PATHWISE UNIQUENESS FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH SINGULAR DRIFT AND NONCONSTANT DIFFUSION. Katharina von der Lühe

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1 Dieraion PATHWISE UNIQUENESS FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH SINGULAR DRIFT AND NONCONSTANT DIFFUSION Kaharina on der Lühe Fakulä für Mahemaik Unieriä Bielefeld Augu 26

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3 Conen Conen. Inroducion.. Brief urey of he problem in finie dimenion Mehod of E. Fedrizzi and F. Flandoli Aim and progre of he hei Fuure direcion Srucure Preliminarie and main reul 3. Tranformaion of he SDE Iô formula for mixed-norm Sobole funcion Tranformaion of he SDE Some helpful lemma Krylo-ype eimae for condiional expecaion Uniform exponenial eimae for he ranformed diffuion Conergence of he ranformed drif Bounded fir and econd momen Pahwie uniuene On mall ineral Exenion from mall o arbirarily large ineral A. Appendix 55 A.. Approximaion by coninuou funcion A.2. Mean-alue ineualiy for weakly differeniable funcion and Sobole embedding A.3. Krylo-ype eimae Reference 89

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5 INTRODUCTION. Inroducion.. Brief urey of he problem in finie dimenion We conider he following ordinary ochaic differenial euaion SDE): X = x + b, X ) d + σ, X ) dw, [, T ], where x, b, σ are meaurable funcion from [, T ] o, repeciely m, and W i an m-dimenional andard Wiener proce. I i well known ha we hae rong exience and uniuene for hi euaion under Lipchiz coninuiy of he coefficien, which wa hown by K. Iô, ee [Iô46], who fir rigorouly deeloped he heory of ochaic inegraion. Since Lipchiz coninuiy i a raher rong aumpion and hi kind of SDE arie in many eing ha do no necearily proide Lipchiz coninuou coefficien, e.g. ineracing paricle, i i naural o ak if i i alo poible o ge a uniue rong oluion under weaker properie. I urn ou ha hi hold under much more general aumpion on he drif erm b, neiher coninuiy nor he abence of ingulariie i neceary. So wo ueion hae o be anwered, namely fir wheher here i a rong or a lea a weak oluion and econd if here i ome oluion wheher i i uniue a lea in ome ene. A grea ool in hi heory wa found by T. Yamada and S. Waanabe. They proed, ha exience of a weak oluion and pahwie uniuene imply he exience of a uniue rong oluion, ee [YW7]. There are many work which ineigae he problem of exience or uniuene under weaker aumpion han Lipchiz coninuiy. Beide [KR5], [FF] and [Zha] on which we will hae a cloer look laer, we wan o menion here ome of hee reul. Srong exience and uniuene could be obained for example under local weak monooniciy and weak coerciiy condiion on he coefficien. A proof can be found in he book Sochaic Parial Differenial Euaion: An Inroducion of W. Liu and M. Röckner [LR5], which i baed on [Kry99]. Furhermore, in heir work A udy of a cla of ochaic differenial euaion wih non-lipchizian coefficien, [FZ5], S. Fang and T. Zhang relaxed he Lipchizian condiion mainly by a logarihmic facor. Thi mean ha he Lipchiz conan i muliplied wih a funcion, depending on he diance, wih pecial properie which are ypically fulfilled by log/), log/) log log/) and o on. Moreoer, A. Yu. Vereenniko proed rong exience and uniuene for bounded meaurable coefficien if he diffuion marix i nondegeneraed, coninuou and Lipchiz coninuou in x, ee [Ver78]. In [GM] I. Gyöngy and T. Marínez relaxed hi o R + ) and b almo eerywhere bounded by a conan plu ome nonnegaie funcion in L d+ R + ). In heir work Srong oluion of ochaic euaion wih ingular ime dependen drif [KR5]) N. Krylo and M. Röckner proed he exience of a uniue rong oluion in he whie noie cae, i.e. he diffuion coefficien σ i he uni marix. The drif coeffi- locally unbounded drif, namely b L 2d+) loc

6 INTRODUCTION cien, defined on an open e Q +, i uppoed o fulfill n pn b, x) pn dx d < R {x :,x) Q n } for ome p n 2, n > 2 uch ha d p n + 2 n < and a euence Q n ) n of bounded open ube of Q wih Q n Q n+ and n Qn = Q. In 2 E. Fedrizzi and F. Flandoli, [FF], inroduced a new mehod o proe he pahwie uniuene under uch condiion. The aim of hi hei i o exend heir reul o nonconan diffuion. Therefore, we will hae a look on hi mehod in deail in he nex ecion. Alo if he diffuion i no conan i i poible o ge exience and uniuene reul under imilar condiion on he drif. The mo general reul can be found in he work of X. Zhang Sochaic homeomorphim flow of SDE wih ingular drif and Sobole diffuion coefficien [Zha], repeciely [Zha5] for he cae p =. There, he drif i in L loc R +, L p )) for ome p, >, fulfilling d p + 2 <. ) The diffuion coefficien i uniformly coninuou in x, locally uniformly wih repec o, nondegeneraed, bounded and he gradien i alo in L loc R +, L p )). The idea of he proof i o remoe he drif by he o-called Zonkin ranformaion, ee [Zo74], and ue known reul for SDE wih zero drif. Thi ranformaion i baed on he oluion u o he euaion d u + b i xi u + d d σσ ) ij x 2 2 i x j u =, ut, x) = x. i= i= j= One difficuly i o how ha hi i a diffeomorphim o ge a one-o-one correpondence beween he oluion X for he original SDE and he oluion u, X ) for he ranformed euaion. The mehod of E. Fedrizzi and F. Flandoli o proe pahwie uniuene for conan diffuion i more inuiie. A cenral poin of hi work i o exend hi proof and ome reul of [FF] o nonconan diffuion coefficien. Therefore, he following ecion i deoed o preen heir mehod in ome deail..2. Mehod of E. Fedrizzi and F. Flandoli Le X ), X 2) be wo rong oluion o he euaion X = x + b, X ) d + W, [, T ]. 2

7 INTRODUCTION For b L, T ), L p )) here exi a uniue oluion, ee [Kry], o he euaion u + u = b on [, T ], ut, x) =. 2) 2 Denoe hi oluion by U b and apply Iô formula o U b, X i) ). Since U b i a oluion o he aboe euaion we ge he following expreion for he drif erm: b, X i) ) d = U b, x) U b, X i) ) + x U b, X i) )b, X i) ) d + x U b, X i) ) dw. Now, he SDE may be rewrien by replacing he drif: X i) = x + U b, x) U b, X i) + ) x U b, X i) )b, X i) ) d + x U b, X i) ) + I dw. 3) The adanage of hi reformulaion i ha he new drif erm x U b b i in ome way more regular han before. The oluion U b of 2) i an elemen of he Sobole pace W,, T ), W 2,p )) and herefore ha nice properie, e.g. x U b i Hölder coninuou. If we define T b), x) := x U b, x)b, x),, x) [, T ], and ake a oluion U T b) of he euaion u + u = T b) on [, T ], ut, x) =, 2 an applicaion of Iô formula for U T b), yield an expreion for he ranformed drif erm: T b), X i) ) d = U T b), x) U T b), X i) ) + x U T b), X i) )b, X i) ) d + x U T b), X i) ) dw. 3

