ❞✐str✐❜✉s✐ ✐❞❡♥t✐❦ ❞❡♥❣❛♥ r❛t❛✲r❛t❛ ✵ ❞❛♥ ✈❛r✐❛♥s✐ ∞✱ ❞❛❧❛♠ ❛rt✐✿ t D s e dx Z i (t−s)x
E(D ) = = δ(t − s) R ✭s❡❝❛r❛ ♠❛t❡♠❛t✐❦❛✱ ❦❡❞✉❛ ❞❡✜♥✐s✐ ❞✐ ❛t❛s ❜❡❧✉♠ ❜✐s❛ ❞✐t❡r✐♠❛ ✶✵✵ ♣❡rs❡♥✮ ❚❡♦r✐ ❞❡r❛✉ ♣✉t✐❤ ②❛♥❣ r✐❣♦r s❡❝❛r❛ ♠❛t❡♠❛t✐❦❛ ❛❞❛❧❛❤ ♠❡❧❛❧✉✐ t❡♦r✐ ❞✐str✐❜✉s✐ ✭❣❡♥❡r❛❧✐③❡❞ ❢✉♥❝t✐♦♥✮ st♦❦❛st✐❦ ♣❛❞❛ s❡❜✉❛❤ r✉❛♥❣ ✈❡❦t♦r t♦♣♦❧♦❣✐ ❜❡r❞✐♠❡♥s✐ t❛❦❤✐♥❣❣❛✳ ✭❚✳ ❍✐❞❛✱✶✾✼✻✮✳
P❡rs❛♠❛❛♥ ❉✐❢❡r❡♥s✐❛❧ ❙t♦❦❛st✐❦
❈♦♥t♦❤
P❡rs❛♠❛❛♥ ▲❛♥❣❡✈✐♥ dX t t dt t ,
X = −bX + a dB = x
✵ ✵
❙♦❧✉s✐ P❉❙ ✐♥✐ ❛❞❛❧❛❤ ♣r♦s❡s ❖r♥st❡✐♥✲❯❤❧❡♥❜❡❝❦ t u −bt −b (t−u) Z t X = e x + a e dB
✵ ✵
P❡rs❛♠❛❛♥ ▲♦❣✐st✐❦ ❙t♦❦❛st✐❦ t N dN t t dt t dB t , N
= rN + αN = Y > ✶ − ✵ ✵
K ❙♦❧✉s✐ P❉❙ ✐♥✐ ❛❞❛❧❛❤ ♣r♦s❡s ▲♦❣✐st✐❦ (rK − )t+αB t ✷ ✶ ✷ α e
N t = t − (rK − )s+αB s R ✶ ✷ α
✶ ✷
Y + r e ds
✵
❙❡❝❛r❛ ✉♠✉♠✿ P❡rs❛♠❛❛♥ ❞✐❢❡r❡♥s✐❛❧ st♦❦❛st✐❦ t t t t , dX = f (t, X ) dt + σ(t, X ) D dt X = Y
✵
❞✐t✉❧✐s❦❛♥ s❡❜❛❣❛✐ dX t t t t ,
X = f (t, X ) dt + σ(t, X ) dB = Y
✵
❞❛♥ ❞✐✐♥t❡r♣r❡t❛s✐❦❛♥ ✭❞✐♠❛❦♥❛✐ s❡❝❛r❛ ♠❛t❡♠❛t✐s✮ s❡❜❛❣❛✐ ♣❡rs❛♠❛❛♥ ✐♥t❡❣r❛❧ st♦❦❛st✐❦ Z Z t t X t f s σ s s
= Y + (s, X (s, X ) dB + ) ds | {z } | {z }
✵ ✵
✐♥t❡❣r❛❧ ❞❡t❡r♠✐♥✐st✐❦ ✐♥t❡❣r❛❧ st♦❦❛st✐❦
✐♥t❡❣r❛❧ ❞❡t❡r♠✐♥✐st✐❦ ✿ ✐♥t❡❣r❛❧ ❘✐❡♠❛♥♥✱ ✐♥t❡❣r❛❧ ▲❡❜❡s❣✉❡✱ ✐♥t❡❣r❛❧ ❍❡♥st♦❝❦✱ ❞s❜ ✐♥t❡❣r❛❧ st♦❦❛st✐❦✿ ✐♥t❡❣r❛❧ ■t♦✱ ✐♥t❡❣r❛❧ ❙tr❛t♦♥♦✈✐❝❤✱ ✐♥t❡❣r❛❧ ❘✉ss♦✲❱❛❧❧♦✐s✱ ❞s❜
❚❡♦r❡♠❛ ❊❦s✐st❡♥s✐✲❑❡t✉♥❣❣❛❧❛♥ ❙♦❧✉s✐ ❞❛❧❛♠ ❦❛❧❦✉❧✉s ■t♦ ❚❤❡♦r❡♠
