A Probabilistic Numerical Method for Fully Non-linear Parabolic Partial Differential Equations

A Probabilistic Numerical Metod for Fully Non-linear Parabolic Partial Differential Equations Aras Faim To cite tis version: Aras Faim. A Probabilistic Numerical Metod for Fully Non-linear Parabolic Partial Differential Equations. Matematics [mat]. Ecole Polytecnique X, 2010. Englis. <tel-00540175> HAL Id: tel-00540175 ttps://pastel.arcives-ouvertes.fr/tel-00540175 Submitted on 11 Dec 2010 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. Te documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés.

P P PP P t t t t t P t2 t t s 2 r s P P P r2 s s s r 3 r rs s st t t t r r t q t t t q t s s P rs té P r s P r s s s s r 3 r P 2t q P r s Pr s t rs té P r s P r s 1 t rs rs té P r s P r s st r rs t2 t r rs t2 r t r rs t2 2 r

ts r t t t 3 r 3 r s r s s rt r 2 P t st P r t r str t s rts r 2 P t rs r 1 rs r t t t 2 s t s r t 2 t s s t s s t s r t r s s st t t s r s r r s r t s st s t str t r t t s s st t t r s r t t s r t st 2 s 2 sr t 1tr 2 t t t t t s s s t r

Pr st r t r 2 r P r P rt r t q t s str t r ç s t t s 1 trt rt r r tr t t r st r q r s Ps r q t t t r t s s r s r r t s s2 t t q s r t t 1 r t ss 2s rr r à r 1 t s ér t r ét t2 t r s Ps èt t r s r ss t s s rs t s é r t t s r 1 r r t r t r r r 2 r è s t rt ss rt t P t t é r st st t é r s éq t s é t trô t st st q s rt s s 1 st s s t s s s ss q P r séq t t s t s s té st t sé r s P t t é r s r s q s t 1 t s r s s t s s s é s r 1 t s t s très rt ts P r tr r r ét t sé s tt t ès été tr t r r s t s rs tr 1 r ss t rés t t r rs s s t s s s té r s t r é t à rt r ér t t t st ré t r t 1 r s s s sé q P é r té t2 tr s t r s é r té st r ér r s ér t rs é r s t ès t sé ét r2 s ts s é t r 1 t r s2stè éq t s é s r t r s r s s r s t 1 r s æ r s é s r q rt tr r r 1 t s s ér s t s P r ss st t rs s s t r ér r s r r s é t q rés r t ss r t t é ér s t ét à s éq t s té r ér t s st s t t t s r s ê s r ts q s s r t r r t t 1 r t s t q s tr t té s é t r r 1 t s t r s 1 r èr rt s r t r é st stré r q q s 1 ér s ér q s ét st t sé r rés r r è é étr q s 1 r r 2 r è sé t s r rt t s t té st st q s è st t r è sé t rt 1 t s à s t té st st q s t s t è st t tr è 1 è rt t ès tr t t q r t t s r é s t s s r s é ss r r é s r s é ss s r st r r é r ttr æ r r t 2 t s s é r t t s s q 2 s t r é q 2 t r é s s s r r t r r q 2 r s r t r q st s t r r é t q 1 st r é r r t r s s tré q

s s r rs r t t st t rs é t s r r s s s tré q r s r t r t é é r r é t r r sq t r r s s r t t tt rt st stré r q q s 1 ér s ér q s q tr s s q r r t r t é é r r t t

str t s s t s s s t t rts rst rt tr s r st r t r 2 r r P s s r ts s2 t t r rt s r r t r t rr r 2s s t r 1 t t 1 t t 2 r P s r 2 t s r s s r rt s t r t r rt t ss 2 r P s s t q t s r s st st t tr st s s t r 1 sts s t ss s s r r t t s s t2 s t s s r t 2 r P s t t s r s t 2 t s t r 1 t s s s t r t t r t r 1 t s t t t s s t2 s t 2 r P s t r t s t s s t t t r r s t s tr 2 r s s t r ss s r r r s t r r s t t s s t2 s t s r 2 r 1 t s t t r s st t t st s r r t r t r s s t t t P s r t2 t2 t r r s t r t2 s r r t rs t s s s t r2 t s ts s t s2st r 1 t q t s t t r r t s r t t t s s t t 1 t t s s t t t r 2 r r t st t r r ss st t rs t r t r s t s r s r s t r t r t r r 3 t t t t P s s str t r r s t s r ts s t s t t r t r t r r s 1 t s rs r s t t r r 1 t t r t r s s 2 s s t s r ss rst rt 2 s r 1 r ts t s s t s t tr r r t r t r rt s t ss t t st st t t2 st t r rt s t t ss ts t t st st t t2 s t s s st t t r s rt t t s s s t t t r t 2 r t r ss r t r ss r t s r t r t t 2 t r t t t t r t s s t r s s r t t t t 2 r r r r t r s r r r s t r t r t r s r r r r t r s t t s s s t t r t s 2s ss t t rst s t t r s t s ss t t s s t t r s t t t r t r t r 1 st 2 t r t r t 2 t r s r t r t r 1tr r t s r r t r r s tr st st t 3 t r r r r ts t 1 3 r t t2 r r t r t s sts

t rts s rt r t r ss rs t t r r r t s 1 t t t r t s t t t t2 ss t t 3 t r s t t r t s t s t st st t 3 t r 2 t s q t s t t r t q ss r s t r t q t s rt s s 2 s r 1 r ts s s s s t r r r t r 1tr r t

t ts tr t Pr st r t r 2 r P r P s s r t 3 t s2 t t s t s r t t r 1 t r s ts Pr t r r s t r t t r t r r t r t r s Pr st r r s ts r t r r t s t rt t 3 t Pr st r t s r 2 r P r P s Pr r s t r s r P s s r 2 r r P s t r r t r t é 2 s r t é 2 s r s2 t t r s ts r t r s t Pr t P 2 r t r ss r t r r t r r ss r t r r r t r r ss r t r r ss t t r s r r r ss t t s tr s r r r s ts r q r t 1 r s s ts q ss r t 1 st t r t 2 r 2

t ts t t s r s rs a, b R a b := min{a, b} a b := max{a, b} a + := max{a,0} R d := R d \ {0} C d s t t r t s R d Mn, d s t t n d tr s t r tr s t s2 tr tr s s 3 d s t 2 S d ts s s t t s2 tr tr s s t 2 S + d 2 t t rt r r 2 t s t S + d r tr 1 A Mn, d A T s t tr s s A r A, B Mn, d A B := Tr[A T B] rt r r d = 1 A B r t rs R n A B r s t t s r r t A s t s rs t tr 1 A r s t 2 s t t ϕ Q T := 0, T] R d ϕt, x ϕt, x ϕ := sup ϕt, x ϕ 1 := ϕ + sup. t,x Q T Q T Q T x x + t t 1 2 2 t L p r r R s t 2 R p := E[ R p ] 1/p

t r tr t t r s r t t s t t r t s r 2s r rr t s t t t t s s t r s t r 1 t t s t s P s t t r t s 2 rt t r s 2 t s r s r t r t t t s s 2 r t t r s s r t t r t s r r 2 ss s s t t r s t t s r t s s t r t s r P s st rts 2 t s 2 r r r P s 1t s 2 t t r P s t s 2 2 s t 1 t t r t 1t r t s r r q t s t r r t st r t q t s s r r t s s r r q t s t r r L X vt, x Ft, x,vt, x, σ Dvt, x = 0 [0, T R d vt, = g R d, r L X ϕ := ϕ t + µ Dϕ + 1 2 a D2 ϕ s t t s r t r s r ss X a := σσ t r r 1 t t s t t s r q t s 2 s2st s sts t st st r t q t r st st r t q t dy t = Ft, X t, Y t, Z t dt + Z t dw t Y T = gx T. r r s 2 ss s t r r t2 r t s t P s t rr s vt, X t = Y t Dvt, X t = Z t r t s r t s r t 2 2 t s t ss s t s s r r P s t t r t t rs s r str t r r t2 t r ts s t 1 st ss s t s r t s r P s r r t s t s t r 1 t t s t P s rs t t s s t r2 s r s 1t s 2 t t s r s r 2 t r t r s r s t s r t 3 t t r r s r ss X t t s t r s r t 3 r t t t s r s t t t s s t

t r tr t r 1 t t s t s r r P s r st r P 3 2 P ès r 3 rt r t tt r rs r t r t t r s r t t r 1 t t t ts rt s r t t t s L 2 rr r r r r s t t t st s r t 3 t s t t t t 1 t t s t rt r r 1 t r r t r s t t t s r r t r t s2 t t 2s s t r 1 t t r r ss rr r r r st s P t r 1 t t s t s t r 1 t t s t P r r s 2 r t s r t 3 t {t i } N i=0 [0, T] t r 1 t r Y Z 2 Ŷ N t N = g ˆX T N Ẑt N 1 i = E i [Ŷ t N t i+1 W i+1 ] i+1 Ŷ N t i = E i [Ŷ N t i+1 ] t i+1 Ft i, ˆX N t i,ŷ N t i,ẑn t i, r E i = E[ F ti ] t i+1 = t i+1 t i W i+1 = W ti+1 W ti r r t s rr r 2s s s r t 3 t s r r t t rr r t s s r t 3 t s t s s r r r s π 1/2 r π := sup{ t i i = 1,, N} r 2 r r q t s t st rt t s r t 2 r s s2st s r r rr s t t r L X vt, x F t, x,vt, x, σ Dvt, x, D 2 vt, x = 0, [0, T R d, vt, = g, R d, r L X ϕ := ϕ t + µ Dϕ + 1 2 a D2 ϕ. µ σ r t s r R + R d t Md, d R d a := σσ T s r R + R d t S + d F : t, x,r,p, γ R + R d R R d S d Fx, r,p, γ R. s s2st s 2 dy t = Ft, X t, Y t, Z t,γ t dt Z t dw t dz t = A t dt + Γ t dw t Y T = gx T, r st s r tr t t r s t t s t q r Y t, Z t, A t,γ t s t s s t q t s r t r r t2

t s t t t r t rr s t t 2 r P t s2st s 2 Y t = vt, X t Z t = σ Dvt, X t Γ t = D 2 vt, X t A t = L X Dvt, X t. 2 s r t 3 t r s t s ˆΓ N t i = 1 E i t [ẐN t i+1 W i+1 ] i Ẑt N i = 1 E i [Ŷ t N t i+1 W i+1 ] i+1 Ŷ N t i = E i [Ŷ N t i+1 ] t i+1 Ft i, ˆX N t i,ŷ N t i,ẑn t i, ˆΓ N t i. s t t s t s s s t tr r st r t r t 2 r r P s t 2 r P s r s 2 r s t t s r r 1 t r t 2 r t r rt t 3 t r r t t2 str ts t r r q t2 st t 2 r P s r s r st st t 3 t r s r tr s r ss s r rt t 3 t é 2 r ts r r 2 1 s t 1 t q s 1 t s t r 1 s r s r P s P s s s r t 2 t t rs r 2 s ss t t ts t r t t r t r t r t t r t r s t r t s t r s r t t t r r st s tr t r 2 t t t t t r st st r t q t s s s s t r t s s s t t r st s t r st s sts s t t r r t r s r 2 s r ss s s t s t t P t t s r rt r r t t P t r 2 s r ss t t r s r t t r 1 t 2 s t r s r 1 t r t s t r r rt t q t 2 t r s r t t r 1 t t 2 t st s s t t 1 t t r t r s t t 1 t r t r st r s t r r r q r s

t r tr t t r s t 1 t t s 2 t r 1 t s rt r t r t2 rr r t r s t t r t r q r t 2 r P t s t s t 2 ss t t t s t s s r s r s t t s s s s t2 s t s r t s t st s t r t s r 1 t t r s t q s s t2 s t t 2 r P t r s2 t t 2s s t r 1 t rr r r r s ts r t rst r t r t s r t t r 1 t r r r P s r s t rr s r 1 t rr r r ss t P s s r t t s t t r rr r s r 2 r r r s t r r r P s r s t rr r r 1 t r t P s s r t t r r r s ts s r str t s t t s t r 2 s r ss rst t t2 t s t t t s t t r 1 s t r r r rt t 2 t s t s t t t rt r t t r r t2 t r s t t ts ss t t t r t r s t t t t s t s ss r2 r r 1 r ts s t t t t t s t s t s r s s r r t t s r r r s r 2 t t s t 2 r s s t t r2 s s t2 s t s t r t t s ts r2 r s s s t tt r t2 t s t t 1t st st s s s t t r t t r r s ts r r t t r t t ss rr r 2s s r s ts t t r2 r st st r t q t s s 2 st 2 t r t r 1 t t t r t s r ts r t r ss t r s t t t s r s r t r t r t s s t r s t t 1t t r P s s tr t r s s t s s t r t t2 t r s s s s t s r ts r r s t s r t 3 t t s r 1 t t r 2 tr r r s r 1 tr 2 tr r t r r 1 t t r t s r t r t s r 2 s s rt r t 3 t r t 3 r t tr 2 t rs s t2 r s t r 2 t r s s s t t r t s s r s t t t2 s s t s t t 2 t tr t s r t 2 t r str t r s r 1 tr t t 2 rt r st t s r t s r t 3 t r t t r s t r t t r s s s t s r t

s s t t t2 r q r t 2 s r t 2 s t r 2 st t t t r r 1 t s r t s s 2 s t s t s s s s r t t rs rr s s t r 3 t r t W r t st t ts r r r r t s t 2 s t t q r q r s t t r t t t t t st s t s s s t t t rs s t2 r s 2 s r t t t r t r r t2 r t r st t r t t r r t t r 2 r r r s s t t s sts t 2 rs t t st tr s t 1 3 r t t r t 3 t 2 s t2 t r t r r t2 t 2 r P rs t t2 s t t r t2 s t t t r r t2 rs t t P s r s r2 s r t t r r s t t r s r rs t P r t r t 2s t r t r 2 st t 2 r t r r s t t r r 3 s t r st r t r r 1 t t s t 2 r r P s t P s r 2 P s t t r rt r t q t s s t s r r rr t s t r rt r t q t s P s t t r s t r t t s r t r r s r r st r t s s st s s s t r t s t s r t t q t t r 2 r rt 2 r rt s t t s r t 3 t s t s r ss t r 1 t t r t s t r t r t r rt s r t t r r rt s r tr r2 t t s t s t s ss t s ss t s r t s r s t r r t t t2 t r t r r t2 t t t s t s t t t rt r t t r r t2 t r s t t ts ss t t r tr t t s t r s t t r t r r 1 t t t r t r s t t é 2 s r rs t P s t s r rr t t s t r s t r r t r tr t t s s r t t t2 s r ss s r t t t2 s r ss s tr t t é 2 s r r 3 r t tr t t s t t s r s tr s r t tr t rr r t r s t t t r t s t r tr t t s t t r s s t r 1 t t é 2 t r s t r t2 2 t s s s r r r st st r t q t s

t r tr t t s r r t tr t t r s t t t st r t s t r r t r s t s t r t r s t r r r t r r s t r r t r t r r t P s t q r t t s t s t3 t t2 r t s r 2 t r st P s q t s r r s 2 t r t2 s t2 t t s s t3 2 é 2 s r s t t r s t s t2 s t ss t s r t 2 t t s s t t t tr t t r s κ s r r 2 t t st t r t r 1 t s t r s t t s t t r rst r s t r s t r t r 1 t s t t r t s t t s s t2 s t t r t2 s t r t s t t t r ss s t t é 2 t r t r s t t t é 2 s r t r t2 t r t2 s r s t3 tr t t é 2 s r t r t2 s s t3 t s tr t t r s t s t 0 t s t3 st t s s t s r t r t t r r t t r s rr s s s t r t t t r 1 t s t s r 1 t s r t r 1 t t r t 2 t s t tr t t r s s r r t 2 s r s t r s r t r t s r t2 r t r t r s s t r s ts r 3 s t r s t t s t s s t r 1 t t s t t q t t r r s s r s t s P t r r s r s r s t t s t s t s st 2 r s t r r s r s t s t t t t t tr t t r s r r t 2 t t st r r t r s r t r t r t r tr t r t r t r s t t t q t t t str t 2 t t2 r t s s r r q r t s t t 2 s t t r r s t s rt 2 t t r t t t r 3 t t s tr r s t r st t r t t r r t t r 2 r r r s s sts t 2 rs t t st r t r t r 3 tr s t 1 3 r t t r t 3 t 2 s t2 t r t r r r s 2 s st rts t s t s s t r p tr 1 Γ t ϕ r tr r2 t r w t t r t p Γ ϕ r t2 t r s 2 r r s t 2 t st t r r t t t r t t r w t t st s r s s t r s t s t st s t 3 r r t t 2 t 2 s t r t t s t 2 r r P

s r t2 s sts t t t t2 t r t r p tr 1 Γ t ϕ r r s t t rst s r t s t s t t r s t 2 t r t r sts r s t s t 2 r t t st tr t 2 r t t t r s s s s r t 2 t r t ss s t s t t 1 t t st t s s r t r s s t t r s t st r t 1 t s2st s s t t t r t st t r t r 2 t r t t s 3 s t 1 t t s s s t s t s rst t r s t r t t st t s s rr t t r t 2 s t r t t r s t r t s r t r t t r s t s t s r t s t t r t t r st t s t r t r 2 t r t r s t t r t t r s s s 2 s t 2 s r tr s r rs t r ss r t Pr r P t t 2 t r t r s t r t t st r t 1 t P r s r ss r r s 2 t tr t ss 2 t r t rs t r tr t rs s t 1tr ss r r s 2 s s r t t t r ss t rt str st t s t t s r t r r s 2 st t ts t r r t r t s s 2 t r r t s t t t r ss r r st t t s s t t r s st t s r r 2 r t r r t t t 2 s t r t r t r t rst s t r r s r r r2 t t t st t s t t r ts r s s t r s t t t ss s r r t s s s r s t r t s t rst s r t t r s t r t t t r t2 t t t r ss r r t s s P r s t r t st t s t t 1 t t rst s ss t t s s 2 2 r s r t s s s t r t t s s t t r s 2 t st r s r t s s s r rr t s r s s t r t r t 1 t t t s t t 1 ss r t r t t r t 1t s r r t s s 2s t r r t r r r ts t s r s s s