8 INTRODUCTION By replacing hi erm in euaion 3), we ge X i) = x + U b, x) + U T b), x) U b, X i) ) U T b), X i) + + x U T b), X i) x U T b), X i) )b, X i) ) d ) + x U b, X i) ) + I dw and by ieraion and he conenion T k+ b) = x U T k b) b, T b) = b, X i) + n k= U T k b), X i) ) } {{ } =:Y i,n) Then one can proe ha [ ] E X ) X 2) CE + E ] [e An) 2 e An) E = x + + n U T b), x) + k k= n k= x U T b), X i) k ) + I } {{ } =:σ n),x i) ) ) T n+ b), X i) ) d dw. X ) X 2) T n+ b), X ) ) T n+ b), X 2) ) d } {{ } I Y,n) Y 2,n), σ n), X ) ) σ n), X 2) ) ) dw } {{ } I 2 2, 4) where A n) := σ n), X ) Y,n) ) 2 ) σ n), X 2) Y 2,n) 2 {Y,n) Y 2,n) } d. By proing ha E[e An) ] i uniformly bounded in n, ha I conerge o for n and ha I 2 i a maringale, one ge pahwie uniuene. 4

9 INTRODUCTION.3. Aim and progre of he hei The aim of hi hei i o generalize he mehod of E. Fedrizzi and F. Flandoli o ime and pace dependen diffuion. Inead of one conider euaion of he form u + u = f 2 u + 2 d d σσ ) ij x 2 i x j u = f i= j= o ranform he SDE. Along wih ome oher echnical iue, he nonconan diffuion lead o addiional erm in he ochaic inegral of he reformulaed SDE. Neerhele he core of he proof remain he ame a in [FF] where we hae o handle he fac ha a oluion o he SDE i in general no a Brownian moion. Thi i he cae if σ = I and i wa a crucial poin in he proof of E. Fedrizzi and F. Flandoli. Thi propery enabled hem o ue Girano formula and exponenial eimae for Brownian moion which are no applicable in our generalizaion. A a compenaion, we uccefully ue Krylo eimae. We herefore proe a erion of Lemma 5. from [Kry86] for differen inegrabiliy in ime and pace aed a Lemma 3. and proed in Secion A.3. The price we hae o pay i ha we hae o aume p, > 2d + ). Since he eimae are baed on oluion o PDE i hould be poible o exend i, maybe up o he cae p, fulfilling condiion ), bu in hi hei we reric o hee onger aumpion on p and. Beide he ordinary Krylo-ype eimae we alo need imilar one on condiional expecaion, which we formulae and proe in Secion 4.. We only hae o aume ha he diffuion coefficien i bounded, nondegeneraed, he drif i in L, T ), L p )) and P T b, X ) d < =. Up o our knowledge hi ha no been done ye under hee general aumpion. For bounded b a erion can be found in [Kry9] and for σ uniformly coninuou in x, local uniformly coninuou wih repec o in [Zha]. Only for an eimae on he linear combinaion of wo oluion a in Propoiion 4.4 coninuiy of he diffuion erm i reuired. For impliciy we will ae our reul under global aumpion, bu here are no difficulie o exend i by localizaion echniue, e.g. in he ame way a in [Zha]. The reul of X. Zhang, [Zha], i cloe o our. The aumpion are more general wih repec o he inegrabiliy of b and x σ ince we hae o aume p, > 2d + ), which come from our Krylo eimae, bu could be poible exended o p, fulfilling ), which alo X. Zhang reuire. Furhermore, he aumpion on drif and diffuion coefficien are he ame excep he coninuiy condiion on σ. Inead of reuiring uniform 5

10 INTRODUCTION coninuiy in x, locally uniformly wih repec o, which gie direcly ee Remark.4 of [KR5]) he olabiliy of euaion of he form u + 2 d d σσ ) ij x 2 i x j u + i= j= d b i xi u = f, i= we aume σ o be coninuou and uch ha here exi a oluion o he euaion u + 2 d d σσ ) ij x 2 i x j u = f, i= j= ee Aumpion 2.2 and 2.3. Beide hee imilariie a far a aumpion are concerned, our mehod of proof i compleely differen and more probabiliic and alo much impler a lea from our poin of iew. We are able o proe pahwie uniuene direcly by eimaing E[ X ) X 2) ] in a imilar way a in 4)..4. Fuure direcion One ep in furher reearch could be an opimizaion of he proof mehod. Maybe i i poible o aoid he exponenial eimae on he ranformed diffuion by opping ime argumen. A echniue in hi direcion wa recenly deeloped by G. Da Prao, F. Flandoli, M. Röckner and A. Yu. Vereenniko in [DPFRV6]. Anoher ery inereing iue i he generalizaion o infinie dimenion. Thi wa a rong moiaion for hi hei, ince i eem achieable and would be a grea ep forward in he heory of ochaic parial differenial euaion. In conra o he finie dimenional cae we do no hae ellipic regulariy for parial differenial euaion on Hilber pace. To aoid difficulie i could be a good approach o ar wih exponenial inegrable coefficien..5. Srucure The econd chaper i deoed o baic definiion, epecially he inoled pace are inroduced. Then we ae our aumpion on he coefficien of he ochaic differenial euaion and he reul abou pahwie uniuene. The hird chaper deal wih he ranformaion of he SDE which i neceary o proe pahwie uniuene. Since he ranformaion i baed on Iô formula, we fir how ha under our aumpion i i applicable for funcion in he mixed norm Sobole pace W,, T ), W 2,p )) before we explain he ranformaion in deail. The fourh chaper ae all neceary ool o proe pahwie uniuene on a mall ineral. Fir we gie ome ueful fac of he inoled funcion and he relaion beween a oluion o he original SDE and he ranformed euaion. Then we proe he Krylo eimae for condiional expecaion, which hen gie u a uniform exponenial eimae for he ranformed diffuion. Afer hi we how he conergence of he difference beween he ranformed drif erm of wo oluion and in he end, we proe ha 6

