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
P❡♥❡r❛♣❛♥ ❧❛✐♥♥②❛ ❞❛r✐ ●❡r❛❦ ❇r♦✇♥✱ ❞❡r❛✉ ♣✉t✐❤ ❞❛♥ P❉ st♦❦❛st✐❦

  ▼❛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✐❤

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