t r tr t t r t r t r r s s t r t t s t r 2 ss r t t ss tr s t s t r 23 t t ss r t r t r ss t r t r t 2 t r t r s r s t s t 1 3 r t t2 r t s t t r t r r t t r r rt r r r t r rt str t 2 s t t t2 1 3 t r rt str t 2 2 t t2 r t t t r t 2 t s t q t s s r t t t r t 2s r s t t r t 2 t s r rs r r rs t t t r s r r r rt s s t r r r r r t r t t r t r s s s t t t t t t r t t t t r s t t t t t ss s 2 t r s q t t r t r r s ss t s r r s 2 t r t r t t r r t r t r t q 2 πq r π str t 2 t π ts r t t π0 = 0 π = π 0+ > 0 t t r t ss t r s 2 t r t r t q 2 eq r e s r s t t s s s ss s t r t q 0 s s t t π q 0 = 0 t st r t 1 t s t t r t q 0 s s t s s π q 0 E Q 0 t [α1 {E q 0 T E 1} ]e q 0 = 0, r E q0 T s t t ss t r E 1 s t t ss t r s t s r s tr s r r t st st s t t r t r 2 t t2 ss t e t s r t t q 0 > q 0 t 1 st t r t s t r t π q 1 +V e 2 e q 1 = 0 r t s r rs r r t t r s t r s t t r s s s t ss t r t r st r 2 V e 2 = S t t s t t q 0 > q 1 s t q 1 q 1 s t t t t2 t r s t r t r r s 1t r t2 r t r r s t t r t t t t t t2 t r r r r rs t t t r s r t r t r s s r 2 π q 2 e q 2 S t V y 2 t, E q2 t, Y q2 t r V 2 s t t t r rr s s t t t 3 t r V y s t s s t t2 t t t r s t t t t t ss 2 Y q t s t t t ss r ss r t t r t t t2 q t r r r s t t V y s s t t r r q 2 q 1 s t t t r r r s r s r t 2 r t t s s r r

r t r r r s t t r s r t r t π q 3 + 1 η λλ q 3 + e q 3 V 3 e + βv 3 y γλ q 3 V 3 y = 0 r λq s t t r s r r t t r t t t2 q r r r V 3 s t t t r V y 3 s t s s t t2 t t t r s t t t t t ss 2 V e 3 s t s s t t2 t t t r s t t t r t 2 γ η β r s t st ts t r r t t r s t r s t t r s s s t r 2 V e 3 = S t t r s q 3 2 q 1 q 2 s t s t t r e q 3 βv y 3 + λ q 3 γv y 3 1 η λq3 r r 1 s t s t t t s s ss t q 3 r t r t q 2

t r Pr st r t r 2 r P r P s s t r s r 3 s s t r t r r s t t t s t t t t t r2 r st st r t q t s t s t t t s2 t t 2s s t s r t t r 1 t t s r rst r r s t t rr s rr r st t t tr t t r s rt r st t t t r rr r r r r s t r 1 t rr r 2 t t s s r r s ts r t r t r q t t s r s t q t r s t r rt t 3 t t t s s r t 3 t t µ σ t s r R + R d t R d Md, d r s t 2 t a := σσ T t r r t r L X ϕ := ϕ t + µ Dϕ + 1 2 a D2 ϕ. F : t, x,r,p, γ R + R d R R d S d Fx, r,p, γ R s r t 2 r L X v F, v,dv, D 2 v = 0, [0, T R d, vt, = g, R d. r s t s st st r r s t t r t s t t s r s r 2 s t 2 tr t s r r r st st r t q t s s rt t t s st st r r s t t s sts r st r s r t 2 r s r s r rt r t r t 3 r 3 r r

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s r t 3 t r r2 t ϕ : Q T R t 1 t r t [ D ϕt i, x = E ϕt i+1, ˆX ] t i,x H t i, x, r H = H 0, H 1, H 2 H 0 = 1, H 1 = σ T 1 W, H 2 = σ T 1 W W T I d 2 σ 1. Pr r t s t s2 s r t t G s ss r r t t r r 2 t f : R R t 1 t r t E[fGH k G] = E[f k G], r f k s t k t r r r t f t s s str t s H k s t s r t 2 r k t ϕ : R d R t t 1 t r t 2 r t t t s r t t [ ] t,x E ϕ ˆX W i = d j=1 [ ] ϕ t,x E ˆX x σ jit, x, j t r r [ ] t,x E ϕ ˆX H 1 t, x [ ] = σt, x T t,x E ϕ ˆX. r i j t s r t t [ ] t,x E ϕ ˆX W i W j = d k=1 = 2 d k,l=1 [ ] ϕ t,x E ˆX x W j σ kit, x k [ 2 ] ϕ t,x E ˆX x k x σ ljt, xσ ki t, x, l r j = i [ ] t,x E ϕ ˆX W i 2 = 2 d k,l=1 [ 2 ] ϕ t,x E ˆX x k x σ lit, xσ ki t, x. l s r s E [ ] t,x ϕ ˆX H 2 t, x [ ] = σt, x T E 2 t,x ϕ ˆX σt, x. t t r t t s v t i,. t v t i+1,. s t t r t t ss t tt r t

t r Pr st r t r 2 r P r P s r r 2 r t r t s r t t r t 2 rts s ss t2 s t t r r s t t D 2 ϕ s D 2ϕt, x = E [ W T /2 t,x ϕ ˆX σt 1 W /2 /2 /2 σ 1 ]. s r r s t t s s t t t r s s r2 s r t t r st r r t s st s r t t t t r t t s ts µ σ t r P s r tr r2 r t s 2 s r r t t r 2 s X r r r s t r s r str t s t t s t s r t r r t r L X s s t r P t r t s t r r t2 2 t ss t r s r 1 t s t t t r s s t t 2 t r t r r t t r r s r s t t s t d = 1 µ 0 σx 1 s r t r r 1 t t r t Ŵ tk := k j=1 w j t k := k, k 1 r {w j, j 1} r t r r s str t s 1 2 δ + δ t s s t r 1 t [ ] D 1 ψt, x := E ψt +, X t,x H 1 ψt, x + ψt, x 2, s t t r t r s r 1 t t r t r 2 s r t tr r r 1 t Ŵt k := k j=1 w j t k := k, k 1 r {w j, j 1} r t r r s s tr t s 1 6 δ { 3} + 4δ {0} + δ { 3} s t t E[wj n] = E[W n ] r t rs n 4 t s s t r 1 t [ ] D 2 ψt, x := E ψt +, X t,x H 2 ψt, x + 3 2ψt, x + ψt, x 3, 3 s t t r t r s r 1 t t ss t t r r t t t r s st t s r s 1 t r t r s r t t r t t s s sts t r 2 s r ss X t r s t t s s sts r 1 t t r r t2 2 s t t r t 2 rts r

s2 t t s t s r t t r 1 t s2 t t s t s r t t r 1 t r s ts r rst r r s ts t r t 2 r s s r q r s t t t r P s t s s r s r s t t s s s s t2 s t s r t t r s t s r s r s t s t v r s v [0, T] R d s s s t2 s s t r s s rs t r 2 t, x [0, T R d 2 s t t ϕ s t s 2 0 = v ϕt, x = max ϕ [0,T] Rdv r s 0 = v ϕt, x = L X ϕ Ft, x, Dϕt, x r s 0. min ψ [0,T] Rdv t s 2 t t s r s r t s r 2 r s t s s s t v 2 r s t s s rs t v [0, T R d s t s 2 v v vt, vt,, r r s s s str r t r s 2 t r t t t s r ss r2 2 r t r t 1t s rs t v s s t v s t s 2 min { L X vt, x FT, x, DvT, x,vt, x gx } 0 max { L X vt, x FT, x, DvT, x, vt, x gx } 0. s r t t 2 t t r r q t 2 t t t s s t v g t s rs t v g t t s t s s s r2 2 r r s t s t t ss r t2 s s t t Ft, x,r,p, γ s r s t r s t t r s r t ϕ t s t s 2 0 = v ϕt, x = max ϕ. [0,T] Rdv ϕ K t, = ϕt, + KT t r K > 0 v ϕ K s s 1 t T, x t s s t r rt2 s t t min { L X ϕt, x FT, x, DϕT, x + K, vt, x gx } 0. r s t 2 r K t s r s t r q r q t2 vt, x gx 0 s r r t s s t t s t t v g 0,

t r Pr st r t r 2 r P r P s t s q t 2 F r F p F γ t rt r ts F t r s t t r p γ r s t 2 s t 2 Fγ t s rs t t s2 tr tr 1 F γ r t t 2 s t3 t s r t ss t r t2 F s s t3 t s t r s t t x, r,p, γ r 2 t F,,0,0,0 < F s t t 2 t s t r r t r L X F p ImageF γ F p F γ F p < + γ F a R d R R d S d ; r ss t s q t t m F < r m F := min w R d { Fp w + w F γ w }. s s t 2 s 2 r t t 2 t s2 tr t r F γ 2 w R d s rt s t w = w 1 + w 2 rf γ F γ 2 t t t2 F γ F p w + w F γ w = F p w 1 + F p w 2 + w 2 F γ w 2 = 1 4 F p F γ F p + F p w 1 + 1 2 F γ 1/2 F p F 1/2 γ w 2 2. r t s s r str t s t t r r t r L X t r P rst F s r q r t r 2 t 2 r t t s tr 1 σ σσ T S + d t s s rt r t t r r s ts t 2 t r r t r r P s t s t r r t r σ s r q r t t t r t2 F s t 2 r t t s σ 1 t s s r t r P t s s v t 1 2 a 2 v xx + b 2 vxx r 0 < b < a r st ts r str t t t s t st t t s r t t t 1 3 a2 σ 2 b 2 s t t a 2 3b 2 t r t rs a b t s t s 2 t tt r t t t s σ s t s t st t t t r r t ss t tr µ 1 σ 1 < σ s rt s ss t t t 2 r P s r s r t s r r2 s t3 t g t r 1 sts t v s t t v v 2 r 2. t v s t q s s t2 s t r

s2 t t s t s r t t r 1 t r r t ss t t ts µ σ t r str t t t r t g t r r 1 2 t r t g t t α 1 t r t r s α > 0 1 s M > 0 t ρ r tr r2 s t s t t t ρx = e α x r x M, s t t t ρx 1 ρx ρx 1 2 ρx r t ut, x := ρx 1 vt, x r t, x [0, T] R d. t r P r s t s 2 v rts t t r P r u r L X u F, u,du, D 2 u = 0 [0, T R d vt, = g := ρ 1 g R d, Ft, x,r,p, γ := rµx ρ 1 ρ + 1 2 Tr[ ax rρ 1 2 ρ + 2pρ 1 ρ T] +ρ 1 F t, x,rρ, r ρ + pρ, r 2 ρ + 2p ρ T + ργ. t t t ts µ σ r ss t t s s2 t s t t F s t s s t s t s s F g s t r r s t t r P r r st t s t t t q t2 s st t t r t2 γ s s t r t r t t r t r s s t t r t s t s ss r2 s t s sts t t t s t s t s r t s r s r r r 1 r ts t r t t t t 2 r r r t s t s t s r 1t r s t r t r t t r t r s s t t 1t r P s t t t2 t s t 1t s ss t s r str r t s s t F 2 t t t r P s t s s r s r s t r t s ss t r t2 F s t s s ss t s t t t2 1 a γ + b p + Ft, x,r,p, γ = inf 2 α A {Lα t, x,r,p, γ} L α t, x,r,p, γ := 1 2 Tr[σα σ αt t, xγ] + b α t, xp + c α t, xr + f α t, x

t r Pr st r t r 2 r P r P s r t t s µ σ σ α b α c α f α s t s 2 µ + σ + sup σ α 1 + b α 1 + c α 1 + f α 1 <. α A ss t r t2 F s t s s r 2 δ > 0 t r 1 sts t s t {α i } M δ i=1 s t t r 2 α A inf σ α σ α i + b α b α i + c α c α i + f α f α i δ. 1 i M δ r ss t s s t s A s s r t s σ α b α c α f α r t s s r A t C 1 2,1 b t s s r s t3 x 1 2 ö r t r t r ss t t t t g s s t3 t s t r s st t C > 0 s t t r ss t v v C 1/4 r t str r t C 1/10 v v C 1/4 s r s s 1 s s t r t r s r t r t r s r t s s r t t t P t r s t r t r t r t r s s 2 st t t r s t s r t 3 t t s r x r t 1t st st r t q t t t t x s r t r r 1/2 r r t r r s t r t r rr s s t t ss r t x 1/2 x 1/5 r s t 2 Pr t r r s t r t r r 2 r r r s s r q r s t s t s st t t st r r s r ss t r s r t q t s t r t t r s r s t s t s s t t r t t s s s s r r t t s s t t t s r r r t ϕ s t t t r t s r t, x [0, T] R d lim t, x t, x, c 0, 0 t + T [c + ϕ]t, x T [c + ϕ]t, x = L X ϕ + F, ϕ,dϕ, D 2 ϕ t, x.

s2 t t s t s r t t r 1 t r s str t r r t tô s r s tt t ϕ, ψ : [0, T] R d R t s t3 t s ϕ ψ = T [ϕ]t, x T [ψ]t, x + C E[ψ ϕt +, t,x ˆX ] r s C > 0 r C s 2 st t K Pr 2 t r t r T r tt s [ ] t,x T [ψ]t, x = E ψ ˆX t,x + F t, x, E[ψ ˆX H t, x]. t f := ψ ϕ 0 r ϕ ψ r s t st t t t t F τ t t rt r t t r s t t τ = r, p, γ 2 t r [ T [ψ]t, x T [ϕ]t, x = E f = E r s θ = t, x, r, p, γ 2 t t H t, x ] t,x ˆX t,x + F τ θ D f ˆX [ t,x f ˆX 1 + F τθ H t, x [ t,x T [ψ] T [ϕ] = E f ˆX 1 + F r + F p.σ T 1 W + 1 F γ σ T 1 W W T Iσ 1], r t θ x s tt r t t s t2 F γ a 2 ss t F 1 a 1 F γ 0 t r r [ t,x T [ψ] T [ϕ] E f ˆX F r + F p.σ T 1 W + 1 F γ σ T 1 W W T σ 1] [ t,x = E f ˆX F r + F p.σ T 1W + F γ σ T 1W W T ] 2 σ 1. ], t m F := max{ m F,0} r t t m F s ss t K := m F < t r F p.σ T 1W + F γ σ T 1W W T 2 σ 1 K r t T [ψ] T [ϕ] [ ] t,x E f ˆX F r K [ ] C t,x E f ˆX r s st t C > 0 r t st q t2 s r s r t s t r r

t r Pr st r t r 2 r P r P s r t t2 r s t t r s s s t 2 r t r t t r q r r s t s s r r t r r t r s r t s r 1 t t t2 r t r s r s r t t t t F s t s s t t F r 1 4 F T p F γ F p 0, t t st r t t2 t ϕ ψ = T [ϕ]t, x T [ψ]t, x s s t r t r t q t 2 tr t ut, x := e θt t vt, x s s r P s t s 2 r t t s s t t t P r t 2 u s L X u F, u,du, D 2 u = 0, [0, T R d ut, x = gx, R d, r Ft, x,r,p, γ = e θt t Ft, x,e θt t r, e θt t p, e θt t γ + θr t s s 2 s t t F s t s s t s t s s F t t r t r s t 2 r θ t ϕ, ψ : [0, T] R d R t L t s t r 1 sts st t C > 0 s t t T [ϕ] T [ψ] ϕ ψ 1 + C rt r g s L t 2 v s L r 2 Pr t f := ϕ ψ r s t r s r [ T [ϕ] T [ψ] = E f ˆX 1 a 1 F γ + A 2 + F r 4 ] F Tp F γ F p. r A = 1 2 F γ 1/2 F p Fγ 1/2 σ T 1W. 1 Tr[a 1 F γ ] 0 F r < F T p F γ F p < 2 ss t F t s t t T [ϕ] T [ψ] f 1 a 1 F γ + E[ A 2 ] + C t E[ A 2 ] = 4 F T p F γ F p + a 1 F γ r r 2 ss t T [ϕ] T [ψ] f 1 + 4 F p T Fγ F p + C f 1 + C.