11 INTRODUCTION under our aumpion oluion of 5) hae finie fir and econd momen. Chaper 5 i deoed o he proof of pahwie uniuene. Since we aed ome neceary ool only up o ome poible mall T, we fir proe i on [, T ], before we how ha i i exendable o arbirarily large ineral. Therefore, in he end we ge pahwie uniuene on he whole ineral of he original SDE. In he appendix we li ome mall lemma which we need in he proof before. For he ake of compleene hey are all gien wih proof alo if ome of hem are eay and ju lile generalizaion of well known reul. We ar wih ome fac abou our mixed norm pace, epecially approximaion by mooh funcion. I eem ha hi ha no been done ye rigorouly and herefore we proe hem in deail. Then we ae an eay mean-alue ineualiy and proe a Sobole embedding. In he end we gie ome Krylo eimae. The proof for he ineualiie of expecaion are ery imilar o he one for condiional expecaion. Bu ince we need ome of hee reul for he proof in he condiional cae, we alo gie here he proof in deail. 7

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13 Acknowledgmen Fir and foremo, I hank Prof. Dr. Michael Röckner for uperiing hi work, for hi coninuou uppor and adice. Furhermore, I am graeful o Nora Müller and Daniel Alemeier for giing me ome aluable feedback on hi hei a well a o Sarah Gerz for her adice on grammar and phraing. I would alo like o hank my friend and family for heir uppor in he la few year. Financial uppor of he DFG hrough he IRTG 32 Sochaic and Real World Model and he CRC 7 Specral Srucure and Topological Mehod in Mahemaic i graefully acknowledged, a well a of he Bielefeld Young Reearcher Fund hrough he docorae compleion cholarhip.

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15 2. Preliminarie and main reul 2 PRELIMINARIES AND MAIN RESULT In hi chaper, baic concep for hi hei a mixed-norm Sobole pace are inroduced. In paricular, we formulae he reuired aumpion on he coefficien of he SDE and preen our reul abou pahwie uniuene. Beide, here will be ome hor noaion inroduced which will be ued hroughou he hei. Definiion 2.. For p,, ) we define f L p T ) := T f, x) p dx p d, where denoe he Hilber-Schmid norm. We define L pt ) o be he pace of meaurable funcion f : [, T ] repeciely m ) uch ha f L p T ) <. Furhermore { } W,p,2 T ) := f : [, T ] f, f, x f, xf 2 L pt ), where, x, 2 x denoe weak deriaie wih repec o ime, repeciely pace. The aociaed norm i gien by f W,2,p T ) := f L p T ) + f L p T ) + x f L p T ) + 2 xf L p T ). If we omi he T, we mean ha we ake R + inead of [, T ]. We conider he SDE X = x + b, X ) d + σ, X ) dw, [, T ], 5) where W i an m-dimenional andard Wiener proce on a filered probabiliy pace Ω, F ), P), wih F ) fulfilling he uual condiion, x and b : [, T ], σ : [, T ] m are meaurable funcion wih he following properie: Aumpion 2.2. For ome p, > 2d + ) we hae c) b L pt ), c2) σ i coninuou in, x), c3) σ i nondegeneraed, i.e. here exi a conan c σ > uch ha σσ, x)ξ, ξ c σ Iξ, ξ ξ, x) [, T ], where σ denoe he ranpoed marix of σ,

16 2 PRELIMINARIES AND MAIN RESULT c4) σ i bounded by a conan c σ, c5) x σ L pt ). Aumpion 2.3. Le σ be uch ha for all f L pt ) here i a oluion u W,2,p T ) o he parial differenial euaion u + 2 d d σσ ) ij x 2 i x j u = f on [, T ], ut, x) =, i= j= uch ha u L p T ) C f L p T ), where C i independen of f and increaing in T. Remark 2.4. Aumpion 2.3 eem o be maie rericie, bu in fac i i proen for a large cla of funcion. If σ i independen of he reul can be found in [Kry]. Baed on hi one can proe ha i hold alo for σ uniformly coninuou in x, uniformly coninuou wih repec o, ee Remark.4 in [KR5]. If p and σ aifie a anihing mean ocillaion condiion he aumpion i alo fulfilled, ee [Kry7]. Definiion 2.5 weak/rong oluion). A weak oluion for euaion 5) i a pair X, W ) on a filered probabiliy pace Ω, F ), P) uch ha X i F -adaped, W i an F -Brownian moion and X, W ) ole euaion 5). Gien a Brownian moion W on a probabiliy pace, a rong oluion for euaion 5) i a coninuou proce which i adaped o he filraion generaed by W and ole euaion 5). Definiion 2.6 Pahwie Uniuene). We ay ha pahwie uniuene hold for euaion 5) if for wo weak oluion X, W ), X, W ), defined on he ame probabiliy pace, we hae ha X = X and W = W imply P X = X ) [, T ] =. Theorem 2.7 Main reul). Under Aumpion 2.2 and 2.3, we hae pahwie uniuene in he e of coninuou procee which fulfill P T b, X ) d < =. 6) Noaion 2.8. For wo oluion X ), X 2) o SDE 5), defined on he ame probabiliy pace, wih iniial alue x ), x 2) and he ame Brownian moion, we define for all 2

17 2 PRELIMINARIES AND MAIN RESULT λ [, ], R > and [, T ] X λ := λx ) + λ)x 2), x λ := λx ) + λ)x 2), b λ, X ), X 2) ) := λb, X ) ) + λ)b, X 2) ), σ λ, X ), X 2) ) := λσ, X ) ) + λ)σ, X 2) ), τ λ R := inf { : τ R := inf { : X ) X λ > R }, > R or X 2) } > R, and a λ := 2 ) λσ, X ) ) + λ)σ, X 2) ) λσ, X ) ) + λ)σ, X 2) )). In he following, wheneer we peak of wo oluion, we mean wo weak oluion defined on he ame probabiliy pace wih he ame Brownian moion. Furhermore by C > we alway denoe ariou finie conan, where we ofen indicae he dependence of parameer by wriing hem in bracke. 3

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19 3. Tranformaion of he SDE 3 TRANSFORMATION OF THE SDE Thi chaper coni of a deailed udy of our before menioned ranformaion of he SDE. To hi end, an appropriae erion of Iô formula for funcion in W,p,2 T ) will be eablihed in he fir ecion. The econd coer he formulaion and udy of he ranformaion. 3.. Iô formula for mixed-norm Sobole funcion We formulae a erion of Iô formula for funcion in W,2,p T ) in Propoiion 3.3. The proof relie on wo auxiliarie a Krylo-ype eimae and a Sobole embedding. Thoe wo are formulaed a Lemma 3. and 3.2 wih boh proof deferred o he appendix. Lemma 3.. Le c), c3), c4) of Aumpion 2.2 be fulfilled and X be a oluion o 5) uch ha condiion 6) hold. Then we hae for eery, r d + and any nonnegaie meaurable funcion f : [, T ] R E T f, X ) d CT, d, p,,, r, c σ, c σ, b L p T )) f L r T ). Lemma 3.2. For all u W,2,p T ), here exi a erion of u uch ha up,x) [,T ] u, x) C u W,2,p T ), where C i independen of u. In paricular hi erion i coninuou. Propoiion 3.3. Iô formula) Le u W,2,p T ), c), c3), c4) of Aumpion 2.2 be fulfilled and X a oluion o 5) uch ha condiion 6) hold. Then here exi a erion of u uch ha for T we hae u, X ) = u, X ) + ur, X r ) dr + x ur, X r )br, X r ) dr + x ur, X r )σr, X r ) dw r + 2 d d σσ r, X r )) ij x 2 i x j ur, X r ) dr i= j= P-almo urely. Proof. By Lemma A.5 here exi a euence u n ) n in C, T ) ) which conerge 5