▼✐s❛❧❦❛♥ f (t, x) ❞❛♥ σ(t, x) ❛❞❛❧❛❤ ❢✉♥❣s✐✲❢✉♥❣s✐ t❡r✉❦✉r ♣❛❞❛ [✵, T ] × R ②❛♥❣ ♠❡♠❡♥✉❤✐ ❦♦♥❞✐s✐ ▲✐♣s❝❤✐t③ ❞❛♥ ❦♦♥❞✐s✐ ♠❡♠❜❡s❛r s❡❝❛r❛ ❧✐♥❡❛r ❞❛❧❛♠ ♣❡✉❜❛❤ x✱
✷
) < ∞ ❞❛♥ Y ❛❞❛❧❛❤ ♣❡✉❜❛❤ ❛❝❛❦ ②❛♥❣ t❡r❛❞❛♣t❛s✐ t❡r❤❛❞❛♣ F ❞❡♥❣❛♥ E(Y ✳
✵
▼❛❦❛ ♣❡rs❛♠❛❛♥ ✐♥t❡❣r❛❧ t s s s Z Z t t X = Y + f (s, X ) ds + σ (s, X ) dB
✵ ✵
♠❡♠♣✉♥②❛✐ s❡❜✉❛❤ s♦❧✉s✐ ❦♦♥t✐♥✉ ②❛♥❣ t✉♥❣❣❛❧✳ ▲❡❜✐❤ ❧❛♥❥✉t s♦❧✉s✐ ✐♥✐ ❛❞❛❧❛❤ s❡❜✉❛❤ ♣r♦s❡s ▼❛r❦♦✈✳ ❆❧❛t ♣❡♥t✐♥❣ ❧❛✐♥♥②❛✿ ❘✉♠✉s ■t♦ ❉✐❜❡r✐❦❛♥ s❡❜✉❛❤ ❢✉♥❣s✐ ❦♦♥t✐♥✉ f (t, x) ❞❡♥❣❛♥ t✉r✉♥❛♥✲t✉r✉♥❛♥ ♣❛rs✐❛❧ ②❛♥❣ ∂f ∂f ∂ f ✷ ✷ ❦♦♥t✐♥✉ ∂t ✱ ∂x ✱ ❞❛♥ ∂x ✱ ♠❛❦❛ Z Z t t
✷
∂f ∂f ∂ f ✶ t s s s s + f (t, B ) = f ( ) + (s, B ) dB (s, B ) + (s, B ) ds
✵, B
✵ ✷
∂x ∂t ∂x ✷
✵ ✵
❇❡❜❡r❛♣❛ P❡♥❡r❛♣❛♥ P❡♥②❛r✐♥❣❛♥ ❙t♦❦❛st✐❦ ✭st♦❝❤❛st✐❝ ✜❧t❡r✐♥❣✮ t
❑❡❛❞❛❛♥ s✐st❡♠ ✭♣r♦s❡s ✐♥♣✉t✮ X ♣❛❞❛ s❡t✐❛♣ ✇❛❦t✉ t✿ t t t dX = α(t) dB + β(t)X dt , X ♣❛❞❛ s❛❛t t = ✵, t ✵
α (t), β(t) ❢✉♥❣s✐ ❞❡t❡r♠✐♥✐st✐❦✱ B ❣❡r❛❦ ❇r♦✇♥✱ ❞✐str✐❜✉s✐ ❛✇❛❧ X s❛❧✐♥❣ t
✵
❜❡❜❛s ❞❡♥❣❛♥ B t ❖❜s❡r✈❛s✐ ✭♣r♦s❡s ♦✉t♣✉t✮ Z ❞❛r✐ s✐st❡♠ ♣❛❞❛ ✇❛❦t✉ t✿ dZ t t t dt , Z
= f (t) dW + g (t)X =
✵ ✵,
f t t (t)
✱ g(t) ❢✉♥❣s✐ ❞❡t❡r♠✐♥✐st✐❦✱ W ❣❡r❛❦ ❇r♦✇♥ ②❛♥❣ s❛❧✐♥❣ ❜❡❜❛s ❞❡♥❣❛♥ B ❞❛♥ X ✵ s
▼❛s❛❧❛❤ ♣❡♥②❛r✐♥❣❛♥✿ ❜❡r❞❛s❛r❦❛♥ ♥✐❧❛✐✲♥✐❧❛✐ ②❛♥❣ t❡r❛♠❛t✐ Z ✱ ✵ ≤ s ≤ t✱ X t t
❜❛❣❛✐♠❛♥❛ ♠❡♥❡♥t✉❦❛♥ ❡st✐♠❛t♦r t❡r❜❛✐❦ ˆ ❞❛r✐ ❦❡❛❞❛❛♥ X ❞❛r✐ s✐st❡♠ ♣❛❞❛ ✇❛❦t✉ t❄ P❡♥②❡❧❡s❛✐❛♥ ❞❡♥❣❛♥ ♠❡♥❣❣✉♥❛❦❛♥ ♠❡t♦❞❡ ❦❡s❛❧❛❤❛♥ r❛t❛✲r❛t❛ ❦✉❛❞r❛t t❡r❦❡❝✐❧ ✭❧❡❛st ♠❡❛♥ sq✉❛r❡ ❡rr♦r ♠❡t❤♦❞✮✿