s2 t t s t s r t t r 1 t r t t t 2 v s r 2 r t 2 t ss t t v T,. = g s L 1t 1 s i < n ss t t v t j,. C j r r2 i + 1 j n 1 Pr s t r t ϕ v t i+1,. ψ 0 s t t v t i,. Ft, x,0,0,0 + C i+1 1 + C. Ft, x,0,0,0 s 2 ss t F t s r t s r t r q t2 t t v t i,. Ce CT r s st t C t r r 1 t t v 2 s 2 {i i = 0,, N} R d r t 2 r q r s t 1t t t 2 t [0, T] s 2 2 t r t s s t r r t2 r rt2 v t s r s r r st 2 s 2 s r t r t t v s s t3 x r 2 Pr r rt t t t s s d = 1 r r t s 2 t r s t t r 1 t [0, T ] r s t r t s t t r x, x R d t x > x v t, x v t, x = A + B, ˆX t,x ˆX t,x r t δ k := D k v t +, Dk v t +, r k = 0, 1, 2 A := E [ δ 0] + F t, x, Dv t,x t +, ˆX F t, x, Dv t,x t +, ˆX = E [ 1 + F r δ 0 + F p δ 1 + F γ δ 2], B := F t, x, Dv t,x t +, ˆX F t, x, Dv t +, ˆX t,x Fx x x, 2 ss t 2 r t r k = 1,2 E [ δ k] = E [ δ 0 Hk t, x + v t,x ] t +, ˆX H k t, x H k t, x = E [ k 1 δ 0 Hk t, x + Dv t,x t +, ˆX W σt, x k σt, x k σt, x ]. t s s 2 x x t s s r t q t s t t v t, x v t, x lim sup x x ց0 x x v E[ t +, lim sup x x ց0 +Dv t +, ˆX t,x v t +, x x t,x ˆX 1 + F r + F p W t,x ˆX 2σ x t, x σ x t, x W F γ σt, x 2 + F p σt, x σt, x + F W 2 γ σt, x 2 ] + C.

t r Pr st r t r 2 r P r P s ss v t +,. s s t3 t st t L t+ v t, x v t, x lim sup x x ց0 x x L t+ E[ 1 + µ x t, x + σ x t, x N N 1 + F r + F p σt, x + F N 2 γ σt, x 2 F γ σt, x 2 + ] 2σ x t, x σ x t, x NF γ σt, x 2 + F p σt, x + C. s r t t F p σ x F p σ = σ Fγ x Fγ σ 1 Fγ 0. t r s t r t s r r r ss t s t s t t F p σ x σ < s 3 t t t tr str t r s ss t r s t s r s t 2 s v t, x v t, x lim sup x x ց0 x x E[ 1 L t+ + µx t, x + σ x t, x N N 1 + F p σt, x + F N 2 γ σt, x 2 F γ σt, x 2 + 2σ x t, x ] NF γ σt, x 2 + C + C. t P t r t2 s r q t t P 2 t s t2 lim sup x x ց0 Z := 1 α + αn 2 r α = F γ σt, x 2. v t, x v t, x [ x x L t+ E P 1 + µ x t, x + σ x t, x N N 1 + Z 1 F p σt, x ] +Z 1 NF γ 2σ x t, x σt, x 2 + C + C. 2 2 rt3 q t2 v t, x v t, x [ lim sup x x ց0 x x L t+ E P 1 + µ x t, x + σ x t, x N N 1 + Z 1 F p σt, x 2]1 +Z 1 2σ x t, x 2 NF γ σt, x 2 + C + C 2 r t t 1 t t t r s r t2 P v t, x v t, x [ lim sup x x ց0 x x L t+ E Z 1 + µ x t, x + σ x t, x N N 1 + Z 1 F p σt, x 2]1 +Z 1 2σ x t, x 2 NF γ σt, x 2 + C + C.

s2 t t s t s r t t r 1 t 2 1 t q r t t r s t 1 t t s r t t 1 t t t t r s s 3 r r r lim sup x x ց0 v t, x v t, x [ x x L t+ E P 1 + µ x t, x + σ x t, x N N 1 + Z 1 F p σt, x 2]1 2 + C +Z 1 NF γ 2σ x t, x σt, x 2 L t+ 1 + C 1 2 + C + C, + C s t v t, x v t, x lim sup x x ց0 x x Ce C T/2, r s st ts C, C > 0 2 r t t t t r t s r s r 2 r s s t t st s r s t 3 r r x R d t k = k t k = 1,, n v t k, x gx CT t k 1 2. Pr 2 t s r t s t r r j i v t j, ˆX [ t i,x t j = E tj v t j+1, ˆX t i,x t j+1 1 α j + α j Nj 2 ] + F j 0 + F j r E tj [v t j+1, ˆX t i,x t j+1 ] + F j p E tj [Dv t j+1, ˆX t i,x t j+1 ] r F j 0 := Ft j, ˆX t i,x t j,0,0, 0 α j Fr j Fp j r F tj t r r s s t r t t j N j = Wt j+1 Wt j s st r ss str t t r r j r i t n 1 s t t [ v t i, x = E g ˆX t i,x T P i,n ] n 1 +E F j 0 +F r j E tj [v t j+1, ˆX t i,x t j+1 ]+Fp j E tj [Dv t j+1, ˆX t i,x t j+1 ], j=i r P i,k := k 1 j=i 1 α j + α j Nj 2 > 0 s r 1 i < k n P i,i = 1 s 2 {P i,k, i k n} s rt r i n r rt2 s t r F,,0,0,0 < + s ss t s v t i, x gx, E [ g ˆX t i,x T gx P i,n ] + CT ti.

t r Pr st r t r 2 r P r P s t {g ε } ε t 2 s t t s t r g 2 t t 2 rs {ρ ε } g ε = g ρ ε t t t g ε g Cε, Dg ε Dg D 2 g ε ε 1 Dg. [ E g ˆX ] [ t i,x g T gx P i,n E i,x ˆXt T g ε ˆX ] t i,x T P i,n [ + E g ε ˆX ] t i,x T g εx P i,n + gε g [ Cε + E g ε ˆX ] t i,x T g εx P i,n T Cε + E[ 1 P i,n Dg εˆb + t i 2 Tr[ D 2 g ε â ] s, ˆX t i,x s ds ] T + E[ P i,n Dg ε ˆX t ] i,x s ˆσsdW s, r t ˆbs = bt j, ˆX t i,x t j ˆσs = σt j, ˆX t i,x t j r t j s < t j+1 â = ˆσ T ˆσ 1t st t t r s r t 2 rst s {P i,k, i k n} s rt E [ T P i,n Dg ε ˆX t ] n 1 i,x s ˆσsdW s = E [ tj+1 P i,n t i = t t t E tj [ P j,j+1 tj+1 t j n 1 E [ tj+1 P i,j+1 j=i t j j=i t i t j Dg ε ˆX t ] i,x s ˆσsdW s n 1 E [ [ tj+1 P i,jˆσt j E tj Pj,j+1 j=i ] Dg ε ˆX t i,x s dw s = E tj [ t j = E tj [ tj+1 s t s r s Dg ε ˆX t ] i,x s ˆσsdW s Dg ε ˆX t i,x s dw s ]]. W tj+1 W tj 2 tj+1 t j t j 2W s Dg ε ˆX t i,x s ds ] Dg ε ˆX t i,x s dw s E [ T P i,n Dg ε ˆX t ] i,x s ˆσsdW s t i n 1 2 E [ P i,j+1ˆσt j 2α j j=i E [ tj+1 t j n 1 Cε 1 C T t i ε 1. j=i t j ] sd 2 g ε ˆX t i,x s ds ]],.

s2 t t s t s r t t r 1 t 2 t ss b σ s st t t t Dg ε ˆX t i,x s ˆbs, ˆX t i,x s + 1 [ 2 Tr D 2 g ε ˆX t i,x s âs, ˆX t i,x s ] C + Cε 1. P t t [ E g ε ˆX ] t i,x T g εx P i,n CT ti + CT t i ε 1, 2 r s v t i, x gx Cε + CT t i ε 1 + CT t i. r q r r s t s r t ε = T t i r r2 t v s ö r t s t r 2 Pr r 1 2 ö r t t2 t r s t t t s 2 r 2 r g v t k, t ss rt r s t 2 2 v t, v t, s r t s r 0 t t t t t st q t r r r t v t, x v t, x Ct t 1 2, r C s t t r t T r t t r t r r r s s r s s s s s t s2st s r 1 t t r2 t s ts r s r s t r t s s F s t s t s 2 t st r t t2 t r s s t tr t r t2 F r s t t F s t s s t u t 2 t s 2 u T,. = g u t i, x = T [u ]t i, x, r r t ψ r [0, T] R d t R t 1 t r t [ ] t,x T [ψ]t, x := E ψt +, ˆX + F, D ψ t, x, s t v t i, x := e θt t i u t i, x, i = 0,...,n. r s t s s t t t r v v s r r r t s r s t rr r st t r t t 2s s t r v v

t r Pr st r t r 2 r P r P s r ss t F lim sup 1 v v t,. <. ց0 Pr 2 t F r t2 t t t v t, x = e θ 1 + θe[v t,x t +, ˆX t ] + F +, x, D v t, x. 1 + θ = e θ + O 2 t s s s t t v t, x = T [v ]t, x + O 2 t t 2 v v t, 1 + C v v t +, + O 2, s s 2 t r q t2 t t v v t, O r t T 2 r t r t r T s t s s t st r t t2 t ϕ ψ = T [ϕ] T [ψ]. 2 r t r t r t t rr r st t s t r s r s t r t s Pr s t t ss t F s tr s t β := F r t r tr r2 t s ϕ ψ s t s 2 s r 1 ϕ T [ϕ] g 1 1 ψ T [ψ] g 2 r s t s g 1 g 2 r r2 i = 0,, n ϕ ψt i, x e βt ti ϕ ψ + T, + T e βt ti g 1 g 2 +. r t s r s r s t t str t t t t2 t t ss t F tr t β := F r r r2 a, b R + r2 t s ϕ ψ t t δt := e βt t a + bt t s t s s T [ϕ + δ]t, x T [ψ]t, x + δt b, t T, x R d. Pr s δ s t x D [ϕ + δ] = D ϕ + δt + e 1 r e 1 := 1,0, 0 t s r t r r t2 F t t t r 1 st s ξ s t t F t +, x, D [ϕ + δ]t, x = F t +, x, D ϕt, x + δt + F r t +, x,ξe1 + D ϕt, x,

s2 t t s t s r t t r 1 t t,x T [ϕ + δ]t, x = δt + + E[ϕt +, ˆX ] + F t +, x, D ϕt, x +δt + F r t +, x,ξe1 + D ϕt, x = T [ϕ]t, x + δt + { 1 + F r t +, x,ξe1 + D ϕt, x } T [ϕ]t, x + 1 + β δt +. T s t s s t st r t t2 t t s r s T [ϕ + δ]t, x T [ψ]t, x + δt + ζt, r ζt := 1 + β δt + δt. t r s t r t t ζt b r t s t ss δ δt + δt = δ t r s t [t, t + s δ s r s t s t t 1 ζt = δ t + βδt + δ t + βδ t be βt t, t r q r st t s r t r str t b 0 Pr Pr s t 2 r r r t 2 t t s r r s t r r t 1t t s r r s r t t 2 ss t t ss r t2 t t t r s s r t t ϕt, ψt, g 1 g 2. ψ := ψ + e βt t a + bt t r a = ϕ ψ + T,, b = g 1 g 2 +, β s t r t r t r s s t t ψt, ϕt, 2 ψt, x T [ ψ]t, x g 1 g 2 s tr r ϕ ψ r t r q r r s t 2 t rst ϕt, ψt, 2 1t ss t t ϕt+, ψt+, r s t+ T T s t s s t st r t t2 t t s r t t T [ϕ]t, x T [ψ]t, x. t t r r t 2 t s s t s ϕt, x T [ϕ]t, x ψt, x T [ψ]t, x. ϕt, ψt, Pr r r t t s ss t t ts 2 s s t v ε t r P 2 t t s t ts s s t3 x 1/2 ö r t s t r 1 t s r 2 t s t v v ε v ε v.

t r Pr st r t r 2 r P r P s t ρt, x C s t t s rt {t, x : t [0,1], x 1} t t ss w ε t, x := v ε ρ ε r ρ ε t, x := 1 t x ε d+2ρ ε 2, ε s t t r t 1 t2 t r t r F w ε s s s t w ε v 2ε. r r s v ε s s t3 x 1/2 ö r t s t w ε s C, β 0 t D β w ε Cε 1 2β 0 β 1 r 2 β 0, β N N d \ {0}, r β 1 := d i=1 β i C > 0 s s st t s s q t s st 2 r s t t t R [w ε ]t, x := wε t, x T [w ε ]t, x + L X w ε t, x + F, w ε, Dw ε, D 2 w ε t, x r s t 0 s 0 1t 2 r t s t st t t r t r R [w ε ] t 3 r r 2 {ϕ ε } 0<ε<1 s t t s s t s 2 R [ϕ ε ] R, ε := C ε 3 r s st t C > 0. r t s r s t s r rt t t t s s t r t r s st t t t r t t s s t r rt2 w ε s t t w ε T [w ε ] + C 2 ε 3 t s r Pr s t t t w ε v C w ε v T,. + Cε 3 Cε + ε 3. s t t t v v v w ε + w ε v Cε + ε 3. 3 t r t s st t r t ε > 0 t s s t r t rr r v v Pr r v v C 1/4. r s ts t r s s t t t r t t r r ss t t 2 t s t s2st t r s s r s t r t rr r v v inf ε>0 {Cε1/3 + R, ε} = C 1/10,

s2 t t s t s r t t r 1 t r s st ts C, C > 0 r q r r t r s r st t s t t t r v v s t 2 t r t r Pr t t t t t t r r 1 t t t t + s r 2 st t r t µt, x st t s σt, x Dϕ ε s t s r tô s r t t 1 [ Eϕ ε t +, ˆX x ϕ εt, x ] L X ϕ ε t, x = 1 E t+ t ˆX t,x L ˆX t,x ϕ ε u, ˆX u x L X ϕ ε t, x du, r L ˆX t,x s t 2 r t r ss t t t r s L ˆX t,x ϕt, x = t ϕt, x + µt, xdϕt, x + 1 2 Tr[ at, xd 2 ϕt, x ]. 2 tô s r s t t t t L ˆX t,x Dϕ ε s s t 1 [ Eϕ ε t +, ˆX ] x ϕ εt, x L X ϕ ε t, x = 1 t+ E t u t L ˆX t,x L ˆX t,x ϕ ε s, ˆX x s dsdu. s t ss t ts µ σ t s r t t r ε 0, 1 Eϕ ε t +, ˆX x ϕ εt, x L X ϕ ε t, x R 0, ε := C ε 3. t s s t t R [ϕ ε ]t, x Eϕ ε t +, ˆX t,x ϕ εt, x L X ϕ ε t, x + Fx, ϕ ε t, x, Dϕ ε t, x, D 2 ϕ ε t, x F, D [ϕ ε ]t, x 2 R 0, ε + C ED k t,x ϕ ε t +, ˆX Dk ϕ ε t, x k=0 2 t s t3 t t2 t r t2 F 2 s r t s t s t t ED i ϕ ε t +, ˆX t,x Dϕ εt, x Cε 1 i, i = 0, 1, 2, t t r t r s t r q r r s t

t r Pr st r t r 2 r P r P s r t r t r s t s s s t s 3 t s ss t t r s s Fγ = cγ, r s c > 0 t s s d > 1 s r 2 s s t t g s t r P s q s t vt, x = E [ g x + 1 + 2c W T t ] r t, x [0, T] R d. s s r t t t s s t v s C [0, T R t [ D k vt, x = E g k x + ] 1 + 2c W T t, t < T, x R. s s s rt r t t v s r t s 2 r r r t t r t g s C s r t s 2 r r rs s t ss t r st t t r r 1 t t t v t t s s t s t 23 t rr r t r s t t r s s t s 2 v T, = g, v t i 1, x = E [ v t i, x + W ] + ce r σ = 1 µ = 0 r s t r t t s [ v t i, x + W H 2 ], i n. Pr s t s r t r F ss t t D 2k+1 v s r r2 k 0 lim sup 1/2 v v <. 0 Pr v s rst r t t r s t t x t s r tô s r t t [ ] vt, x = E[vt +, x + W ] + ce vt + s, v + W s ds, t rr r u := v v s t s s ut n, X tn = 0 r i n 1 u t i, X ti = E i [ u ti+1, X ti+1 ] + c Ei [ u ti+1, X ti+1 ] +ce i 0 0 [ v i + s, Xi+s v ] i + 1, X i+1 ds, r E i := E[ F ti ] s t 1 t t r t r t F ti t t [ ] a k i := E k u t i, X ti, b k i := E 0 [ k v ] t i 1 + s, X ti 1 +s k v t i, X ti ds,

s2 t t s t s r t t r 1 t tr t tr s 1 1 0 0 0 1 1 0 A := 1 1 0 0 1, B := 0 1 0... 0 0 1 0 0, s r t t s t t t t rs a k := a k 1,...,ak n T b k := b k 1,...,bk n T s t s 2 Aa k = cba k+1 + cbb k r k 0 t r r 1 1 1 a k = ca 1 Ba k+1 + ca 1 Bb k r A 1 0 1 1 =. 0 0 1 2 r t t s t t t rs A 1 B k r 2 j i 1 A 1 B k i,j = 1 {j i+k} k 1 r k 1 i, j = 1,...,n. rt r s a k n = 0 A 1 B n 1 a k = 0 t r t t s r s a 0 = ca 1 Ba 1 + ca 1 Bb 0 =... = n 2 c k+1 k A 1 B k+1 b k, k=0 t r r n 2 u0, x = a 0 1 = c c k A 1 B k+1 1,j bk. k=0 s j 2 A 1 B k 1,j = 1 {j 1+k} k 1 r k 1 j = 1,...,n, r t n 2 u0, x = c c k k=0 n j=k+2 j 2 k b k 1 j. 2 t r r t s t s t t t u0, x = c n j 2 j 2 c k k j=2 k=0 b k 1 j.