20 3 TRANSFORMATION OF THE SDE o u in W,2,p T ). Then Iô formula, ee e.g. [KS9] Chaper 3 Theorem 3.6, yield u n, X ) = u n, X ) + u n r, X r ) dr + x u n r, X r )br, X r ) dr + x u n r, X r )σr, X r ) dw r + 2 d d σσ r, X r )) ij x 2 i x j u n r, X r ) dr i= j= P-almo urely for eery n N. From Lemma 3.2 we know ha here exi a erion of u uch ha u, X ) u n, X ) C u u n W,2,p T ). Therefore, u n, X ) conerge P-almo urely o u, X ) a n. Wih Lemma 3. we obain E ur, X r ) dr u n r, X r ) dr E ur, X r ) u n r, X r ) dr C u u n L p T ). And uing Hölder ineualiy wice lead o E E x ur, X r )br, X r ) dr x u n r, X r )br, X r ) dr x ur, X r )br, X r ) x u n r, X r )br, X r ) dr E br, X r ) x ur, X r ) x u n r, X r ) dr E br, X r ) 2 dr 2 E x ur, X r ) x u n r, X r ) 2 dr 2. 6

21 3 TRANSFORMATION OF THE SDE One more applicaion of Lemma 3. yield hen E x ur, X r )br, X r ) dr C b 2 2 L /2 p/2 T ) xu x u n 2 2 C x u x u n L p T ). x u n r, X r )br, X r ) dr L /2 p/2 T ) For he la deerminiic inegral, we receie a imilar eimae: E 2 E 2 d d ) σσ r, X r )) ij x 2 i x j ur, X r ) x 2 i x j u n r, X r ) dr i= j= σr, X r ) 2 xur, 2 X r ) xu 2 n r, X r ) dr 2 c2 σe 2 xur, X r ) 2 xu n r, X r ) dr C 2 xu 2 xu n L p T ). Finally, for he ochaic inegral, we hae by imilar eimae and he Iô-Iomery E E = E x ur, X r ) x u n r, X r ))σr, X r ) dw r x ur, X r ) x u n r, X r ))σr, X r ) dw r x ur, X r ) x u n r, X r ))σr, X r ) 2 dr c 2 σe x ur, X r ) x u n r, X r ) 2 dr 2 C x u x u n 2 2 L /2 p/2 T ) = C x u x u n L p T ). 7

22 3 TRANSFORMATION OF THE SDE Therefore, here exi a ubeuence u nk ) k uch ha u nk r, X r ) dr + k + 2 d i= x u nk r, X r )br, X r ) dr + d σσ r, X r )) ij x 2 i x j u nk r, X r ) dr j= ur, X r ) dr + x ur, X r )br, X r ) dr + x u nk r, X r )σr, X r ) dw r x ur, X r )σr, X r ) dw r + 2 i= j= d d σσ r, X r )) ij x 2 i x j ur, X r ) dr P-a.. and herefore, we hae Iô formula for funcion in W,2,p T ) Tranformaion of he SDE We may ranform SDE 5) by mean of oluion o a paricular PDE, which i aed below in 7). Then an applicaion of Iô formula i ued o replace he drif erm. By ieraion, we are able o reformulae he euaion a aed in 4). Aume ha b and σ fulfill Aumpion 2.2 and 2.3, hen for eery f L pt ) here exi a oluion u W,p,2 T ) o he euaion u + 2 d d σσ ) ij x 2 i x j u = f, on [, T ], ut, x) = 7) i= j= uch ha u W,2,p T ) C f L pt ), 8) where C doe no depend on f and i increaing in T. Then we hae up x u, x) up x u, x) x ut, x) + x ut, x),x) [,T ],x) [,T ] = up,x) [,T ] x u, x) x ut, x). By he Hölder coninuiy of x u, ee [KR5] Lemma.2, hi ogeher wih 8) lead o up x u, x) up Cp,, ε) T ε 2 u W,2,x) [,T ],x) [,T ],p T ) Cp,, ε, T )T ε 2 f L p T ) 9) 8

23 3 TRANSFORMATION OF THE SDE for eery ε, ), which fulfill ε + d p + 2 <. Since Cp,, ε, T ) i increaing in T, we can aume he conan in fron of f L p T ) o be a mall a we wan by chooing T appropriae. Thi will be of imporance in Lemma 4.. Now, le U b a oluion o he euaion u + d d σσ ) ij x 2 2 i x j u = b on [, T ], ut, x) =. ) i= j= Uing Iô formula for funcion in W,2,p T ) Propoiion 3.3), we ge U b, X ) = U b, x) + + U b, X ) + 2 x U b, X )b, X ) d + d i= x U b, X )σ, X ) dw d σσ, X )) ij x 2 i x j U b, X ) d. j= Here, we ue ha U b i a oluion o PDE ), o obain U b, X ) = U b, x) + x U b, X )b, X ) d + x U b, X )σ, X ) dw b, X ) d. Tha implie b, X ) d = U b, x) U b, X ) + x U b, X )b, X ) d + x U b, X )σ, X ) dw. Now, we define T b) := x U b b and ranform SDE 5) by replacing he drif erm: X = x + U b, x) U b, X ) + T b), X ) d + x U b, X )σ, X ) + σ, X ) dw. ) 9

24 3 TRANSFORMATION OF THE SDE Noe, ha T b) L pt ) ince x U b i bounded and b L pt ). Nex, le U T b) be a oluion o he euaion u + 2 d d σσ ) ij x 2 i x j u = T b) on [, T ], ut, x) =. i= j= Uing again Iô formula Propoiion 3.3), we ge U T b), X ) = U T b), x) + x U T b), X )b, X ) d + x U T b), X )σ, X ) dw + U T b), X ) + 2 d d σσ, X )) ij x 2 i x j U T b), X ) d i= j= = U T b), x) + x U T b), X )b, X ) d + x U T b), X )σ, X ) dw T b), X ) d, and herefore T b), X ) d = U T b), x) U T b), X ) + x U T b), X )b, X ) d + x U T b), X )σ, X ) dw. Again, we define T 2 b) := x U T b) b and replace he drif in he ranformed SDE ): X = x + U b, x) + U T b), x) U b, X ) U T b), X ) + T 2 b), X ) d + x U b, X )σ, X ) + x U T b), X )σ, X ) + σ, X ) dw. 2