X t ❉✐❝❛r✐ ❡st✐♠❛t♦r ˆ ②❛♥❣ ♠❡♠✐♥✐♠❛❧❦❛♥ ❦❡s❛❧❛❤❛♥ r❛t❛✲r❛t❛ ❦✉❛❞r❛t✿
✷ ✷
R t X t X t t − ˆ (X − Y )
:= E ≤ E
✷
✉♥t✉❦ s❡t✐❛♣ ♣❡✉❜❛❤ ❛❝❛❦ Y ∈ L (P) ②❛♥❣ t❡r✉❦✉r t❡r❤❛❞❛♣ ❛❧❥❛❜❛r✲σ Z F := σ {Z s : s ≤ t} . t Z t t ✷
X (F t )
❊st✐♠❛t♦r ˆ ❛❞❛❧❛❤ ♣r♦②❡❦s✐ ♦rt♦❣♦♥❛❧ ❞❛r✐ X ❦❡ r✉❛♥❣ ❍✐❧❜❡rt L ❞❛♥ ❜❡r❧❛❦✉ Z
ˆ t t
X F t = E X t
❏❛❞✐✱ ❡❦s♣❡❦t❛s✐ ❜❡rs②❛r❛t ❛❞❛❧❛❤ ❡st✐♠❛t♦r t❡r❜❛✐❦ ✉♥t✉❦ ❦❡❛❞❛❛♥ X ❞❛r✐ s s✐st❡♠ ❜❡r❞❛s❛r❦❛♥ ♦❜s❡r✈❛s✐ Z ✱ ✵ ≤ s ≤ t✳ t Z F t
❇❛❣❛✐♠❛♥❛ ♠❡♥❡♥t✉❦❛♥ E X ❄
✳
R
, ❞❡♥❣❛♥ R t ❛❞❛❧❛❤ ♣❡♥②❡❧❡s❛✐❛♥ ♣❡rs❛♠❛❛♥ ❘✐❝❝❛t✐ dR t dt
= α(t)
✷
− g (t)
✷
f (t)
✷
✷ t
= µ
, R
✵
= σ
✷ ✵
. ▲❡❜✐❤ ❧❛♥❥✉t✱ R t = E
(X t − ˆ X t
)
✷
✵
✵
❚❤❡♦r❡♠ ✭❑❛❧♠❛♥✲❇✉❝②✮
= E X t F Z t ❛❞❛❧❛❤ ♣❡♥②❡❧❡s❛✐❛♥ ❞❛r✐ ♣❡rs❛♠❛❛♥
❏✐❦❛ ✶ ❦❡❛❞❛❛♥ X t ❞❛r✐ s❡❜✉❛❤ s✐st❡♠ ❞✐❜❡r✐❦❛♥ ♦❧❡❤ dX t
= α(t) dB t + β(t)X t dt ✱ ✷
❞✐str✐❜✉s✐ ✐♥✐t✐❛❧ X ✵ s❛❧✐♥❣ ❜❡❜❛s ❞❡♥❣❛♥ ❣❡r❛❦ ❇r♦✇♥ B t ❞❛♥ ♠❡♠✐❧✐❦✐ r❛t❛✲r❛t❛ µ ✵ ❞❛♥ ✈❛r✐❛♥s✐ σ
✷ ✵ ✸
❖❜s❡r✈❛s✐ Z t ❞❛r✐ s✐st❡♠ ❞✐❜❡r✐❦❛♥ ♦❧❡❤ dZ t = f (t) dW t + g (t)X t dt
✱ Z ✵ =
✵✱ ❞❡♥❣❛♥ ❣❡r❛❦ ❇r♦✇♥ W t s❛❧✐♥❣ ❜❡❜❛s ❞❡♥❣❛♥ B t ❞❛♥ X ✵ ✱
♠❛❦❛ ❡❦s♣❡❦t❛s✐ ❜❡rs②❛r❛t ˆ X t
❞✐❢❡r❡♥s✐❛❧ st♦❦❛st✐❦ d ˆ X t = g
X
(t)R t f (t)
✷
dZ t + β (t) − g (t)
✷
R t f (t)
✷
ˆ
X t dt , ˆ
- ✷β(t)R t