t r Pr st r t r 2 r P r P s t r r ss t t t D 2k+1 v s L r r2 k 0 t s t t [ ] ti b k j E k vs, X s k vt j, X tj ds C 3/2 t i 1 r s st t C t r t t n j 2 j 2 u0, x cc 3/2 c k. k j=2 k=0 n u0, x cc 3/2 1 + c j 2 = cc 3/2 1 + cn 1 1 c j=2 C. Pr st r r r t t t r s st t s ss t r t t t t 1 t t s t t t r t rs T t r t r t r ss X t s t 1 t t s r t s r r ss s t t 2 t r r t s t t s r s t s tr t t r t r r t r r 1 t t s r r ss s r r t r t s ss t t s s t s t st t t s2 t t r rt s r s st r t t 1 t t r t r E s r 2 s st t r ÊN rr s t s s 3 N T [ N [ψ]t, x := ÊN ψt +, ˆX ] x + F, ˆD ψ t, x, ˆT N [ψ]t, x := K [ψ] T N [ψ]t, x K [ψ] r ˆD ψt, x := ÊN [ ψt +, ] t,x ˆX H t, x, K [ψ] := ψ 1 + C 1 + C 2, r C 1 = 1 4 F p F γ F p + F r C 2 = Ft, x,0,0,0. s r r t r s s r r 2 s r t t s t t s t t r s s ˆv Nt, x,ω = ˆT N [ˆv N]t, x,ω,

Pr st r r ˆT N s t r s ω t r t t s s t s 3 s t r st t r t r 2 s t R b t 2 r r s R t r ψw H i W r ψ s t t ψ b H i s r t r t 2 s H 0 x = 1, H 1 x = x and H 2 x = x T x x R d. ss t r 1 st st ts C b, λ,ν > 0 s t t ÊN [R] E[R] p C b λ N ν r r2 R R b r s p 1 1 s r t r r ss r 1 t s t t r t 2 rts s tr s r r 3 23 t t 1t t s t r st st r t q t s 2 ss t s s t s r r2 p > 1 t t st ts λ = d 4p ν = 1 2p s r 1t r s t st s s t s t s s 3 N t t st r t t r ˆv N t r s v r t ss t s tr ss t t t 2 r P s r s t r t q s t t t lim 0 λ+2 N ν =. ss t t t t g s s t3 t ts µ σ r r st r2 ω ˆv N, ω v 2 r 2 r s t q s s t2 s t Pr t t r t t t r s t st st t 1t 2 r 2 ss t t ss r t2 t t t str t t t2 s 2 s t t ˆv s r 2 ˆv t, x := lim inf t, x t, x 0 ˆv t, x ˆv t, x := lim sup t, x t, x 0 ˆv t, x. r t s t r t t ˆv ˆv r r s t 2 s s t2 s r rs t s s t 2 t r s ss t s t t t t 2 r t q t t q s s t2 s t t r s 1 st s 2 r rt r t 2 r t t r st t s s 2 r rt t r t s rs t r rt2 t s s t r rt2 s r t s t2 r t

t r Pr st r t r 2 r P r P s r r t r t t ˆv s s rs t s r t 0, x 0 [0, T R n t t r t t st t ϕ C 2 [0, T R n s t t 0 = min{ˆv ϕ} = ˆv ϕt 0, x 0. 2 ss t s s q t n, x n, n t 0, x 0, 0 s t t ˆv n t n, x n ˆv t 0, x 0 ˆv n ϕt n, x n = min{ˆv n ϕ} =: C n 0. ˆv n ϕ + C n t s r t t t2 t r t r T t t T n [ˆv n ] T n [ϕ + C n ]. 2 t t ˆv n t s r s ˆv n t, x T n [ϕ + C n ]t, x T n ˆT n [ˆv n]t, x, r r s t t s t N s r ˆv n t n, x n = ϕt n, x n + C n t st q t2 s s ϕt n, x n + C n T n [ϕ + C n ]t n, x n + n R n 0, R n := 1 n T n ˆT n [ˆv n ]t n, x n. t t R n 0 P s s s s q t r ss t t s s q t s s 2 n s n t s r t t L X ϕ F, ϕ,dϕ, D 2 ϕ 0, s t r q r s rs t r rt2 t r s t s st rt 2 R n t r s t t t rr r st t t 1 t t 2 T n [ˆv n ] K n s 2 r t T n ˆT n [ˆv n ]t n, x n T n T n [ˆv n ]t n, x n. 2 t s t3 t t2 F T n ˆT n [ˆv n ]t n, x n C E 0 + n E 1 + n E 2. r E i = E Ê[ˆvn t n + n, X xn n H n i t n, x n ] T n ˆT E n [ˆv n ]t n, x n C Ê[Rn] 0 + E Ê[R1 n] + 1 E Ê[R2 n]. n

Pr st r r R i n = ˆv n t n + n, x n + σxw Hi W i = 1,2,3 H i s r t 2 r i s s t st t r t rr r R n R n C E Ê[R 0 n] + E Ê[R1 n] + 1 E Ê[R2 n]. n s R i n R b t t 2 ss t R n p Cn λ 2 N ν n, n s 2 R n p 0 s 2 s ss t t s s 3 s s t t s r t r t rr r r t t r t2 F s ss t s r r r ss r t r s t s 2 ss t t t s s 3 N s t t lim 0 λ+21 10 N ν > 0. r 2 s t3 t g t L p s t r t r v ˆv p C 1/10. Pr 2 r 2 ss t t ss r t2 t t t str t t t2 s tr r s t r r t s t t v ˆv v v + v ˆv = ε + R, ε + v ˆv. ˆv s t s s ˆv 1 T [ˆv ] R [ˆv ] r R [ϕ] := 1 T ˆT [ϕ], r t r s t t 1t R [ˆv ] s 3 r st st t r 2 Pr s t t s r t st q t2 t t v ˆv C ε + R, ε + R [ˆv ], r t st t C > 0 s 2 t s t3 t F β t st t r 2 t r t t r r t s t t r s tr t r r v ˆv C ε 1/3 + R, ε + R [ˆv ],

t r Pr st r t r 2 r P r P s s r s t st rt t r r t r ss t t t R [ˆv ] p C 1/10. t t s t s s 3 N t rr r st t r s t ˆv v p C ε 1/3 + R, ε + 1/10, t t t r 1/10 s t t t 3 t t r s t t ε 1 t s str t t r r s ts t s s t t t 1t t t r t 2 rts r r ss t r λ = d 4p ν = 1 2p r r2 p > 1 r t r r s t t s N t r r α 0 t α 0 > d 2 + 4p r t Lp r t r r s t t s N t r r α 1 t α 1 d 2 + 21p 5 r s ts t s s t r t t t r t r s s s st t s r t t 1t t r t t2 s r s rst s r t ss r t r q t s t s st r t r t 1 t st r r r st s t 1 r t t t s 2 s r r s2 1 t s t s r t r st tr 1 s s s s s s r 1t s r t t q t r t r 3 t ss t st t r t t s r t st r s s t r 1 t s t s s r r 1 s s s t t t t r 1 s s r t s s t t r t r Ft, x,r,p, γ s t t r r s t r t s r r r t t s s 2 r t t s s v T,. := g v t i, x := E[v t i+1, ˆX x] + F t i, x, D v t i, x r D ψ := D 1 ψ, D2 ψ, D 1 D2 r r r r t t [ D2 2 ϕt i, x = E ϕt i + 2, ˆX t i,x 2 σ T 1 W ti + W ti W ti + W ti T ] I d 2 σ 1 [ = E D 1 ϕt i +, ˆX t i,x σ T ] 1 W ti + W ti

r s ts s r r s t t s t r rt r t r s t r r st s s rst tr s t r r s t t s t r t r s s s t 1 t t r t r E s r 2 r 1 t ÊN q t2 s t 2 r t tt r q t r t N r r 1 s r r s ts r t t s r s s s t rst r s s r r s t t r rr t s s r s rt t t r r 1 r ts s t t s t r s t t s t 2 tt r r r t s r s s s s t r D 1,x ϕt, XT g s s t r 1 s s t t s t t g t s s t r s t t tt r r r 2 s s t t r Z s st 2 t rst s 2 s ss t t r r ss st t r r t 1 s t s s rst t s t s s r t st rt3 s s r r ss t s t 3 s rt s rt t r r ss t s r s r t s 3 t s rt s t t r t t t r str t t r 2 r ss s t s s t t r t 2 rts r s s st rt r rt r t t 1 t 3 t t φ k y = exp η k y r t k s s s s t r t r η k s r s s r t 1 t t k r r 1 r ts r r t t s t r t rs t r s t 2 r r r r t r s ts r t 2 s η k = 5/ t r s k r t r r r t r q t s r s t t s r r t s t r r r t t s r rt t t r t r t t t t s tr r r t r 3 s t 3 r s t St := {x R d : vt, x = 0} t vt, x t s s t s 2 t tr rt r t q t v t v + Dv D2 vdv Dv 2 = 0 v0, x = gx g : R d R s s t3 t s t r r t r r t s t r t r r t rr s st st r r s t t

t r Pr st r t r 2 r P r P s t t s r R d t r s 2R > 0 t gx := 4R 2 x 2 s t t g s s t s t s r t ts rst s t s r r s t s s t s t t t s r St s s r t r s Rt = 2 R 2 t r t 0, R 2 rs t r r t r t 0, T t T = R 2 v t 1 2 σ2 v + Fx, Dv, D 2 v = 0 vt, x = gx, r 1 Fx, z,γ := γ 2 σ2 1 + z γz z 2. t r t r t r s s t r r 1 t ˆv t t v s t r t r t s t r t 2 rts s r s s r t s r r ss s r r t r r s t t t ss t r 1 t ˆv r 1 t t s r Ŝ t := {x R 3 : ˆv t, x = 0} 2 s t r t s t t s t st t t r t D 1 v st t t r s t t 2 s st t s t s 3 t 0.01 r 2 r rs t s t r t s t ss r2 t r s t t 1t r t t St s s r t 2 t t [0, T r t s r s s t 2 t2 tr2 r r r 1 r ts t r t2 F s tr t s t t t s 2 r tr r2 t q t 200 r r r s ts s t t s s r t t s s r r s ts r r r s t s t s s r t t r s t 2 r r t r r r s ts r rt r t s 1 rr s t t s s r t t r r s r s ts t t rt s 10 10 10 s t t st q t 0.0125 s t σ s t t t r 1 r 1.8 s r t t r s ts r tt r t σ = 1 s s r t t t rr r r s s r t 0.25 rr s t r t t 2 s t s st t t t r t st s s t t s t r ts t r t r t t r r r s t t2 r r2 t st rr s r s ts t s r r rt r t t t s ts t t T = 0.25 r ss t r t r t t r s σ = 1.8 s r t t r s ts r s r s t 2 tt r t r s ts r s t σ = 1 t t s s s t r ss r 3 t t t r s t t t t = 0.15 t t r r ss t t t s s s t

r s ts r t t r t r r t s r r t t t t t t t 2 s t t r t t s r2 t t 2 r rt r s r r s ts r t r t r r s t r t r st tr2 t t s r t 3 r s t r v s sts t s s t t r s t t rs s t t t 2 str t t t r s t r t t s r s σ = 1 t st = 0.0125 rt s t t r t 2 rts s r r ss t t s s r t t t 10 10 s s r s r r s ts s 1024 ts t s r t s r t t s t s t t t r 3 t t t r r t s t r r r t ts t s r r t r s ts r 3 t 2 ss P ss P s t s t rt t 3 t 1t r rt t t t t s t rt t 3 t r t t s t {S t, t [0, T]} tô r ss t r t n s r t s st r s s t r ss {θ t, t [0, T]} t s R n r θ i t s t t st t i t s r t2 t t t t t st r s ss t r s 2 s r t2 t r t r rt s t s st r s 2 ss t S 0 s 2 t t r st r t s r ss {r t, t [0, T]} ds 0 t = S 0 t r t dt t [0, 1] t 2 s t t r ss s s r 2 dx θ t = θ t ds t S t + X θ t θ t 1 ds0 t S 0 t = θ t ds t S t + X θ t θ t 1r t dt,

t r Pr st r t r 2 r P r P s r s r t r r r r t t st s r 1 = 1,,1 R d t A t t t r ss s θ t s R d r t r t r s t t S s t t t r ss X θ s r 2 r s t r s rs t η > 0 t rt t 3 t r s 2 v 0 [ ] := sup E exp ηxt θ. θ A r r 2 r t s t s r st st tr r r t r 3 s t q s s t2 s t t rr s q t r s t s s s t s t t r t r t r s s t r r 1 t t s t t 2 r q t tr s t t s r t st t s s s s rst st rt 2 t s 1 r 1 t s t t r s r s t s r s ts s s t t t s r t d = 1 r t = 0 r t [0,1] ss t t t s r t2 r r ss s 2 t st ds t = µs t dt + Y t S t dw 1 t dy t = km Y t dt + c Y t ρdw 1 t + 1 ρ 2 dw 2 t, r W = W 1, W 2 s r t R 2 t s t 1t t s s 2 s t t t rt t 3 t r s t t st t r s t st t t t t r t 2 X t, Y t = x, y t

r s ts r s r t r r r r t t st s s t vt, x, y s s t q t 1 vt, x, y = e ηx 0 = v t km yv y 1 2 c2 yv yy sup θ R 2 θ2 yv xx + θµv x + ρcyv xy = v t km yv y 1 2 c2 yv yy + µv x + ρcyv xy 2. 2yv xx q s 1 t s t t s r s r 2 r vt, x,y = e ηx exp 1 T 2 t µ 2 ds Ỹ s L 1 ρ2 r t r ss Ỹ s 2 Ỹ t = y dỹt = km Ỹt µcρdt + c Ỹ t dw t. r r t t r t r t r s s r r t s v t km yv y 1 2 c2 yv yy 1 2 σ2 v xx + F y, Dv, D 2 v = 0, vt, x, y = e ηx, r σ > 0 t r t2 F : R R 2 S 2 s 2 Fy, z,γ = 1 2 σ2 γ 11 + µz 1 + ρcyγ 12 2 2yγ 11. t t t t r t2 F s t t s t s 2 ss t s r t tr t r t2 F ε,m y,z,γ := 1 1 2 σ2 γ 11 sup ε θ M 2 θ2 y εγ 11 + θµz 1 + ρcy εγ 12,

t r Pr st r t r 2 r P r P s r r t r r r s ε, n > 0 t 2 s t σ s t t ss t s tr r t s r t r r t s s s 2 dx 1 t = σdw 1 t, dx 2 t = km X 2 t dt + c X 2 t dw 2 t. r r t r t t t t2 t s r t t r 1 t t r ss X 2 s t t st s X n 2 = X2 n 1 + km t + c X 2 n 1 ξ n t + 1 4 c2 ξn 2 1 1 + k t r ξ n n 1 s s q t r r t str t N0, 1 r r r s ts rr s t t s t r t r µ = 0.15 c = 0.2 k = 0.1 m = 0.3 Y 0 = m ρ = 0 t t rt s x 0 = 1 t t r t2 T s t q t 2 r t t s r t rs t t s t r t q s 1 t r t v 0 = 0.3534 s s M = 40 r t tr t t r t2 s t r t t r t s t M = 10 r rt t s t r s ts t s s t st t t s s r t t s tt r s t 40 10 s s t s r r r s ts rr s t s rt s r r r s ts s t t t t s s r t t s r r2 s r r s ts r 2 t s rt s t t r r st t s 2 r r t t s

r s ts t r s ts t t 1 t st r t s r t 0.005 r s 1 t r st q t 0.025 t t2 q t 1.2 r st r t t 0.038 0.002 r s t t t t s s r t st s t r s ts t s s r t t 1 t st r t s r t 0.002 r s 0.0009 r s t t t t s s r t st s r r s t ts t rr rs t 2 t t r t 2 rts s r r ss t s t s t s t s t r t s rst s r t t r s s t σ = 0.2 t r r r t r t s r2 r s r r s 2 t rr r r s s s t t st s r s t 3 r t t s s t s s r r s t ts t t t r q r t t t t s s t t r t2 r t s r t s s r t t r r t t r r r s t s s rs st t s r r2 s t st s t 1 ts tt r r t r s t s t s 1 t n = 2 ss t t t t r st r t r ss s 2 t r st r ss dr t = κb r t dt + ζdw 0 t. t r r ss t s s r t2 s 2 st t rst s r t2 s r r ss s 2 s s r

t r Pr st r t r 2 r P r P s r s t t t s s t r s t ds i t = µ i S i t dt + σ i Y i S i dy i t = k i m i Y i dt + c i t t t β i dw i,1 Y i t t, β 2 = 1, dw i,2 t r W 0, W 1,1, W 1,2, W 2,1, W 2,2 s r t R 5 r s t2 s r 3 r rr t t t s r t2 r r ss ts t t2 r ss β 2 = 1 t t t rt t 3 t r s t t s 2 r t st t X t, r t, S 1 t, Y 1 t, Y 2 t = x, r, s 1, y 1, y 2 t t t r t t t v t, x,r,s 1, y 1, y 2 s t s s t q t 0 = v t L r + L Y + L S1 v rxv x sup θ 1,θ 2 { θ 1 µ r1v x + θ 1 σ 2 1y 1 s 2β 1 1 1 v xs1 + 1 2 θ2 1σ 2 1y 1 s 2β 1 2 1 + θ 2 2σ 2 2y 2 v xx } r = v t L r + L Y + L S1 v rxv x + µ 1 rv x + σ1 2y 1s 2β 1 1 1 v xs1 2 2σ1 2y 1s 2β + µ 2 rv x 2 1 2 1 v xx 2σ2 2y 2v xx L r v = κb rv r + 1 2 ζ2 v rr, L Y v = 2 k i m i y i v yi + 1 2 c2 i y i v yi y i, i=1 L S1 v = µ 1 s 1 v s1 1 2 σ2 1s 1 y 1 v s1 s 1. r r t t r t r t r s s r r t s v t L r + L Y + L S1 v 1 2 σ2 v xx + F x, r, s 1, y 1, y 2, Dv, D 2 v = 0, vt, x, r,s 1, y 1, y 2 = e ηx, r σ > 0 t r t2 F : R 5 R 5 S 2 s 2 Fu, z,γ = 1 2 σ2 γ 11 x 1 x 2 z 1 + µ 1 x 2 z 1 + σ1 2x 4x 2β 1 1 3 γ 1,3 2 2σ1 2x 4x 2β + µ 2 x 2 z 1 2 1 2 3 γ 11 2σ2 2x, 5γ 11 r u = x 1,, x 5 1t s r t tr t r t2 F ε,m u, z,γ := 1 { 2 σ2 γ 11 x 1 x 2 z 1 + sup θ µ r1z 1 + θ 1 σ1x 2 4 εx 3 ε 2β1 1 γ 13 ε θ M + 1 2 θ2 1σ 2 1x 3 εx 4 ε 2β 1 2 + θ 2 2σ 2 2x 5 εγ 11 },