25 3 TRANSFORMATION OF THE SDE Ieraion yield afer n + ep n n X = x + U T b), x) U k T b), X k ) + + k= k= k= T n+ b), X ) d n x U T b), X k )σ, X ) + σ, X ) dw 2) wih he conenion T b) = b and T k+ b) = x U T k b) b. We define and herefore, SDE 2) become U n), x) := n U T b), x) k k= X = x + U n), x) U n), X ) + + For wo oluion X ), X 2) we define Then euaion 3) read Y i,n) T n+ b), X ) d x U n), X ) + I ) σ, X ) dw. 3) Y i,n) b n), X i) σ n), X i) ) := = Y i,n) + := X i) + U n), X i) ), ) := T n+ b), X i) ), x U n), X i) ) + I b n), X i) ) d + ) σ, X i) ). σ n), X i) ) dw. 4) 2

26

27 4 SOME HELPFUL LEMMAS 4. Some helpful lemma In hi chaper we preen he neceary ool o proe our main reul. Fir, we gie ome ueful properie of he inoled funcion and a conracion propery beween X i) and Y i,n) in Lemma 4.. Then we proe wo Krylo-ype eimae for condiional expecaion. A erion of Lemma 5. from [Kry86] i aed in Lemma 4.2 under ery general aumpion on he coefficien. I i ufficien o aume b L pt ) and σ o be bounded and nondegeneraed. The price we hae o pay i ha i i only applicable o funcion in L r T ) wih r, d +. The econd Krylo-ype eimae, namely Propoiion 4.4, reuire alo he coninuiy of he diffuion coefficien. Baed on hi ineualiy we proe an exponenial eimae for he ranformed diffuion before we how conergence of he difference beween he ranformed drif erm of wo oluion. In he end, we will proe ha under our condiion eery oluion o 5) ha finie fir and econd momen. The following Lemma i imilar o Lemma 7 in [FF] and o i he proof. Bu we gie i in deail o make clear ha i work in he ame way for our exended ranformaion. Lemma 4.. Le c), c3), c4) of Aumpion 2.2 and Aumpion 2.3 be fulfilled and X ), X 2) be wo oluion o 5) uch ha 6) hold. Then here exi T T uch ha for all T, T ] we hae i) T n b) L p T ) 2 n b L pt ), ii) n up x U T b), x) k,x) [,T ] R 2, d k= iii) 2 xu n) L p T ) C for ome conan C >, independen of n, and i) Y,n) Y 2,n) 3 2 X 2) 2 X ) X 2), Y 2,n) for all, T ]. X ) Y,n) Proof. i) Se and chooe T uch ha ε = 4 b L p T ) + ) up x U f, x) ε f L p T ) 5),x) [,T ] for all T, T ] and f L pt ), where U f denoe a oluion o u + 2 d d σσ ) ij x 2 i x j u = f on [, T ], ut, x) =. i= j= 23

28 4 SOME HELPFUL LEMMAS The poibiliy of chooing uch a T i gien by 9). The + in he denominaor of ε i ju o aoid iue in cae b =. Then we hae up x U b, x) ε b L p T ),,x) [,T ] T b) L p T ) up x U b, x) b L p T ) ε b 2 L pt ),,x) [,T ] up x U T b), x) ε T b) L p T ) ε 2 b 2 L pt ),,x) [,T ] and by ieraing T 2 b) L p T ) up,x) [,T ] x U T b), x) b L p T ) ε 2 b 3 L pt ) T k b) L p T ) ε k b k+ L pt ) 4 k b L p T ) + ) k b k+ L pt ) 4 k b L pt ) 6) which proe i). ii) Applying 5) and 6) yield up x U T b), x) ε T k b) k L p T ) ε,x) [,T ] R 4 b d k L pt ) 4. k+ Therefore, we ge n up x U T b), x) k,x) [,T ] k= and o he econd ineualiy i proed. iii) We hae wih 8) n k= 4 k+ = 4 n k= 4 k 4 k= 4 k 2 2 xu n) L p T ) n xu 2 T b) k L p T ) k= n k= U T k b) W,2,p T ) n C T k b) L p T ). k= And uing i) lead o 2 xu n) L p T ) C n k= for ome C >, independen of n. 2 k b L pt ) C b L p T ) n k= 2 C k 2 C k k= 24

29 4 SOME HELPFUL LEMMAS i) To proe he conracion beween X and Y we ue he mean-alue ineualiy from Lemma A.7 for U T k b). Thu Y,n) Y 2,n) = X ) + U n), X ) ) X 2) U n), X 2) ) X 2) n + U T b), X ) k ) U T b), X 2) k k= X 2) n + U T b), X ) k ) U T b), X 2) k X ) X ) X ) X 2) + k= n up x U T b), x) k,x) [,T ] k= ) ) X 2). X ) Then ii) proide Y,n) Y 2,n) 3 X ) X 2). 2 On he oher hand, we hae wih he ame argumen X ) X 2) = Y,n) U n), X ) ) Y 2,n) + U n), X 2) ) Y,n) Y 2,n) n + U T b), X ) k ) U T b), X 2) k ) k= Y,n) Y 2,n) + X ) X 2), 2 which i euialen o X ) X 2) 2 Y,n) Y 2,n). From now on we denoe T by T. 4.. Krylo-ype eimae for condiional expecaion To ge an exponenial eimae on he ranformed diffuion, which we are going o ae in he nex ecion, we need a Krylo-ype eimae on he linear combinaion of wo oluion of SDE 5) a aed in Propoiion 4.4. For he proof we hae o do ome preparaion, fir Lemma 4.2 which i a erion of Lemma 5. from [Kry86] for condiional expecaion and differen inegrabiliy in ime and pace, and econd Lemma 4.3, where we proe ha he erm on he righ-hand ide of he ineualiy are bounded. Lemma 4.2. Le he condiion c), c3), c4) of Aumpion 2.2 be fulfilled and X ), be wo oluion of 5) uch ha 6) hold. Then, for any nonnegaie funcion X 2) 25