▼❛t❡♠❛t✐❦❛ ❦❡✉❛♥❣❛♥ ✭❞✐♥❛♠✐❦❛ ❤❛r❣❛ s❛❤❛♠✴❛s❡t ❜❡r❤❛r❣❛✴♥✐❧❛✐ ❦✉rs ✈❛❧✉t❛ ❛s✐♥❣✱ ❞s❜✮ ❘❛♥❣❦❛✐❛♥ ❧✐str✐❦ ❞❡♥❣❛♥ ❞❡r❛✉ P❡r❣❡r❛❦❛♥ ❛❝❛❦ ❞❛r✐ ✭♠✐❦r♦✮♦r❣❛♥✐s♠❡ ▼❛s❛❧❛❤ t✉r❜✉❧❡♥s✐ ❞❛❧❛♠ ❞✐♥❛♠✐❦❛ ✢✉✐❞❛ ✭♣❡rs❛♠❛❛♥ ◆❛✈✐❡r✲❙t♦❦❡s st♦❦❛st✐❦✮ P❡♠♦❞❡❧❛♥ ♣♦❧✐♠❡r ♣❛❞❛ ✜s✐❦❛ ■♥t❡❣r❛❧ ❋❡②♥♠❛♥ ❞❛❧❛♠ ♠❡❦❛♥✐❦❛ ❦✉❛♥t✉♠ ❚r❛♥s❢♦r♠❛s✐ ❋♦✉r✐❡r ❞✐♠❡♥s✐ t❛❦❤✐♥❣❣❛ ▼❛s❛❧❛❤ ❉✐r✐❝❤❧❡t ❞❛❧❛♠ ♣❡rs❛♠❛❛♥ ❞✐❢❡r❡♥s✐❛❧ ♣❛rs✐❛❧ ❞s❜
❉❛❢t❛r P✉st❛❦❛
❇✳ ❖❦s❡♥❞❛❧✳ ❙t♦❝❤❛st✐❝ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✻t❤ ❡❞✳ ✱ ❙♣r✐♥❣❡r✱ ✷✵✵✺
■✳ ❑❛r❛t③❛s ❛♥❞ ❙✳ ❙❤r❡✈❡✳ ❇r♦✇♥✐❛♥ ▼♦t✐♦♥ ❛♥❞ ❙t♦❝❤❛st✐❝ ❈❛❧❝✉❧✉s✱ ✷♥❞ ❡❞✱ ❙♣r✐♥❣❡r✱ ✶✾✾✾❏✳▼✳ ❙t❡❡❧❡✳ ❙t♦❝❤❛st✐❝ ❈❛❧❝✉❧✉s ❛♥❞ ❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ✷✵✵✶
P✳ ❑❛❧❧ ❛♥❞ ❏✳ ▼❛②❡r✳ ❙t♦❝❤❛st✐❝ ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✱ ✷♥❞ ❡❞✳ ❙♣r✐♥❣❡r✱ ✷✵✶✶ ❏✳ ❳✐♦♥❣✳ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❙t♦❝❤❛st✐❝ ❋✐❧t❡r✐♥❣ ❚❤❡♦r②✱ ❖❯P✱ ✷✵✵✽ ▼✳ ❇❛❝❤❛r✱ ❡t ❛❧✳ ❙t♦❝❤❛st✐❝ ❇✐♦♠❛t❤❡♠❛t✐❝❛❧ ▼♦❞❡❧s✱ ❙♣r✐♥❣❡r✱ ✷✵✶✸ P✳ ❑❧♦❡❞❡♥ ❛♥❞ ❊✳ P❧❛t❡♥✳ ◆✉♠❡r✐❝❛❧ ❙♦❧✉t✐♦♥ ♦❢ ❙❉❊s✱ ❙♣r✐♥❣❡r✱ ✶✾✾✷ ❘✳ ❑❤❛s♠✐♥s❦✐✐✳ ❙t♦❝❤❛st✐❝ ❙t❛❜✐❧✐t② ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✷♥❞ ❡❞✳ ❙♣r✐♥❣❡r✱ ✷✵✶✷ ❈✳ Pr❡✈♦t ❛♥❞ ▼✳ ❘ö❝❦♥❡r✳ ❆ ❈♦♥❝✐s❡ ❈♦✉rs❡ ♦♥ ❙t♦❝❤❛st✐❝ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ✷✵✵✼❚✳ ❍✐❞❛✱ ❍✲❍✳ ❑✉♦✱ ❏✳ P♦tt❤♦✛✱ ❛♥❞ ▲✳ ❙tr❡✐t✳ ❲❤✐t❡ ◆♦✐s❡✳ ❆♥ ■♥✜♥✐t❡
❉✐♠❡♥s✐♦♥❛❧ ❈❛❧❝✉❧✉s✱ ❑❧✉✇❡r✱ ✶✾✾✸❚❡r✐♠❛ ❦❛s✐❤