r s ts r ε, M > 0 r t 2 s t σ s t t ss t s tr r t s r t r r t s s s 2 dx 1 t = σdw 0 t, dx 2 t = κb X 2 t dt + ζdw 1 t, dx 3 t = µ 1 X 3 t dt + σ 1 X 4 t X 3 β 1 t dw 1,1 t, dx 4 t = k 1 m 1 X 4 t dt + c 1 X 4 t dw 1,2 t, dx 5 t = k 2 m 2 X 5 t dt + c 2 X 5 t dw 2,2 t. t X 2 t s s t r t t 1 t s r t 3 t X 2 t n = b + e k t X 2 1 exp 2κ t t n 1 b + ζ ξ n, 2κ r ξ n n 1 s s q t r r t str t N0, 1 s r t r t ss t r t s t t2 s lnx n 3 = lnx 3 n 1 µ + 1 1 2 σ2 1 X 3 2β1 1 4 n 1 X n 1 t + σ 1 X 3 βi 1 n 1 X 4 1,2 n 1 W n r W n 1,2 := W n 1,2 W 1,2 n 1 t t r t rs µ 1 = 0.10 σ 1 = 0.3 β 1 = 0.5 r t rst ss t k 1 = 0.1 m 1 = 1. c 1 = 0.1 r t s r ss t rst ss t s ss t s 2 t s r t rs s t t s 1 µ 2 = 0.15 c 2 = 0.2 m = 0.3 Y 2 0 = m s r t t r st r t t b = 0.07 X 2 0 = b ζ = 0.3 t s t rt t ss ts r s r s t t r t s t st s rst s t s s r t r r ss t t 4 4 4 4 10 s s t r s rt s r 1 t s s s r t st s r t s t t t s t t 2 s t r t t st s 2 r r s ts r t t t s t rs t st s t t s r t t r t st s r t t t r s st r r r2 t t st t t t r s r s t s s s s t t t t r rt s r s s r 2 t s r t t r t t r t st s t s t s t s r s t r r r rt t r s ts rr s t t rt2 s rt s t 4 4 4 4 40 s s rst t t t t t s s r t 3 t r s ts r r s t st s t 3 r t 1 t s t s s t r2 s t 0.258 r r 1 r ts t t rt2 s rt s t s s 2s r t r2 r t r s ts s t t s s s t t st s r t t rt2 s rt s t r2 r s t s t r ss rs t r t r 2t s r2

t r Pr st r t r 2 r P r P s r s r ts r s ts r r t t t s t s s rt s s r r s ts t r t r s r t s t t t s s s st 2 s t s s r t r r ss t s r r 1 r ts r t t t s s r r s tt r t t r r s ts t t r r rt s t 2 t r r ss t s r r r s ts r r t s t s r t r t t r st s r t t s s ts t r r s ts ts t t t t t t t s st t t r t2 ss t 2 s r t t r s s r t t r s s r q r r t t s s s s r r rt s t t t t 2 t s t t t r s t s t st t t s t ss rt s t t tr 2 1 r t t s r t t s t r t r s ts t s t 2 t r t t s t t t r st

t r Pr st r t s r 2 r P r P s s t r s r 3 s s t t r t t r s 2 r P s s s ss ï r 3 t t t r t r s t r t s t t t r q r t r s r s t s r 2 t r r 1 t é 2 t r t t r t t rr r 2s s t t s t r s ts r s2 t t r rt s t s Pr r s t r s r P s t µ σ t s r [0, T] R d t R d Md, d r s t 2 η t r [0, T] R d R d t Rd a = σ σ s t 2 r L X vt, x F t, x,vt, x, Dvt, x, D 2 vt, x, vt, = 0, [0, T R d, vt, = g, R d. r F : R + R d R R d S d C d R L X 2 ϕ L X ϕt, x := t + µ Dϕ + 1 2 a D2 ϕ t, x + ϕt, x + ηt, x,z ϕt, x 1 { z 1} Dϕt, x ηt, x,z dνz. R d L X s t t s r t r s X t s t s 2 dx t = µt, X t dt + σt, X t dw t + ηt, X t, zjdt, dz + ηt, X t, z Jdt, dz, { z >1} { z 1} r J J r r s t 2 P ss s r ts s t ss t t é 2 s r ν 2 [ ] νa = E J[0,1], dz A Jdt, dz = Jdt, dz dt νdz.

t r Pr st r t s r 2 r P r P s r r t s s r ss s s t r r s t r r t ss r r s r t 3 t r t r ss X s t t = T n t i = i κ 0 t r s r t 3 t s r ss X t t tr t é 2 s r 2 ˆX t,x,κ = x + µt, x + σt, xw + ˆX x,κ ˆX x,κ t i,κ t i+1 = ˆX t i, { z >κ} ηt, x,z J[0, ], dz, ˆX x,κ 0 = x. r µt, x = µt, x + { z >1} ηt, x,zνdz t κ = 0 ν s t s r t Ñκ t Nt κ r s t 2 t P ss r ss r r s r J 2 t s s 3 r t r t κ t t r [0, t] ts s t Nt κ = J[0, t], dz Ñt κ = J[0, t], dz. { z >κ} r t t rt 1 ˆX t,x,κ ˆXt,x,κ { z >κ} s P ss r ss s r = x + µ κ t, x + σt, xw + ηt, x,z i, N κ i=1 r µ κ t, x = µt, x {κ< z 1} ηt, x,zνdz Z is r R d r r s t W N κ 1 str t s 1 { z >κ} λ κ νdz ss s t r t r s t 1 st r t r r t t t s s t2 s t s r r P s r t t t s s t2 s s r s t s r s t s r s t s t v v : [0, T] R d R s t t r 2 t 0, x 0 [0, T R d 2 s t t ϕ t 0 = maxmin{v ϕ} = v ϕt 0, x 0 0 L X ϕt 0, x 0 F, ϕ,dϕ, D 2 ϕ, ϕ t 0, x 0. g vt, vt, t v s t s s t2 s s r s t s s s t2 s t s 2 t t s r s r t s r 2 r s t s s s t2 s r s t v 2 r s t s s s t v s t s 2 v v [0, T] R d vt, vt,,

Pr r s t r s r P s s r 2 r r P s t s s t tr r st s 2 r t 2 t s s t s r t P s s r s r s r t s t t 3 t s 2 t r st t s r r tr rs t s r s r t ss r t s t2 t t s s t r t t t r s v T,. = g v t i, x = T [v ]t i, x, r r r2 t ψ : R + R d R t 1 t r t [ ] T [ψ]t, x:=e ψ t +, + F t, x, D ψ, ψt +,, ˆX t,x D ψ := D 0 ψ, D1 ψ, D2 ψ, r D k ψt, x := E [ ψt +, ˆX t,x,κ H k t, x ], k = 0,1,2, r H 0 = 1, H 1 = σ T 1 W, H 2 = σ T 1 W W T I d 2 σ 1. t s r 1 t r t s t r t 2 s t r r t s t r s s 1t s t 2 2 t ss t s s t ss t t r ss t r t2 F s s t3 t s t r s t t x, r, p,γ,ψ r 2 t F,,0, 0, 0,0 < F s t t 2 t s t r r t r L X γ F a R d R R d S d C d ; F p ImageF γ F p Fγ F p < + r t t t r t2 F s t F s r s t s r t t Ft, x,r,p, γ 1, ψ Ft, x,r, p, γ 2, ψ r γ 1 γ 2. F s r s t t Ft, x,r, p, γ, ψ 1 Ft, x,r,p, γ, ψ 2 r ψ 1 ψ 2.

t r Pr st r t s r 2 r P r P s t r r t ss t tr µ 1 σ 1 < σ s rt s ss t t t 2 r P s r s r t s r r2 s t3 t g t r 1 sts t v s t t v v 2 r 2. t v s t q s s t2 s t r r s str t r r t t t s t t r 1 t ν t s t s r Ft, x, r, p, γ, ψ = Gt, x,r, p, γ, R ψx + ηt, x,zζt, x,zνdz d r s t G s t t s s t s r F t r s r t r st t s t s t t t r r 2 t r st t s r r s t r t r r s t t r ss 2 r P s s t ss q t s r st st tr r s r s 2 t s r t2 q t s t s t s s ss t r r P s t2 ss t s t s F s t r 2 s t3 t r s t t x t t é 2 s r ν s t é 2 s r t r s r F t r 2 s t3 t r s t t ψ t r r rs 2 t s s t 1 t r r r t2 F r s t t 1 r r t t t s r 1 s t t t t t t s r t 2 r q t v t Fx, Dvt, x, D 2 vt, x, vt, = 0 vt, = g, r Fx, p,γ, ψ := { } sup L θ p, γ + ψx + θzνdz θ R + R L θ p, γ := θbp + 1 2 θ2 a 2 γ Ix, ψ θ := ψx + θzνdz. R s 2 r q t s s t r rt t r ss t t s s ss t r r t s

Pr r s t r s r P s s t2 r t t r t t t t t2 s s r t t ν = 0 t ss t r r s t F s t r { Fx, p,γ, ψ := θbp + 1 } 2 θ2 a 2 γ. 1 t r 2 sup θ R + Fx, p,γ, ψ := bp2 2a 2 γ, t s s 2 t s t r r r t 1 s s t t ν 0 1 sts t 1 t r r F s t t s r s r 1 t s r s t t r r t s r 2 r P s t r t s r s s 2 2 t s t t s t s s r ss t r r r t t t t 2 r t t r 1 t t t r s t s r r r s 2 t r s 1 t r r t r t2 s t t t é 2 t r s t s r r θ t 2 s r t s t r 1 t t s r r ss θs r r r s t r r t r t t r 1 t t t r r 2 r st 2 s r t 2 t r t s s s t t ν s t s r t r r 1 s r tt t r Ix, ψ := ψx + θz ψx 1{ z 1} θdψx z νdz. R t s s t r r t 2s t tr t t s r é 2 s r s t tr t é 2 s r r 3 r s r s r t 3 t X t t r s t r 1 t t s s 2 r t t s s t r r r t r 1 t F s { F κ x, r, p,γ,ψ := sup c κ r + θb κ p + 1 } θ R + 2 θ2 a 2 γ + ψx + θzνdz. { z >κ} r c κ := { z >κ} νdz b κ := b zνdz. {1 z >κ} 1 t ss t s r t t F κ s 2 t t r t s F κ t r s t t r p ψ t t2 s κ s s str 2s t r r s t r t s r s t t κ s t s t t t rr s s s t s s t r q r ts r t r r

t r Pr st r t s r 2 r P r P s s t tr t s s t r 1 t r t2 F t r tr t t é 2 s r t t r s2 t t r s ts t r r s t r r t r t s s t r s t r t t t é 2 r t r I[ϕ]x := ϕ x + ηz ϕx 1{ z 1} ηz Dϕx νdz. R d t s r t r t t r 1 t t r r s t r 1 t é 2 t r s t s s t r s t t s s t s t t r s tr t s t r r t t 2 r t r t r t t s t r t 2 t r s t t t, x r t r r s r t s s t2 st r t ηz t t t r r r t r r r t s s t ss t η ηz z 1 C, r s st t C. r s t t r s s t r s t t t r é 2 s r r 3 r t s r { z 1} νdz < t s r s { z 1} ηz νdz < s { z 1} ηz 2 νdz < t é 2 s r é 2 s r s t s κ = 0 t s s tr r s s 2 t r 1 t t é 2 t r r r ϕx + ηzζzdνz, R d t s t s t r 1 t t é 2 t s r t r t J P ss s r t t s t2 2 é 2 s r ν {N t } t 0 t P ss r ss 2 N t = t 0 Jds, dz s t s t2 s R d

t r r t r λ := R d νdz 2 r t ˆX x 2 N t ˆX t x = x + µ 0 t + σw t + ηz i i=1 r Z i s r r r s t 1 λνdz s tr é 2 r ss Y t 2 Y t = N t i=1 ζz i. 1t s s t t r 1 t 2 t r r r 2 r t r t t ˆν η,ζ ϕx := E [ r r2 t ϕ : R d R R d ϕ ˆX x + ηzζzdνz ]. ˆν η,ζ ϕx = 1 E[ϕ ˆX x Y ]. Pr r t s s t2 st tr t t rt r ss ˆX x t t ss r t2 r t ˆX x = x + N i=1 ηy i r t s 1 r ss s [ E ϕ ˆX ] x Y 2 [ E ϕ ˆX ] x Y = e λ λ = e λ λ [ = e λ E ϕ ˆX ] λ x n Y N = n. n! n=0 E ϕ x + n=1 λ n 1 n=1 n! n j=1 n n ηz i ζz j λn 1 n! j=1 [ ] n E ϕ x + ηz i ζz j. t t t t 1 r ss t s t st rts r n = 1 s Y = 0 N = 0 s Z i s r t t [ ] [ ] n n n E ϕ x + ηz i ζz j = ne ϕ x + ηz i ζz 1 j=1 i=1 r t [ ] n [ E ϕ x + ηz 1 + ηz i ζz 1 =E ϕ ηz + ˆX ] x ζz N = n 1, i=2 i=1 i=1 i=1

t r Pr st r t s r 2 r P r P s r Z s t Z i s t s t s s Z i s r r t t [ E ϕ ˆX ] x Y = e λ λ n=1 [ E ϕηz + ˆX ] λ x n 1 ζz N = n 1 n 1!. t t t e λ n=1 r r [ E ϕηz + ˆX ] λ x n 1 [ ζz N = n 1 n 1! = E ϕηz + ˆX ] x ζz [ E ϕ ˆX ] x Y = λe[ϕηz + X x ζz]. s t s t2 Z s νdz λ [ E ϕ ˆX ] x Y = E [ R d ϕηz + ˆX x ζzdνz ] t t r s t r 1 t r I [ϕ]x := ˆν η,1 ϕx νdz Dϕx ηzνdz. R d R d. 1t r rr r r t s r 1 t r 2 s t3 t ϕ I I[ϕ] C Dϕ. Pr s r t s q ˆν η,1 = 1 E[ϕ ˆX xn ]. r r t t I I [ϕ] C Dϕ E [ ˆX ] x x. s E [ ˆX ] x x C R d ηz νdz +, r s t r s t

t r r t r t é 2 s r t s s r é 2 s r tr t é 2 s r r 3 r r t r t t s r t r r s r 2 κ > 0 t tr t r 1 t t r r t r I κ [ϕ]x := ϕ x + ηz ϕx 1{ z 1} ηz Dϕx νdz. { z >κ} s t r s t t r 1 t r I κ, [ϕ]x := ˆν η,1 κ, ϕx dνz ηt, x,z Dϕxdνz, { z >κ} {1 z >κ} r 2 ˆν η,1 κ, := ϕ { z >κ} ˆX x,κ [ + ηt, x,zνdz = 1 E ϕ ] x,κ ˆX Nκ r s t rr r r 1 t t s t é 2 s r t t ϕ s t3 z νdz < t Pr { z 1} I κ, I[ϕ] C Dϕ + z νdz {0< z κ} { z 1} z 2 νdz < t I κ, I[ϕ] C Dϕ + z νdz + D 2 ϕ { z >κ} t t t I I κ, [ϕ] I I κ [ϕ] + I κ I κ, [ϕ]. 2 t tr t rr r s 2 I I κ [ϕ] 2 Dϕ {0< z κ} ηz νdz.. z 2 νdz. {0< z κ} t t r 2 s r t t I κ I κ, [ϕ] C Dϕ ηz νdz + { z >κ} C Dϕ z νdz + { z >κ} t t r t r s t r s t

t r Pr st r t s r 2 r P r P s 2 t tr t rr r s 2 I I κ [ϕ] C D 2 ϕ {0< z κ} z 2 νdz, r 2 t ϕ t r t s t s r r t t r s s t t t t r rr r 2 I κ I κ, [ϕ] C Dϕ z νdz + { z >κ} t s t r s2 t t r s ts s s t s t t t r r s t r t s rst r t t s s t2 s t r t ss t s r q r r t r s ts t t r t t st t t r s ts r t r t r s ts t s r t s s t t s t ss t t r t2 F t t t r r ss t t F s t s s { 1 at, x γ + µt, x p + Ft, x,r,p, γ, ψ:= inf 2 sup L α,β t, x,r,p, γ α A β B } +I α,β t, x,r, p, γ, ψ r s ts A B r L α,β t, x,r,p, γ:= 1 2 aα,β t, x γ + b α,β t, x p + c α,β t, xr + k α,β t, x, I α,β t, x,r,p, ψ:= R d ψ x + η α,β t, x,z r 1 { z 1} η α,β t, x,z p νdz r r 2 α,β A B a α,β b α,β c α,β k α,β η α,β s t s 2 { } sup a α,β 1 + b α,β 1 + c α,β 1 + k α,β 1 + ηα,β, z 1 <. α A,β B z 1 r t2 s t 2 t s t r r t r L X r