30 4 SOME HELPFUL LEMMAS f : [, T ] R wih f L r T ) <, any opping ime γ, T, R > and r, d + he following hold: T τ R λ γ { τr λ γ} E dea λ ) d+ f, X λ ) d F { τ λ R γ} CT, d,, r)b 2 + A) d d+ d 2 f L r T ) P-a... Here we denoe B := E T τ λ R γ b λ, X ), X 2) ) d F, A := E T τ λ R γ ra λ ) d F. Noe, ha A and B depend on, T, R, λ, γ. We refrain from denoing hi in indice ince i will alway be clear wha we mean and hu, would be more confuing han helpful. The proof i rucured a follow. Fir we proe he ineualiy for nonnegaie funcion in C uch ha f > on [, T ] B R, B enoe here he open ball in around he origin wih radiu R and B R he cloure of i. Thi will be done by uing Lemma A.8, Iô formula and he maringale propery of he ochaic inegral. Then, we exend hi o nonnegaie funcion in C. Afer ha we proe ha for hee funcion he ineualiy hold alo for f. The aemen i exended o meaurable bounded funcion by a monoone cla argumen and finally alo o unbounded meaurable funcion. Proof. Noe, ha all he condiional expecaion exi, ince we alway inegrae nonnegaie funcion. Fix a µ > and ake a nonnegaie f C + ) wih f > on [, T ] B R. Obiouly here exi T, R uch ha f = for T or x > R. Then Lemma A.8 enure he exience of a nonnegaie funcion ϕ wih bounded weak deriaie ϕ, x ϕ, 2 xϕ uch ha for any ymmeric, poiie emidefinie d d marix α he following hold: ϕ + d i= d j= x ϕ µϕ, α ij 2 x i x j ϕ µ + rα))ϕ + deα) d+ fe µ, ϕ, x) Cd, )µ d 2 d d+ T ) d+ r e µ f L r. Define ψ := e µ ϕ. Then we hae ψ + d i= d j= α ij 2 x i x j ψ µ rα)ψ + deα) d+ f, 7) x ψ µψ, 8) ψ, x) Cd, )µ d 2 d d+ T ) d+ r f L r. 9) 26

31 4 SOME HELPFUL LEMMAS From [Kry87] Example 6.4.6, we know ha ψ, x ψ, 2 xψ are coninuou on [, T ] B R. Therefore, we may apply Iô formula and ge ψ, X λ ) ψ, x λ ) = ψ, X λ ) d + x ψ, X λ )b λ, X ), X 2) ) d + x ψ, X λ )σ λ, X ), X 2) ) dw + 2 d d σ λ σ λ, X ), X 2) )) ij x 2 i x j ψ, X λ ) d i= j= which how ha κ := ψ, X λ ) x ψ, X λ )b λ, X ), X 2) ) + ψ, X λ ) + d i= d a λ ) ij x 2 i x j ψ, X λ ) d j= i a maringale on [, T τr λ γ). Then, by applying ineualiy 7) we hae for all [, T τr λ γ) on he e { } E dea λ ) E = E d+ f, X λ ) d F µ ra λ )ψ, X λ ) ψ, X λ ) d i= d a λ ) ij x 2 i x j ψ, X λ ) d F j= µ ra λ )ψ, X λ ) d + κ κ ψ, X λ ) + ψ, X λ ) + x ψ, X λ )b λ, X ), X 2) ) d F P-a... 27

32 4 SOME HELPFUL LEMMAS Since ψ i nonnegaie and κ i a maringale, we obain E dea λ ) E d+ f, X λ ) d F µ ra λ )ψ, X λ ) d + κ κ + ψ, X λ ) + x ψ, X λ )b λ, X ), X 2) ) d F = E ψ, X λ ) + µ ra λ )ψ, X λ ) + x ψ, X λ )b λ, X ), X 2) ) d F E ψ, X λ ) + µ ra λ )ψ, X λ ) + x ψ, X λ ) b λ, X ), X 2) ) d F. Then wih 8), we receie ha E dea λ ) E ψ, X λ ) + E d+ f, X λ ) d F µ ra λ )ψ, X λ ) + µψ, X λ ) b λ, X ), X 2) ) d F ψ, X λ ) + up ψ, X λ ) µ ra λ ) + [,] Cd,, r, T ) f L r E µ d 2 d d+ µ b λ, X ), X 2) ) d F + µ ra λ ) + µ b λ, X ), X 2) ) d F, where he la ineualiy follow wih 9). Thi ineualiy i independen of ψ and hold for all µ >, herefore i i alo rue for µ := { A<B 2 }B 2 + {A>,A B 2 }A + {A=B=} c, c >. By Lemma A. A and B are P-almo urely finie, which preen u from echnical iue, e.g. diiding by infiniy. Since all he indicaor funcion and A, B are meaurable wih repec o F, we hae 28

33 4 SOME HELPFUL LEMMAS for he condiional expecaion E µ d 2 d d+ = E + E + E + µ { A<B 2 }B 2d d+ d b λ, X ), X 2) ) d + µ {A>,A B 2 }A d d+ d 2 {A=B=} c d 2 d d+ = { A<B 2 }B 2d d+ d + {A>,A B 2 }A d d+ d 2 + {A=B=} c d 2 d d+ + B + A 2 + c + B E b λ, X ), X 2) ) d + A 2 E + ce ra λ ) d F +B 2 b λ, X ), X 2) ) d +A b λ, X ), X 2) ) d ra λ ) d F ra λ ) d F +c ra λ ) d F b λ, X ), X 2) ) d F +B 2 E b λ, X ) ra λ ) d F, X 2) ) d F +A E ra λ ) d F b λ, X ), X 2) ) d F +ce ra λ ) d F. 29

34 4 SOME HELPFUL LEMMAS Thi lead o E µ d 2 d d+ + µ { A<B 2 }B 2d d+ d + {A=B=} c d 2 d d+ b λ, X ), X 2) ) d + µ 2 + B 2 A ) + {A>,A B 2 }A d d+ d 2 + cb + ca ) { A<B 2 }3B 2d d+ d + {A>,A B 2 }3A d d+ d 2 {A> or B>} 3B 2 + A) d d+ d 2 c 3B 2 + A) d d+ d 2. + {A=B=} c d 2 d d+ ra λ ) d F 2 + A 2 B ) + {A=B=} c d 2 d d+ So, we proed for [, T τr λ γ) { }E dea λ ) d+ f, X λ ) d F { }Cd,, r, T )B 2 + A) d d+ d 2 f L r P-a... Wih Faou Lemma for condiional expecaion, we ge for T T τ R λ γ { τr λ γ} E dea λ ) d+ f, X λ ) d F T τ R λ γ = E { T τr λ γ} dea λ ) d+ f, X λ ) d F = E lim inf { n T τr λ γ n } lim inf n E { T τr λ γ n } = lim inf { n T τr λ γ }E n lim inf n T τ R λ γ n dea λ ) T τ R λ γ n dea λ ) T τ R λ γ n dea λ ) d+ f, X λ ) d d+ f, X λ ) d d+ f, X λ ) d { T τ λ R γ n }Cd,, r, T )B 2 + A) d d+ d 2 f L r = { τ λ R γ} Cd,, r, T )B 2 + A) d d+ d 2 f L r P-a.. F F F 3

