# 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 ( &gt;

✵) = N ✵

✵

dt ❚✐❞❛❦ r❡❛❧✐st✐s✦ P❡rs❛♠❛❛♥ ▲♦❣✐st✐❦ ✴ ▼♦❞❡❧ ❱❡r❤✉❧st ✭✶✽✹✺✮✿ dN N

(t) (t) = rN(t) , N ( &gt;

✶ − ✵) = 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)

, &gt; = rN(t) −

✶ ✶ − ✵ &lt; L &lt; K, N(✵) = N ✵ ✵ dt L K ▲❛❥✉ ♣❡rt✉♠❜✉❤❛♥ t❛❦❦♦♥st❛♥✿ dN (t) N (t)

, N &gt; = 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 &gt;

✵ ✵ 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 &gt; ✶ − ✵ ✵

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✱

✷

) &lt; ∞ ❞❛♥ 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✐❤