s2 t t r s ts 2 t x z α β a a α,β 1 < 0 a α,β a, η α,β, b α,β Imagea α,β sup b α,β T a α,β b α,β <, α A,β B η α,β T a α,β b α,β sup α A,β B 1 z < η α,β T a α,β η α,β sup α A,β B 1 z 2 <. r t F s t s s ss t s t r r tr r2 t, x,r,p, γ, ψ R + R d R R d S d C d t r 2 s r r r t t ψ t r t s t r s t t x Ft, x,ψt, x, Dψt, x, D 2 ψt, x, ψt, s r s t r s r s t s s t r s t r 1 t t r t2 v κ, T,. = g v κ, t i, x = T κ, [v κ, ]t i, x, r r r2 t ψ : R + R d R t 1 t r t [ T κ, [ψ]t, x:=e ψ t +, ˆX t,x,κ D ψ := D 0 ψ, D1 ψ, D2 ψ, ] + F κ, t, x, D ψ, ψt +,, { 1 F κ, t, x,r,p, γ, ψ= inf sup α A β B 2 aα,β t, x γ + b α,β t, x p + c α,β t, xr + k α,β t, x } + ˆν ηα,β,1 ψt, x r η α,β t, x,z p νdz, { z κ} D k ψt, x := E [ ψt +, ˆX t,x,κ H k t, x ], k = 0, 1, 2, r H 0 = 1, H 1 = σ T 1 W, H 2 = σ T 1 W W T I d 2 σ 1. t s r 1 t r t s t t r r r t t r r s t s t s t ss t r F κ, ss t r 2 κ > 0 t [0, T] x x R d 2 s t3 t s ψ ϕ t r 1 sts α, β A B s t t Φ α,β κ [ψ,ϕ]t, x,x = J α,β κ [ψ]t, x J α,β κ [ϕ]t, x

r t r Pr st r t s r 2 r P r P s Φ α,β κ [ψ, ϕ]t, x := inf J α,β α κ [ψ]t, x sup Jκ α,β [ϕ]t, x, β J α,β κ [φ]t, x := 1 2 aα,β D 2 φt, x + b α,β Dφt, x + c α,β φt, x + k α,β t, x η + ˆν α,β,1 φt, x φt, x η α,β t, x,z Dφt, x νdz. { z κ} rst r s t r s t r t r v κ, r κ r r t 2 s t r s t t r r t η µ σ s t3 t s x r 2 t z σ s rt ss t s tr ss t t s r s r t s κ s s t t lim κ = 0 lim supθκ 2 0 = 0 0 r θ κ := sup θκ α,β, α,β t θκ α,β := c α,β + νdz + 1 b α,β { z κ} 4 a α,β b α,β {1> z κ} {1> z κ} η α,β zνdz, η α,β zνdz t v κ, r s t s t v 2 r t v s t q s s t2 s t 2 é 2 s r s t r t κ = 0 t ss rt t r tr r t s 2s ss t s κ s t t s s t s s t s t t t θ κ s r s κ lim κ 0 θ κ = + lim supθ κ <. κ κ := inf{κ θ κ 1 2 } + 2 t t κ θ κ 1 2 s s r t t κ s r s t r s t t lim 0 κ = 0 t r 1 sts q s t t q := lim 0 κ > 0 t r κ < q θ κ = s 2 tr ts t t t t r κ > 0 θ κ < r r κ s t s s

s2 t t r s ts r κ t r s s t r r t r t r s 2 t r r s t Pr s t r t s ss t s r é 2 s r ν s t r r2 s t3 t g lim lim κ 0 0 vκ, = v r v s t q s s t2 s t ss t t t 1 sts Pr t v κ t s t t r L X v κ t, x F κ t, x,v κ t, x,dv κ t, x,d 2 v κ t, x,v κ t, = 0, [0, T R d, v κ T, = g, r F κ : R + R d R R d S d C d R s 2 F κ t, x,r,p, γ, ψ := r Iκ α,β t, x,r,p, γ, ψ:= { z κ} R d. { } inf sup L α,β t, x,r,p, γ + Iκ α,β t, x,r, p, γ, ψ α A β B ψ x + η α,β t, x,z r 1 { z 1} η α,β t, x,z p νdz r a α,β b α,β c α,β k α,β η α,β r s ss t t v κ, t r 1 t s t 2 t s t κ > 0 1 s t tr t é 2 s r s t 2 r v κ, r s t v κ 2 r 2 s 0 t v κ t s t 2 r ss t v v κ C sup α,β C η α,β, z 2 νdz 0< z <κ z 2 νdz 0< z <κ 1 2 1 2. r r s κ > 0 s t t v κ v s s t 0 v κ, r s t v κ t r s s t t t r s t st s rst 2 s κ s t t v κ s r t v t r r 1 t v 2 s 0 t 2 r r v κ, t v κ

t r Pr st r t s r 2 r P r P s t t t t r s t r 2 κ, r t r r s r t κ s s t 2 t r 2 r t r t ss t g r 1 r r t t t r t r r s t s ss t s r str t s t r t s ss t s t s t r t2 F s t s s ss t t B r r r t r t2 F s t s s r t s r s r t β r t F 2 { 1 at, x γ + µt, x p + Ft, x,r,p, γ, ψ := inf L α t, x,r,p, γ 2 α A } +I α t, x,r,p, γ, ψ r L α t, x,r,p, γ := 1 2 Tr[ a α T] t, xγ + b α t, xp + c α t, xr + k α t, x, I α t, x,r, p, ψ := R d ψ x + η α t, x,z r 1 { z 1} η α t, x,z p νdz. t s s t r t2 s t r, p, γ, ψ ss t t r 1 sts t s t {α i } M δ inf 1 i M δ { r t2 F s t s s r 2 δ > 0 i=1 s t t r 2 α A σ α σ α i + b α b α i + c α c α i + k α k α i + η α η α i, z 2 dνz R d } δ. r ss t s s t s A s t s r t s σ α b α c α r t s s r A t C 1 2,1 b [0, T] R d t s s r s t3 x { 1 2 ϕ : [0, T] R d R d ö r t η α s t s s r A t R } R ϕ, d z 2 νdz <

s2 t t r s ts r t r ss t t t t g s s t3 t s t r s st t C > 0 s t t r ss t v v κ, C r ss t C 1 4 + θ 2 κ + ε 3 + 3 4θ κ + θ κ + 1 4 1/10 + 7 10θ κ + θ κ + 3 10 t t s ss t κ s t t lim κ = 0, 0 lim sup 3 4 θ 2 κ < 0 t t r s st t C > 0 s t t { z κ} z 2 νdz { z κ} z 2 νdz v v κ, lim sup 1 2 0 r ss t v v κ, C 1/4 r ss t C 1/10 v v κ, 1 r t é 2 s r νdz = 1 R d z d 1 dz, z 2 νdz <, 0< z <κ 2s κ s t t t t r s s t s t t r r s t s 2s t s κ s t t lim sup 1 2 κ = 0. 0 r s s t t ss t s r s tr t r t t s s s t rst t t s t r str t t t2 2 t s r s r t r t u κ, t s t u κ, T, = g u κ, t i, x = T κ, [u κ, ]t i, x, r [ T κ, [ψ]t, x:=e ψ t +, ] t,x,κ ˆX + F κ, t, x, D ψ, ψt +, { 1 F κ, t, x,r, p, γ, ψ=supinf α β 2 aα,β γ + b α,β p + c α,β + θ κ r + e θκt t k α,β t, x η + ˆν α,β,1 ψ r 1 { z 1} η α,β z p } νdz. κ, { z κ}

t r Pr st r t s r 2 r P r P s r ss t s s tr r J α,β κ 2 J κ α,β [ψ]t, x = 1 2 aα,β D 2 φt, x + b α,β Dφt, x + c α,β + θ κ φt, x + e θκt t k α,β t, x η + ˆν α,β,1 φt, x φt, x η α,β t, x,z Dφt, x νdz. { z κ} r s str t r r t s s t t r r r θ κ t s s str t 2 t t θ κ s ϕ ψ : [0, T] R d R t t s ϕ ψ = T κ, [ϕ] T κ, [ψ]. Pr t f := ψ ϕ 0 r ϕ ψ r s t st t t t r s t2 r t t, x t s t ss r2 2 ss t r t T κ, [ψ] T κ, [ϕ] = E[ft +, ˆX ] + inf sup α β Jκ α,β r φt, x := E[φt, ˆX x ] r φ = ϕ r ψ r r [ ψ]t +, x inf sup J α,β α κ [ ϕ]t +, x β, r T κ, [ψ] T κ, [ϕ] Φ α,β κ s 2 Φ α,β κ [ψ, ϕ]t, x := inf α E[ft +, ˆX ] + Φ α,β κ [ ψ, ϕ]t +, x,x, Jκ α,β [ψ]t, x sup β κ [ϕ]t, x. J α,β 2 ss t t r 1 sts α, β s t t T κ, [ψ] T κ, [ϕ] E[ft +, ˆX ] + J κ [ ψ]t +, x J κ [ ϕ]t +, x. s r t t 2 t r t2 J α,β κ r t [ J κ α,β [ˆφ]t +, x = E J α,β κ [φ]t +, ˆX ]. 2 t t J α,β κ t r [ T κ, [ψ] T κ, [ϕ] E f ˆX 1 + c α,β κ + 1,β 2 aα σ T 1W W d 2 σ 1 + θ κ + b α,β κ σ T 1W ], + ˆν ηα,β,1 f

s2 t t r s ts r b α,β κ = b α,β {1> z κ} ηα,β zνdz c α,β κ = c α,β { z κ} νdz r r 2 t s r t s t r r t [ T κ, [ψ] T κ, [ϕ] E f ˆX 1 1,β 2 aα a 1 + A α,β 2 + c κ + θ κ 1,β 4 bα κ ] a α,β b α,β κ + ˆν ηα,β,1 f, r A α,β := 1 σα,β 1/2 σ T 1 W + 1 2 σα,β 1/2 b α,β κ. r r 2 s t t2 f ss t [ T κ, [ψ] T κ, [ϕ] E f ˆX c κ + θ κ 1 ],β 4 bα κ a α,β b α,β κ 2 t θ κ T κ, [ψ] T κ, [ϕ] 0. s ε t 3 r r s t r s t r r2 s s t t t2 s r r2 t ϕ, ψ : [0, T] R d R t t s ϕ ψ = T κ, [ϕ] T κ, [ψ] θ2 κ 2 2 e θκ E[ψ ϕt +, ˆX t,x,κ ]. rt r κ s t s s t ϕ ψ = T κ,[ϕ] T κ,[ψ] + CE[ψ ϕt +, ˆX t,x,κ ] r s st t C Pr t θ κ s ϕ κ t, x := e θκt t ϕt, x ψ κ t, x := e θκt t ψt, x 2 T κ, [ϕ κ ] T κ, [ψ κ ]. 2 t 2 t s s 2 e θκt t e θκ t,x,κ 1 + θ κ 1 E[ϕt +, ˆX ] + T κ, [ϕ] e θκ 1 + θ κ 1 E[ψt +, t,x,κ ˆX ] + T κ, [ψ]. T κ, [ϕ] e θκ 1 + θ κ 1 E[ψ ϕt +, t,x,κ ˆX ] + T κ, [ψ].

t r Pr st r t s r 2 r P r P s t e θκ 1 + θ κ 1 θ2 κ 2 2 e θκ T κ, [ϕ] θ2 κ 2 2 e θκ E[ψ ϕt +, t,x,κ ˆX ] + T κ, [ψ]. r s t r s t r r t r r v κ, u κ, t r s t t θ κ s t t ϕ ψ : [0, T] R d R t L t s T κ, [ϕ] T κ, [ψ] ϕ ψ 1 + C + θ κ r C = sup α,β c α,β rt r g s L r 1 κ t 2 u κ, t, s L r 2 2 C + g e C+θκT t i. Pr t f := ϕ ψ 2 ss t t s r t s t r [ T κ, [ϕ] T κ, [ψ] E f ˆX 1 a 1 a α,β + A α,β 2 + c α,β + θ κ νdz 1 b α,β η α,β Ta α zνdz,β { z κ} 4 {1> z κ} ] b α,β η α,β zνdz + ˆν ηα,β,1 f, {1> z κ} r A α,β s 2 t t r ˆν ηα,β,1 f f νdz { z κ} r r 1 T κ, [ϕ] T κ, [ψ] f E[ a 1 a α,β + A α,β 2 + c α,β + θ κ 1 b α,β η α,β Ta α zνdz,β ] b α,β η α,β zνdz. 4 {1> z κ} {1> z κ} 2 ss t 1 a 1 a α,β c α,β + θ κ 1 4 b α,β η α,β Ta zνdz α,β b α,β η α,β zνdz {1> z κ} {1> z κ}

s2 t t r s ts r s t r r r t T κ, [ϕ] T κ, [ψ] f 1 a 1 a α,β + E[ A α,β 2 ] + c α,β + θ κ 1 b α,β η α,β Ta α zνdz,β b α,β η α,β zνdz. 4 {1> z κ} {1> z κ} t t t t E[ A α,β 2 ] = 1 a 1 a α,β + 1 b α,β 4 η α,β Ta zνdz α,β 1 b α,β {1> z κ} 2 r E[ A α,β 2 ] t t s t C = sup α,β c α,β T κ, [ϕ] T κ, [ψ] f 1 + c α,β + θ κ f 1 + C + θ κ, η α,β zνdz. {1> z κ} 2 t r ϕ ψ t t s r t t s T κ, [ϕ] T κ, [ψ] f 1 + C + θ κ. r t t t 2 u κ, s r 2 r t s t r 2 s t rst rt t r ϕ ū κ, t i+1,. ψ 0 s t t u κ, t i, Ce θκt t i + u κ, t i+1, 1 + C + θ κ, r C := sup α,β k α,β t s r t s r t r q t2 t t u κ, t i, CT t i + g e C+θκT t i. v κ, := e θκt t u κ,. 1t r r2 r s r v κ, r 2 κ r r2 v κ, s r 2 κ s κ s t s s t v κ, v κ, Kθ 2 κ r s st t K. lim 0 vκ, v κ, = 0.

t r Pr st r t s r 2 r P r P s Pr 2 r 1 κ r r u κ, t,. C + g e C+θκT t. v κ, t,. C + g e CT t. r t 1t rt ū κ, t, x = e θκt t v κ, t, x r t t s s s t t [ ] ū κ, = e θκ 1 θ κ E ū κ, t,x,κ t +, ˆX + F κ, t, x, D ū κ,,ū κ, t +,. 2 r t s r t t r u κ, ū κ, t, 1 2 θ2 κ 2 ū κ, t +, +1 + C + θ κ u κ, ū κ, t +,, r C s s 2 r t t r r ū κ, ū κ, t, C + g e C+θκT t 1 + θ κ 2. 2 t 2 2 e θκt t v κ, v κ, t, 1 2 Cθ 2 κ 2 e CT t 1 + θ κ 2 e θκ +e θκ 1 + C + θ κ v κ, v κ, t +,, r s st t C s e θκ 1 + C + θ κ e C r s r t r q t2 t t v κ, v κ, t, Kθ 2 κ, r s st t K t κ r s t s rt t t r t t t s st 2 t ϕ s t t t t r t s r t, x [0, T] R d ϕt, x T κ, [c + ϕ]t, x lim = L X ϕ + F, ϕ,dϕ, D 2 ϕ, ϕt, t, x. t,x t,x,c 0,0 t + T

s2 t t r s ts Pr r s str t r r 2 s t r r t t r r t t r t t r 1 t s t v κ, r t t t s t κ s t s 2 t v κ, s r 2 s t3 t r s t t x Pr r rt t t t s s d = 1 r r t s 2 t r s t t r 1 t [0, T ] r s t r t s t t r x, x R t x > x [ u κ, t, x u κ, t, x = E u κ, t +, ˆX t,x u κ, t +, ˆX t,x ] + inf sup J α κ α,β [ûκ, ]t +, x inf sup J β α κ α,β [ûκ, ]t +, x β [ E u κ, t +, ˆX t,x u κ, t +, ˆX t,x ] + sup β Jκ α,β [ûκ, ]t +, x inf α J α,β κ [ûκ, ]t +, x s r t t 2 r t [ ] u κ, t, x u κ, t, x E u κ, t +, ˆX t,x u κ, t +, ˆX t,x + Φα,β [ûκ,,ûκ, ]t +, x,x, r Φ s t r 2 ss t t r 1 sts α, β s t t Φ α,β [ûκ,,ûκ, ]t +, x,x = J α,β κ [ûκ, ]t +, x J α,β κ [ûκ, ]t +, x. r r [ u κ, t, x u κ, t, x E u κ, t +, ˆX t,x u κ, t +, ˆX t,x ] + J κ [ûκ, ]t +, x J κ [ûκ, ]t +, x r t t r q t2 t s 1 t t t.. u κ, t, x u κ, t, x A + B + C, r [ A := E u κ, t +, ˆX t,x u κ, t +, ˆX t,x ] + J α,β κ [ûκ, ]t +, x J α,β κ [ ũ κ, ]t +, x,

t r Pr st r t s r 2 r P r P s t ũ κ, y = u κ, y + x x B := J α,β κ [ ũ κ, ]t +, x J α,β κ [ûκ, ]t +, x, C := ˆν α,β,1 u κ, t +, x ˆν α,β,1 u κ, t +, x. t t r t st s t [ C = 1 E u κ, t +, ˆX,x u κ, t +, ˆX ],x N κ, r ˆX,x := x+ N κ i=1 ηα,β x, Z i t Z i s r r r s str t s νdz λ κ t 2 t t J α,β κ B = 1,β 2 aα x a α,β x D 2 uκ, t +, x + b α,β κ x b α,β κ x D 1 uκ, t +, x + c α,β x c α,β x D 0 uκ, t +, x +k α,β x k α,β x, r b α,β κ x := b α,β x {1> z κ} ηα,β x, zνdz t t r D k = E [ Du κ, t +, [ 1,β B E 2 aα x a α,β x Du κ, t +, ] k 1 x ˆX W σ 1 x, r k = 1,2. x ˆX W σ 1 x +b α,β κ x b α,β κ x Du κ, t +, ˆX x +cα,β x c α,β x u κ, t +, ˆX x ]+f α,β x f α,β x. t 2 t t J α,β κ J α,β κ s r t t [u κ, ]t +, x J α,β κ [ũ κ, ]t +, x = 1 2 aα,β xδ 2 + b κxδ1 + c κxδ0 r c κ b κ r t r δ k [ = E D k u κ, t +, ˆX x Dk u κ, t +, ˆX x ] r k = 0, 1,2.