35 4 SOME HELPFUL LEMMAS for all nonnegaie f C + ) wih f > on [, T ] B R. Now, le f C + ) wih f. Take a mooh funcion χ : + [, ] wih χ > on [, T ] B R, for example χ from Lemma A.9. Then we hae on he e { τ λ R γ} E T τ R λ γ dea λ ) = E = lim ε E T τ R λ γ dea λ ) T τ R λ γ d+ f, X λ ) d d+ lim dea λ ) F f + εχ), X λ ) d ε F d+ f + εχ), X λ ) d F by dominaed conergence. A f + εχ i ricly poiie on [, T ] B R, we hae, for a uiable T > T τ R λ γ E dea λ ) d+ f, X λ ) d F lim Cd,, r, T )B 2 + A) d d+ d 2 f + εχ L r ε = Cd,, r, T )B 2 + A) d d+ d 2 f L r P-a... The nex ep i o ge rid of he dependence on T. To hi end, conider he mooh funcion { ) c exp g) := if <, ele, where c i choen uch ha g)d =. Then we hae for he conoluion R ) [ 2,T + 2 ] g ) = R [ 2,T + ] )g) d = [ 2,T + ] )g) d 2 which i for [, T ] and for / [, T + ]. Since [ 2,T + ] g) f i mooh and 2 3

36 4 SOME HELPFUL LEMMAS eual o f on [, T ], we hae on he e { τr λ γ} T τ R λ γ E dea λ ) d+ f, X λ ) d F T τ R λ γ ) = E d+ [ 2,T + 2 ] g )f, X λ ) d dea λ ) Cd,, r, T + )B 2 + A) d d+ d 2 Cd,, r, T )B 2 + A) d d+ d 2 f L r R + F ) [ 2,T + 2 ] g )f, x) dx P-a... r d r Now, le f C + ). Since f i coninuou and compacly uppored, here exi a euence f n ) n of nonnegaie funcion in C + ) which conerge uniformly o f le ψ be a mollifier on + and ake f n := ψ /n f, ee Appendix for he definiion of mollifier and ψ /n ). Therefore, on he e { τr λ γ} we hae E T τ R λ γ dea λ ) d+ f, X λ ) d F = lim n E T τ R λ γ dea λ ) d+ fn, X λ ) d and ince he ineualiy i rue for nonnegaie funcion in C + ) T τ R λ γ E dea λ ) d+ f, X λ ) d F lim Cd,, r, T )B 2 + A) d d+ d 2 fn L r n = Cd,, r, T )B 2 + A) d d+ d 2 f L r To proe ha he ineualiy i alo alid for bounded meaurable funcion, define X := f : Rd+ R f i meaurable, bounded and fulfill P-almo urely { τr λ γ} E T τ R λ γ dea λ ) d+ f, X λ ) d F F P-a... { τr λ γ} Cd,, r, T )B 2 + A) d d+ d 2 f L r. 32

37 4 SOME HELPFUL LEMMAS Noe, ha he lef-hand ide exi, ince we inegrae nonnegaie funcion. The righhand ide of he ineualiy maybe be infinie, which i feaible ince he ineualiy i hen riially fulfilled. Le f f 2... f n... in X wih f n f poinwie and f bounded, hen he ineualiy hold for f, becaue wih monoone conergence we obain E T τ R λ γ dea λ ) = E = lim n E T τ R λ γ dea λ ) T τ R λ γ d+ f, X λ ) d d+ lim dea λ ) F f n, X λ ) d n F d+ fn, X λ ) d lim Cd,, r, T )B 2 + A) d d+ d 2 fn L r n = Cd,, r, T )B 2 + A) d d+ d 2 lim n f n L r = Cd,, r, T )B 2 + A) d d+ d 2 f L r F P-almo urely on he e { τr λ γ}. Since f i again meaurable, we hae f X. Therefore X i cloed under bounded monoone conergence. And by imilar mean i can be alo hown ha X i cloed under uniform conergence. Since C + ) i an algebra and here exi a euence f n in C + ) uch ha f n, he monoone cla heorem i applicable in he erion of [Del78] 22.2) and hi yield ha X conain all meaurable bounded funcion. Now, le f be a nonnegaie meaurable funcion wih f L r T ) <. Since [,T ] f n) X we obain on he e { τr λ γ} wih monoone conergence E T τ R λ γ = lim dea λ ) n E T τ R λ γ d+ f, X λ ) d F dea λ ) d+ [,T ] )f n), X λ )d F lim Cd,, r, T )B 2 + A) d d+ d 2 f n L r n T ) = Cd,, r, T )B 2 + A) d d+ d 2 f L r T ) P-a... 33

38 4 SOME HELPFUL LEMMAS Lemma 4.3. Le c), c3), c4) of Aumpion 2.2 be fulfilled and X ), X 2) be wo oluion o 5) uch ha condiion 6) hold. Then we hae for all T, λ [, ], T E ra λ ) d F CT, c σ ) and E P-almo urely. T b λ, X ), X 2) ) d F Cd, p,, T, c σ, c σ, b L p T )) The idea of proing hi, epecially he econd eimae i aken from [GM], Proof of Corollary 3.2. Proof. Uing c4) we may eimae he race of a λ a in 4): ra λ ) 2 c 2 σ. Then, monooniciy of he condiional expecaion reul in T E ra λ ) d F 2 c 2 σt. To proe ha he econd condiional expecaion i P-almo urely finie, we will ue Lemma 4.2 for X ) and X 2). Noe ha all he eigenalue of σσ are bounded from below by c σ becaue of c3). Since a ymmeric marix ha only real eigenalue and he deerminan i he produc of hem, we hae in cae λ = And he ame hold for dea ). Define γ n := inf : E and dea ) = 2 d deσσ, X ) )) 2 d cd σ. B n) := E A n) := E T τ R γn T τ R γn b, X ) ) d F > n b, X ) ) d F, ra ) d F. 34