s2 t t r s ts 2 t r r k = 1 2 δ k [ = E u κ, t +, ˆX x uκ, x t +, ˆX H k t, x +u κ, x t +, ˆX Hk t, x 1 σk x ] σ k x [ = E u κ, t +, ˆX x uκ, x t +, ˆX H k t, x +Du κ, t +, ˆX x W k 1 σx σ k x σ k x ]. r r r t A [ E u κ, t +, ˆX x uκ, t +, ˆX x 1 ā + ā N 2 + c κ + b κn x +b κx Du κ, x t +, ˆX σx σ 1 x σ 1 x +a x Du κ, x t +, ˆX Nσx σ 2 x σ 2 x ], r a := 1 2 aα,β ā := 1 2 a 1 a α,β c := c α,β c κ := c + θ κ b κ := b α,β κ t 2 t s s 2 x x t t t Du κ, t, x E [Du κ, t +, ˆX x 1 + µ κ + σ N + J κ, 1 ā + ā N 2 + c κ + b κn + b κ b σ 1,β κ + σ 2 aα σ 1 a α,β σ N σ 2 +Du κ,,x t +, ˆX 1 + µ + J ] κ, N κ + Ce θκt t, r J κ, := { z >κ} ηz J[0, ], dz J κ, := { z >κ} η z J[0, ], dz N κ s P ss r ss t t s t2 λ κ := { z >κ} νdz t L t := Du κ, t, [ E Du κ, t +,,x ˆX 1 + µ + J ] κ, N κ L t+ C λ κ + λ κ, r λ κ := { z >κ} η zνdz t G := N + b κ σ 2 2 t s r dq dp := exp b κσ 2 + b κσ N, 4 2

t r Pr st r t s r 2 r P r P s G N0,1 r Q r t Du κ, t, x E Q[ dp dq Duκ, t +, ˆX x 1 + µ κ b κσ 2 + σ G + J κ, 1 ā + ā G 2 + c κ b κσ 2 2 + b κ b σ κ σ b κσ 1,β + 2 2 aα σ 1 a α,β σ ] G σ 2 +L t+ C λ κ + λ κ + Ce θκt t, t t t t 1 ā + a G 2 + c κ b κ σ2 2 s s t t r r t s s t2 r t s r Q Z Z E Q [Z] Du κ, t, x E QZ[ dp dq Duκ, t +, ˆX x 1 + µ κ b κσ 2 + σ G + J κ, +Z 1 b κ b σ κ σ b κσ 1,β + 2 2 aα σ 1 a α,β σ ] G σ 2 +L t+ C λ κ + λ κ + Ce θκt t. Du κ, t, x E QZ[ dp 2 Du κ, t +, dq ˆX x 2 2]1 E QZ[ 1 + µ κ b κσ 2 + σ G + J κ, +Z 1 b κ b σ κ σ b κσ 1,β + 2 2 aα σ 1 a α,β σ 2 ]1 2 G σ 2 +L t+ C λ κ + λ κ + Ce θκt t. t t t E QZ[ dq 2 Du κ, t +, dp ˆX x 2] L 2 t+ exp1 4 b κσ 2. t t r E QZ[ dq dp [ = E Z 1 + µ κ b κσ 2 + σ G + J κ, + Z 1 b κ b σ κ 1 2 aα,β σ 1 a α,β σ σ 2 G 2 ] + 1 + µ κ b κσ 2 + σ G + J κ, + Z 1 b κ b κ 1,β + 2 aα σ 1 a α,β σ 2 ] G σ 2. σ b κσ 2 σ σ b κσ 2 2 t t r t s t q t2 s r t t

s2 t t r s ts t t r s r r s E QZ[ dq 1 + µ κ b κσ dp 2 + σ G + J κ, + Z 1 b κ b σ κ σ b κσ 2 1,β + 2 aα σ 1 a α,β σ 2 ]1 2 G σ 2 1 + c + θ κ b κ 2 4a b κσσ + b κ b κσ σ b 1 κσ 2 + Oθ2 2 κ. r r 2 t κ r s 1 L t L t+ exp 2 C + θκ bκ σσ + b κ bκ σ b κ σ σ 2 +2λ κ +2λ κ L t+ exp C + θκ +Ce θκ T t. 2 s r t r q t2 r r 2 t v κ, L t Dg + CT te θκ +CT t. D v κ, 1 e CT t Dg + CT t. t κ s t s s t lim t T vκ, t, x = gx. +Ce θκ T t Pr t s t t s s t r t r s r t [ u κ, t, x = E u κ, t +, ˆX ] t,x + inf sup J α κ α,β [ûκ, ]t +, x β [ E u κ, t +, ˆX ] t,x + sup J κ α,β [ûκ, ]t +, x. β s r t t 2 r t [ u κ, t, x E u κ, t +, ˆX ] t,x + Φα,β [ûκ,,0]t +, x,x 2 ss t t r 1 sts α, β s t t [ u κ, t, x E u κ, t +, ˆX ] t,x + J α,β [ûκ, ]t +, x + C, κ + sup f α,β, α,β r C := sup α,β f α,β r r r 2 j = i,, n 1 r t u κ, t j, ˆX [ t i,x t j E Q t j u κ, t j+1, ˆX ] t i,x t j+1 1 a j + a jg 2 j + Cj + C.

t r Pr st r t s r 2 r P r P s r a j := ā t j, ˆX t i,x t j Cj := c κ b κ σ2 2 t j, ˆX t i,x t j G j s r t st r ss r r s r t q t s r Q 2 t s t s t q t2 t t t t 1 a j + a j G2 + Cj s s t r t u κ, t i, x E Q[ g ˆX t i,x T ] 1 a j + a jg 2 + Cj n 1 j=i + C n 1 j=i e θκt j. t t t t q t2 s t t t t ] E Q t j [1 a j + a jg 2 j + Cj = 1 + E Q t j [Cj ] 1 + θ κ. t t r Z := n 1 j=i 1 a j + a j G2 j + C Z j s s t t r r E Q [Z] s r s s t2 s r Q Z t r s t t P r r u κ, t i, x E Q [Z]E QZ[ g ˆX ] t i,x T + C n 1 e θκt j. 2 t t v κ, r t r r v κ, t i, x e θκt ti E Q [Z]E QZ[ g ˆX ] t i,x T j=i n 1 + e θκt t i C e θκt j. v κ, t i, x gx e θκt t i E Q [Z]E QZ[ g ˆX t i,x T gx ] + C gx T t i + e θκt t i CT ti. t t t g ˆX t i,x T gx r s t 3 r P s t r r QZ s s t i, T, 0 2 s t r r lim sup v κ, t i, x gx 0. t i, T,0 2 t s r r t r t t t s t r lim inf t i, T,0 vκ, t i, x gx 0, j=i r 2 1t t r s t r r2 t r r t t v κ, t, x gx CT t 1 2. s s r t t 2 t s r r t s t r v κ, s 1 2 ö r t r 2 x t r 1 t s t v κ, t s t s s t r q r t t r st s r s t t v 2 r 2 r r v s t q s s t2 s t 2 r r2 t s ss rt s tr r v κ,

s2 t t r s ts t r r t r t r r t s s t s s t s s t r r s 2 r 3 t t t s s t r t r t r t s s r t s s t s st t s t s s r s r r r t s s t 2 r s t t r s ts s t 2 t r 3 t t t s r r s st 2 r s r r s t r t s s t t tr t rr r 2 t r t s r r r s 2 v v κ r s t s r s t 2 t 2 r v v κ C z 2 νdz 0< z <κ r r 2 s κ s t t 0< z <κ z 2 νdz C 1 2 st tr t t r t r v κ, t v κ s t t v κ, s s r r t str t 2 t s t r t r r v κ, r r2 s s t t t s s t t t t r t r r r2 t F s t s s Ft, x,0,0,0,0 = 0 v κ, v κ, Cθ 2 κ. 1 2. t κ s s t t t lim sup 3 4 θ 2 κ <, 0 v κ, v κ, C 1 4 Pr r s str t r r 2 t r r tr t t r 1 t s t v κ, s t r str t 2 t s t r r r t r t r s t t s t s st 2 t s r t r r r 1 t s t s t r s r r t s r s s r t r t u κ, r r r 1 t s t s t R κ, [ψ]t, x := ψt, x T κ,[ψ]t, x + L X ψt, x +F κ, ψ,dψ,d 2 ψ, ψt, t, x.

t r Pr st r t s r 2 r P r P s r 2 {ϕ ε } 0<ε<1 s t t s s t s 2 β 0 t D β ϕ ε Cε 1 2β 0 β 1 r 2 β0, β N N d \ {0}, r β 1 := d i=1 β i C > 0 s s st t R κ, [ϕ ε ] R, ε := C ε 3 + θ κ ε 1 + θ κ + ε 1 z 2 νdz, { z κ} r s st t C > 0 t κ t lim supθκ 2 < 0 lim sup z 2 νdz <, 0 { z κ} R κ,[ϕ ε ] R, ε := C ε 3 + ε 1. Pr R κ, [ϕ ε ] s 2 { E [ 1 sup ϕε t +, X t,x,κ α ϕ ε t, x + 1 [ ] 2 Tr a α D 2 ϕ ε t +, X t,x,κ D 2 ϕ ε t, x +b α Dϕ ε t +, X t,x,κ Dϕ ε t, x + θ κ + c α ϕ ε t +, X t,x,κ ϕ ε t, x ] } +I α [ϕ ε ]t, x Iκ, α [ϕ ε]t +, x r t é 2 t r t r 2 I α [ϕ ε ]t, x Iκ, α [ϕ ε]t +, x C Dϕ ε + + t D 2 ϕ ε + D 2 ϕ ε z νdz { z >κ} { z κ} z 2 νdz C ε 3 + θ κ + ε 1 z 2 νdz. { z κ} 2 t s r t s t r t t r t r s r 2 ε 3 1 t θ ϕ ε t +, X t,x,κ ϕ ε t, x s 2 θ ε 1 s ss rt t s str t r r 1t t 1 r r s t t t t r s tr r s t β = θ κ +C r C = sup α c α r r Pr s t t r s tr r s r r s 2 t Pr s t

s2 t t r s ts Pr s t t ss t s tr s r t r tr r2 t s ϕ ψ s t s 2 1 ϕ T [ϕ] g 1 1 ψ T [ψ] g 2 r s t s g 1 g 2 r r2 i = 0,, n ϕ ψt i, x e θκ+c ϕ ψ + T, + T e θκ+ct ti g 1 g 2 + r C = sup α c α r 1 t t s t P 2 t r2 t s ts s t s2st s r s t r s t r t r r t s s r t P s s t s 2 ss t t r r t2 r s t r r 2 t s r t t v i s s t3 t r s t t x 2 ö r t s t r s t t t r t s t s t s t r 2 1 2 ö r t s t t s s t r r r 1 t s t s t r r2 t s t s t t s t s 2 r r t r s t r t r r t ss t t t 1 2 ö r t s t r t s t t s t s2st r r t t s s s t 2 tr t s t s2st P s t t r r t2 r s t r t s t t s s2st t k t st t s t s2st P s max { L X v i t, x F i t, x,vi t, x, Dv i t, x, D 2 v i t, x, v i t,, v i M i v } = 0 r i = 1,, M F i t, x,r,p, γ, ψ := v i T, = g i, inf α A i {L α t, x,r, p, γ, γ + I α t, x,r,p, γ, ψ} L α t, x,r, p, γ, γ := 1 2 Tr[aα t, xγ] + b α t, x p + c α t, xr + k α t, x I α t, x,r, p, γ, ψ := ψ t, x + η α t, x,z r 1 { z 1} η α t, x,z p dνz R d M i r := min j i r j + k. t s 3 t t g i s t s t s 2 g i M i ḡ 0 r ḡ = g 1,, g M r i g i = g t s 2 g i M i ḡ 0 1 st r s r r s t r t s t s2st s r Pr s t s t s r r t t v 1,, v M v r s t 2 t s t s t A = M i=1 A i A i s r s t s ts t 0 v i v Ck 1 3 r i = 1,, M. r t r ö r t t2 r v i

t r Pr st r t s r 2 r P r P s ss s r i t v i t s s t2 s t t r 1 st st t C s t t r 2 i = 1,, M v i 1 C. Pr s t3 t t2 t r s t t x s t t r 1/2 ö r t t2 t r s t t t 2 t t r 2 s ss t 1 y R d t > 0 t t R + s t t t t r i = 1,, M ψ i t, x := λ L [ ] e At t x y 2 + Bt t + Kt t + λ 1L 2 2 + vi t, y r L = 1 2 v 1 λ a γ t r t ψ i t, x = λ L Ae At t x y 2 + B K 2 Dψ i t, x = 2λLe At t x y D 2 ψt, x = λle At t I d d. { } t ψ i inf L α t, x,ψ i, Dψ i, D 2 ψ i + I α t, x,ψ i, Dψ i α A { 1 = λl Ae At t x y 2 + B + K inf t α A 2 λleat Tr[a α t, x] +2λLe At t b α t, x x y + c α t, xψ i + k α t, x + λ L t 2 eat x + η α t, x,z y 2 x y 2 21 { z 1} η α t, x,z x y } dνz. R d 2 s K λ s t t a α K, b α K, c α K, k α K, K 1 λ K v K, η α t, x,z K1 z. t t ss r t2 t t s r r t s r s s t t r 2 α c α 0 2 s s t r A t r 1 sts t st ts C 1 C 2 C 3 C 4 s t t t ψ i inf α A { } L α t, x,ψ i, Dψ i, D 2 ψ i + I α t, x,ψ i, Dψ i A K 1 2 λle At t K x y 2 C 1 x y + C 2 B C 3 C 4. r r r B A s t r t s t { } t ψ i inf L α t, x,ψ i, Dψ i, D 2 ψ i + I α t, x,ψ i, Dψ i 0. α A

s2 t t r s ts t t r ψt, x = L 2 3 t r s t t λ λ x y 2 + λ 1 + v i t, y. ψt, x L x y + v i t, y v i t, x. t t ψ i s s r s t 2 r s r ψ i t, y v i t, y. L λbt t + λ 1 + v i t, y v i t, y. 2 r r r λ = t t 1 2 v i t, y v i t, y C t t. t r q t2 s r 2 2 s ψ i t, x := λ L [ ] e At t x y 2 Bt t Kt t λ 1L 2 2 + vi t, y. r t t t t r s t s t s2st s rr t r s t s 2 2 s 2 s tt M = 1 k = 0 r r 2 t r r r r t s w κ ε w κ ε r r s t 2 t r r s s r s t r L X u κ t, x F κ t, x,u κ t, x, Du κ t, x, D 2 u κ t, x, u κ t, = 0, [0, T R d, u κ T, = g, R d. F κ t, x,r,p, γ, ψ := inf α A {Lα t, x,r, p, γ + Iκ α t, x,r, p, γ, ψ} r sup inf 2 inf sup r L α t, x,r,p, γ := 1 2 Tr[ σ α σ αt t, xγ ] + b α t, xp + c α t, x + θ κ r, Iκ α t, x,r,p, γ, ψ:= ψ x + η α t, x,z r 1 { z 1} η α t, x,z p νdz. { z >κ}

t r Pr st r t s r 2 r P r P s 2 Pr s t r Pr s t u κ u κ, t, x u κ w κ ε + w κ ε u κ, t, x Ce θκ+c 1T t ε + ε 3 + θ κ ε 1 + θ κ + ε 1 z 2 νdz { z κ} u κ, u κ t, x u κ, w κ ε + w κ ε u κ t, x Ce θκ+c 1T t ε 13 + ε 3 + θ κ ε 1 + θ κ + ε 1 z 2 νdz. { z κ} t t t v κ t, x = e θκt t u κ t, x v κ v κ, C ε + ε 3 + θ κ ε 1 + θ κ + ε 1 z 2 νdz { z κ} v κ, v κ C ε 13 + ε 3 + θ κ ε 1 + θ κ + ε 1 z 2 νdz. { z κ} t t r s 2 t s rt r s r t r t ε s s r s t t s t r s t rst r st r t r 2 r r s s s t r t r s t t s s t2 s t t r r t r s r t 1 r t s r r t t s r t s t t r t r r 1 t s 1 t t s s t s t t t s2 t t r s ts r s s s rr r 2s s r s s t t t r r t r 1 t s r ss t P ss r ss s r t 3 t r r t t r t r r s t r s r t s s t t s t r t2 s t2 s t t r r s

t r t Pr t P 2 r t r ss r t t s t r 23 t t ss r t r t r ss t r t r t 2 t r t r s r s t s t 1 3 r t t2 r t s t t r t r r t t r r rt r r r t r rt str t 2 s t t t2 1 3 t r rt str t 2 2 t t2 r t t t r t 2 t s t q t s r r t r r ss r t t Ω, F, P t r t2 s t s r t W t 2 F = {F t, t 0} t t tr t t r t W 2 E t := E[ F t ] t t 1 t t r t r F t s r r t r t r r s s r 2 t t t2 t U : R R { } ss t str t 2 r s str t 2 C 1 r {U < } t 2 π t ω, q t r t t r t r t t r r r t r t q r π : R + Ω R + R s F r r ss 2 s r s s s t ω r t t t s r 1 t, ω ss t t t t π t := πt, s str t 2 C 1 q s t s s π t0+ > 0 π t < 0. t s t 2 e t q t t r t r ss s r t 2 r t r t q r e.. : Ω [0, T] R + s F r r ss 2 s r C 1 q R + t t t q t t2 r ss s 2 r t 2 {q t, t [0, T]} s 2 T E q T := e t q t dt. 0 t r ss r t s t r t s st t t r r s s t t r r t t r ss s s r s r rt r t r t r 3 r 3

t r t Pr t P 2 r t r ss r t r 23 t t t r s t r ss r t t t tr s r r t t r ss r t tr s r st t r Y 2 t 2 s dy t = µ t dt + γ t dw t, r µ γ r t F t r ss s γ > 0 ss t t t r s s r [0, T] r t r ss r t s t t t 0 t r r Y t t s t r t t t r ss s t t T Y T κ r s Y T < κ s t t t t t t ss r s t 1 t q t s κ 1 2 t tr s t α t t2 r t r ss t t r ss tr t t t T s S T := α1 {YT κ}. r ss s r t s r t2 2 t 2 r ss r t s r tr t s tr t t s t t r t t t r [0, T] ss t t t r t s r t ss t s r t ss r tr t t r2 t t t r t r ss tr t t t t s 2 S t := E Q t [S T] = αq t [Y T κ], r Q s r t2 s r q t t P t s q t rt s r E Q t Q t t t t 1 t t r t2 F t r t r s t r s t r s tr s r 2 rr r t r t r s t r t s r t ss t t t s r s E max 1 r ss q t 2 t r s t r s S 0 E max t r s t t 1t tr st t t st r t 1 t r r t r s r t t r ss s s t 2 t ss t2 t s t r s t ss r t t r t s t 3 t r ss s t r s t t r r t s t t r t 1 s s r t t r r rs t t r t t t t t r t r s t ss r t r r2 t t2 t r s t r t 2 t r t q t t t t s r t s π t q t r s t r ss s r 2 e t q t t t t r t 1t r t2 s t r t t r t t t t r [0, T] s 2 T 0 T π t q t dt S T e t q t dt. 0

r r t r r ss r t t t t r t t t2 t 2 tr s t s 2 t r ss s r t t {θ t, t 0} F t r ss s S t r r r2 t 0 θ t t s t r tr ts r ss s 2 t 2 t t t r t s t t t t 2 tr t ss r t s T x + θ t ds t, 0 r x s t s t t t t 2 t r t ts r ss s tr ts 2 t t r t t r t 2 rts t t t t t r t t T s X θ T + B q T r T XT θ := x + θ t ds t, 0 B q T := T 0 T π t q t S t e t q t dt E q t ds t, 0 t E q t := e u q u du, r 0 t [0, T]. ss t t t r s t tr t t 2 str t t t t r s { [ ] } V 1 := sup E U XT θ + B q T : θ A, q Q, r A s t t F t r ss s s t t t r ss X s r 2 rt Q s t t t F t r ss s t t t t st st t r s t r s t t S t t t r t 1 r ss XT θ + Bq T s A s r s s t s t t t 1 3 t t r s t t q θ r t 2 t s r s s 2 s 2 t 3 s ss 2 t r s t t q θ rt 1 3 t t r s t t q r s t r t q 1 2 t rst r r t π t q q1 t = S t e t q q1 t. s t ss t s π t. e t. t 2 t t q 1 t s ss t t t r t t r t s 2 r str t t ss t t t ss r t s t r t t r t t r r r t t r ss s

t r t Pr t P 2 r t r ss r t 1t t r t t t tr str t 2 2 s [ ] sup E U X x,θ Eq1 T + B q1 θ r B q := T 0 π t q t S t e t q t dt. t r s t t 1t t r t t s t s 2 T x + θ 1 t 0 E q1 t ds t + B q1 = U 1 y 1dQ dp r t r t r y 1 s 2 [ E Q U 1 y 1dQ ] [ = x + E Q B q1]. dp t s s t r s t t 1t s r t tr t t2 t 2 s t ts t r t 2 t r s t r t q 1 s s r t t t t s ss s s s t t s t t t ss r t s t r t r t r ss s t ss r t ss s r t t 1t r t2 t t t r r s r r t t 3 s r t s r t s 1 t s r t r s ss t2 t tr t r ss s s s t st r t 1 t s2st r α s t t t 1 t t t r r t r ss ss t t t r s r 3 s t t s r ts t s [ T V 0 := sup E U π t q t dt α ] E q T Emax + q. Q 0 r E max s t r s t r t r t t s t t r t r 3 t t t r t π t q q 0 t = α e t q q 0 t E Q0 t [1 R+ ] E q0 t T E max r dq 0 dp = E U T 0 π tq 0 t [ U T 0 π tq 0 t dt α dt α + E q0 t T E max + E q0 t T E max ].

r r r t r r ss r t t r t r r t t s t t t r t r ss s r t ts ss s ] S t := αe Q0 t [1 R+ E q0 t T E max, t 1 t t t t 1 t r t s r Q 0 2 r r t t2 s s t2 r t2 s r Q 0 s t s r s tr s r t t s r t st st s t t r t r t s t t r t 3 s r st r t t π t q e t qs t π t q q0 = e t q q0 S t. t 2 t t t r t 2 2 ss t t t r s t t r t r t t2 t s t s2st q t s s st tr r 1 t r s r s r 2 s t r r t t r t r r r ss s t rs t r s t t 1t t s s s t r s t q t 2 r t r s t s t t s ss t 1 t 2 rs t t r ss s 2 E max t t r t t s t 1 s r s r ss s t E max t t s t s s t s ss s s s t t t t 1 t s t tr t t r t r ss s s s q t 2 2 t t r r ss s t E max s t r t t r r s ss s r ts r ss s s s 2 ts r t t r rs ss r t r s t t 1t r t2 r ss s 2 r s t s r t t r s r t r s 2 t t t2 t t r r r t s t r s2st q t r t 1t r t2 s t r2 r t rs s t s t t tt r t 3 t r r t s r r r t r r ss r t t s s t s r t s r r tt r t r s s t t t s s t r t s r t s s t tr t t2 t t r t 2 t 2

t r t Pr t P 2 r t r ss r t t s s t t 2 ss t t t st t r Y s t 2 t r t 2 t r dy q t = µ t + βe t q t dt + γ t dw t r β > 0 s t t r r ss S t r ss s s s t r s s t 2 t r tr t r S q t = αq q [ t Y q T κ], s s t 2 t r t 2 q q t rt s r Q q s 2 dq q T = exp λ t q t dw t 1 T λ t q t 2 dt dp 0 2 0 FT r λ : R + Ω R + R s F r r ss 2 s r 2 s t r r ss S r 2 ds q t S q t = σ q t dw t + λ t q t dt, t < T, r t t t2 t σ q t s r r ss 2 s r s t tr r ss {q s,0 s T } s t r s s t t t r ss t 2 s 2 X x,θ T := x + T 0 T θ t ds q t B q T := π t q t dt S q T 0 T 0 e t q t dt r r ss t t r s r t s s s t r str t r tt t t t s r tt r t t t r s r λ t q s t q r 2 t 0. t t r tt r s [ ] V 2 0 := sup E U X x,θ T + B q T. q Q, θ A Pr s t ss t t t r t s t t q r s tr s r Q t t r t 2 s t t t t2 t t r r U t 2 s sup E Q [ BT] q. q Q r r q 2 s t r t s t t t st t str t 2 θ 2 s r t r 3 2 X x,θ2 T + B q2 T = U 1 y 2dQ [ ], x + E Q B q2 T = E [U Q 1 y 2dQ ]. dp dp

r r r t r r ss r t Pr rst 1 s r t str t 2 q t r t s t t rt 1 3 t t r s t t θ r r 2 t ss t2 t X x,θq T + B q T = U 1 y q dq, dp r t r t r y q s 2 [ E Q U 1 y q dq ] = x + E Q [ B q dp T]. s r s t r t sup E [U U 1 y q dq ]. q. 0 dp t t t U U 1 s r s t s t2 dq dp > 0 r s t inf {y q : q 0}. U 1 s s r s rts t r t { sup E Q [ } BT] q : q Q. 2 t t str t 2 q 2 t t st t 2 s r t r 3 2 r r t s rt r t r t r 3 t t t r t 2 q 2 s 3 t s ss t t r s 2 ss t t π t q = πt, q t e t q = et, q t λ t q = λt r s t r st t s π, e : R + R + R C 0,1 R + R + λ : R + R + R C 0 R + dy q t = µt, Y q t + βet, q tdt + γt, Y q t dw t, r s t s t r st t s µ, γ : R + R R st t r E s 2 t 2 s de q t = et, q t dt r r s t t r ss s t 2 2 rs t r r r s 2 [ T ] V 2 t, e,y := sup E Q t,e,y πt, q t dt αe q T 1 {Y q T q Q >0}. V 2 s s t 2 r r q t t 0 = V 2 t + max q 0 + µ λγv y 2 + 1 2 γ2 V yy 2 { πt, q + et, qv e 2 + βet, qv y 2 },

t r t Pr t P 2 r t r ss r t t t r t t t r t V 2 T, e, y = αe1 {y>0}. r t t ss t t t t V 2 s s t t t str t 2 s 2 π q t, q 2 = e q t, q 2 V e 2 + βv y 2 t, e,y. 2 t t t t V 2 1 t t t V 2 e t, E t, Y t = S t. π t, q 2 t q = e t, q 2 t q S t V 2 y t, E q2 t, Y q2 t s t s r t t V 2 s r s y r t r s 1 r ss t t s r t ss t π e t t q 2 < q 1. t r r s t t t r t r t r s r ss s r s s t st t 1t r t t2 r t r s t 2 ts t r t t t r s t r s t t r t t r s t ss r t s 2 s t r t r s r t r ss s r s t s s t t r rt ss t s t r t s r r s t t µ t s t s t r st γ s st t λq = λ 0 eq = e 1 q + e 0 r λ 0 e 1 e 0 r t st ts ss t t π s C 0,1 [0, T] R + str t 2 q V 2 e π t, 0+ > 0 π q t, < 0. q 1 sts s tr t r s s t C 1,1,2 [0, T R + R t t r 1 sts t r t str t 2 s t s 2 Pr 1 st V e s t t t t t V s e Pr s t r s r t st ss rt t r t t t 2 V s t q s s t2 s t r r 2 t ss t t r V C 1,1,2 [0, T R + R s t 2 r r r t q 2 t r s t str t 2

r r r t r r ss r t r r ss t t s tr s r s r t r s r t r s r r ss s t 2 t ss s t r t r dq q dp FT T = exp λq t dw t 1 T λq t 2 dt. 0 2 0 rt 1 3 t t r s t t θ s t r Pr s t s st t s t 1t r s t r t r s r t ] sup E [U U y 1 q dqq q Q dp r y q s 2 [ ] E Qq U 1 y q dqq dp = x + E Qq [ B q T]. s ss t t t r r s t r t r r 2 1 t t t2 t Ux := e ηx, x R. U U 1 y = y/η r s t ] inf [y E q dqq q. 0 dp = inf q. 0 yq. 2 t t str t s t r s t s [ x + E Qq B q ] T = 1 [ y q dq q ] η EQq ln η dp = 1 { [ ]} y q dq q ln + E Qq ln, η η dp s t t t t 3 t r s q t t [ sup E Qq B q T + 1 ] dq q q Q η ln dp [ T = sup E Qq q Q 0 π + λ2 2η t, q t dt S q T T 0 ] e t q t dt. t t r t t t 3 t r t r s t t r t 2 t r t r t r t r s t s t t r t r s r r ss s t 2 t r ss s t r t r s t 3 t r t r s 3 2 t tr 2 t r s tr s r t r s t t t st t st s r

t r t Pr t P 2 r t r ss r t r s t r t r s st r st st tr r t r s ss 2 s r t r s tr t 2 rs [ T V 3 t, e,y := sup E Qq t,e,y q Q t r t tr st t 2 s s 2 ] π + λ2 t, q t dt E qt 2η α1 {Y qt 0}, dy q t de q t = µt, Y q t + βet, q t γt, Y q t λt, q tdt + γt, Y q t dw q t, = et, q t dt, W q s r t r Q q µ, e,γ, λ r s 2 ss r ts t s t t V 3 s s t 2 r r q t 0 = V 3 t + max q R + + µv y 3 + 1 2 γ2 V yy 3 { πt, q + 1 2η λt, q2 + et, qv e 3 + βv y 3 γλt, qv y 3 } t t r t t t r t V 3 T, e, y = αe1 {y>0}. t r s t t V 3 t t r t 2 s t s t 1 3 r t q t r t t ss t t r r 1 rs V 3 s r r t t rst r r t s π q q3 + 1 η λ λ q q3 + e q q3 V 3 e + βv y 3 γ λ q q3 V y 3 = 0, r t 2 t r s t t t, e, y s tt r s t2 s r 1 t t t t t s r r t t t r t r ss s tr t s s r t ss r t s 2 S t = V 3 e t, E t, Y t. t s t t t t r t 2 t r s 2 π q t, q3 = e q t, q3 + λ q t, q3 S t βv y 3 t, Y t, E t γv y 3 t, Y t, E t 1 η λt, q3. tt r 1 r ss s t r r r t r r t t r s s q t r 1 r ts tr st t t r s s r t

r r s ts r s r r ss s t t 2 t r ss s t r r t r t r t t q 3 s s r t q 1 r t t t t r t 2 t s r t 2 π q t, q1 = e q t, q1 S t. s s t t t t t t r t r e q t, q3 βv y 3 t, Y t, E t + λ q t, q3 γv 3 y t, Y t, E t 1 η λt, q3 s s t r s r t s rt t t t s s s s t t t s t r s t t t r r r 2 t t s t t ss r t 2 t t r s s s t r t r ts tr t t2 s t s r r t t t2 r r r ss s t r s t s t t s t t t ss r t s t t t r ss s t r r 2 t 2 s t r s ts r ss s t s r s ts r t 2 s ts t2 t t t r t 1t r s s t t r s t ts s tr t r t r s t t µ t s t s t r st γ s st t eq = e 1 q + e 0 λq = λ 1 q + λ 0 π t q := π t q + λq2 s t r st str t 2 q t V 3 e π t0 > 0 π t < 0. 1 sts s tr t r s s t C 1,1,2 [0, T R + R t t r 1 sts t r t str t 2 s t s 2 2η Pr r s t s r t s t r r r r s ts r q r t 1 t r r s ts s t rst t r t t str t 2 π q t, q3 = e q t, q3 + λ q t, q3 S t βv y 3 t, Y t, E t γv y 3 t, Y t, E t 1 η λt, q3

t r t Pr t P 2 r t r ss r t r r s 2 1 r q 3 > q 1 s r t 2 Pr r q t V t + µv y + 1 2 γ2 V yy + max q 0 θq, V e, V y = 0 r θ s 2 θq, V e, V y = πt, q + 1 2η λt, q2 + et, qv 3 e + βv 3 y γλt, qv 3 y, t t t r r2 t r s r s s r V T, e, y = αe1 {y 0}. πq = q1 q, eq = λq = q, β = 1, α = 1. t t t t s 1 s t s s t ss t r V e = S t t r r r q 1 q 2 q 3 t s t t θq, V e, V y = 1 1 q 2 + 1 + V e + 1 γv y q. 2η 1t ss t t η > 1 2 s t t t t θ s str t 2 t q r t s r t rst r r t t t t t r t 2 s 2 q 3 = 1 2ρ 1 + V e + 1 γv y t ρ = 1 1 2η max q 0 θq,v e, V y = 1 4ρ 1 + V e + 1 γv y 2. t 2 Pr r q t r s t V t + µv y + 1 2 γ2 V yy + 1 4ρ 1 + V e + 1 γv y 2 = 0. t t t r r t t r t q 1 t str t 2 r tt s π q 3 = e q 3 S t τe, y, r t rr t t r τe, y s 2 τe, y = 2η1 γ 2η 1 V y + 1 2η 1 1 + V e. t r r t t s t 1 t 1 s r t rs τe, y < 0 r q t 2 t r s t t str t 2 q 3 > q 1

r r s ts r s rst st s t s t t t [0, L e ] [ L y, L y ] r t e, y s s r t 3 t t t 2 t r {e i, y j } i,j t r t s t s t r t s r r2 t s t t t t st t k = k t r k = 0,, n := T t s t t s r t t r t Vij tn = e i 1 {yj 0} t2 s t q t s t s r t r s r r t r t s t2 s t s tt s r t 3 t s r s t t st s t s t t r s s t s t s rt t s s t t t st [t n, t n+1 ] s V t + 1 2 γ2 V yy = 0. t s t t t t rt t t r ts V t + µv y + 1 4ρ 1 + V e + 1 γv y 2 = 0. t s rt t rt s r 1 t s tr 2 ss s s str t s r r t s t s2st t q t s V t + µv y + 1 4ρ 1 + V e + 1 γv y ϕ = 0, ϕ = 1 + V e + 1 γv y r s s r s t r t t r s s s s s t t r t t s s s s t st 2 r tr t t t r t2 t r s r t 1 t2 s t s r t 3 t s ts r r t rs µ = 0.1 γ = 0.65 η = 5 t t s T = 10 r t r s ts

t r t Pr t P 2 r t r ss r t r r r2 t V 3 T = 10, e,y r t = 0.2 s t t 2 r r q t V 3 e, y t t