39 4 SOME HELPFUL LEMMAS Wih Jenen ineualiy for he condiional expecaion and he Lebegue meaure on [, T ], we receie on he e { τr γ n} ) T τ R γn 2 B n) 2 E b, X ) ) d F T E = T E 2 c σ T τ R γn T τ R γn b, X ) ) 2 d F ) dea d+ ) b, ) X dea ) ) d T τ R γn d+ T E dea ) d+ ) 2 d F b, X ) Applying he ineualiy from Lemma 4.2 wih = p 2, r = 2, proide ) 2 d F. B n) ) 2 2 c σ ) d d+ T Cd, p,, T ) B n) ) 2 + A n)) d d+ d p b 2 L pt ) 2 c σ ) d d+ B ) ) 2d T Cd, p,, T ) n) d+ 2d p + 2 c 2 σt ) d d+ d p b 2 L pt ) P-almo urely. Wih Young ineualiy we hae for ε > and z := B n) ) 2z = ε ε B n)) 2z z)ε z + zε z ε ) z + ε z B n) 2. B n) ) 2 Le ε be mall enough uch ha ) d 2 d+ T Cd, p,, T )ε z b 2 L pt ) <. c σ Noe, ha we may chooe ε independen of ω, n and R. Then we ge d d <, d+ p ) ) d B n) 2 2 d+ T Cd, p,, T ) 2 c 2 c σt ) d d+ d ) ) p + ε z + ε z B n) 2 b 2 L pt ) P-a.. σ which i euialen o ) 2 B n) 2 ) d d+ c σ T Cd, p,, T ) 2 c 2σT ) ) d d+ d p + ε z ) d 2 d+ c σ T Cd, p,, T )ε z b 2 L pt ) b 2 L pt ) P-a.. 35

40 4 SOME HELPFUL LEMMAS on he e { τ R γ n}, which i finie and independen of n and ω. If we ake he limi n we obain ha E T τ R b, X ) ) d F Cd, p,, T, c σ, c σ, b L p T )) P-a.. on he e { τr }. Analogouly, we can proe ha he ame hold for X2). Furhermore, he bound i alo independen of R. If we ake he limi R we ge E T Therefore, we obain E T λe b, X i) ) d F < Cd, p,, T, c σ, c σ, b L p T )) λb, X ) ) + λ)b, X 2) T b, X ) ) Cd, p,, T, c σ, c σ, b L p T )) P-almo urely, for eery λ [, ]. ) d F d F + λ)e T b, X 2) ) P-a... d F Propoiion 4.4. Le c)-c4) of Aumpion 2.2 be fulfilled and X ), X 2) be wo oluion o 5) uch ha 6) hold. Then for arbirary R > here exi an ε > uch ha for eery nonnegaie meaurable funcion f : [, T ] R wih f L r T ) <, r, d +, and eery T, λ [, ] we hae on he e { τ R τ ε } E T τ R τ ε f, X λ ) d F Cd, p,, r, T, b L p T ), c σ, c σ ) f L r T ) P-almo urely, where τ ε := inf { : X ) X 2) } > ε. Proof. Since σ i uniformly coninuou on [, T ] B R here exi an ε > uch ha σ, x) σ, y) < c σ 4 c σ, x),, y) [, T ] B R wih, x), y) ε. 36

41 4 SOME HELPFUL LEMMAS Tha implie for all ξ, T τ R τ ε ) σ, X ) ) σ, X 2) ) σ, X 2) )ξ, ξ ) σ, X ) ) σ, X 2) ) σ, X 2) )ξ ξ σ, X ) ) σ, X 2) ) σ, X 2) ) ξ 2 and herefore, σ λ σ λ, X ), X 2) c σ 4 c σ c σ ξ 2 = 4 c σ ξ 2 2) )ξ, ξ = λ 2 σσ, X ) )ξ, ξ + λ) 2 σσ, X 2) + 2λ λ) σ, X ) )σ, X 2) )ξ, ξ = λ 2 σσ, X ) )ξ, ξ + 2λ + λ 2 ) + 2λ λ) σ, X ) )σ, X 2) )ξ, ξ = λ 2 σσ, X ) )ξ, ξ + λ 2 ) σσ, X 2) + 2λ 2 2λ) σσ, X 2) )ξ, ξ + 2λ λ) σ, X ) )σ, X 2) )ξ, ξ = λ 2 σσ, X ) )ξ, ξ + λ 2 ) σσ, X 2) ) + 2λ λ) σ, X ) ) σ, X 2) ) Togeher wih eimae 2) and c3) we obain ha σ λ σ λ, X ), X 2) )ξ, ξ σσ, X 2) )ξ, ξ )ξ, ξ σ, X 2) )ξ, ξ )ξ, ξ )ξ, ξ λ 2 c σ ξ 2 + λ 2 )c σ ξ 2 2λ λ) 4 c σ ξ 2 2 c σ ξ 2.. Thi how ha for T τ R τ ε all he eigenalue of σ λ σ λ below by c 2 σ and herefore, we can eimae he deerminan: are bounded from dea λ ) = 2 d deσλ σ λ, X ), X 2) )) 2 2d cd σ. Noe, ha τ R τ λ R ince X) R and X 2) R imply ha λx ) + λ)x 2) λ X ) + λ) X 2) R. 37

42 4 SOME HELPFUL LEMMAS So, we obain on he e { τ R τ ε } T τ R τ ε E f, X λ ) d F = E 4 c σ 4 c σ T τ R τ ε ) dea λ d+ ) f, X λ dea λ ) ) d F d+ f, X λ ) d F ) d T τ R τ ε d+ E dea λ ) ) d T τ R λ τε d+ E dea λ ) d+ f, X λ ) d F P-a... Wih Lemma 4.2 and Lemma 4.3, we deduce ha T τ R τ ε E f, X λ ) d F Cd, p,,, r, T, b L p T ), c σ, c σ ) f L r T ) P-a.. on he e { τ R τ ε } Uniform exponenial eimae for he ranformed diffuion In hi ecion we proe ha for σ n), X ) A n) := Y,n) ) 2 ) σ n), X 2) Y 2,n) 2 {Y,n) Y 2,n) we hae ha E[e An) T ] i uniformly bounded in n. To hi end we need a Khaminkiype eimae, a aed in Lemma 4.5, o ge he exponenial eimae ia condiional expecaion. Thi i done in Propoiion 4.6. Lemma 4.5. Le f : [, T ] R be a nonnegaie meaurable funcion and γ an arbirary opping ime. Aume ha X i an adaped proce uch here exi a conan α < wih T γ { γ}e f, X ) d F α P-a.. T. Then we hae E exp T γ f, X ) d α. } d 38

43 4 SOME HELPFUL LEMMAS Proof. We hae E exp T γ f, X ) d = E n! n= T γ f, X ) d By inducion one can proe ha T n T T T f, X ) d = n!... f, X )f 2, X 2 )... f n, X n ) d n... d 2 d, n and hu E exp = = = = T γ E n= E n= n= n= f, X ) d T γ T γ T T T T T T T n 2 T γ n 2 T n 2 T γ n f, X )f 2, X 2 )... T n. f n, X n )f n, X n ) d n d n... d 2 d n { γ}f, X ) {2 γ}f 2, X 2 )... {n γ}f n, X n ) {n γ}f n, X n ) d n d n... d 2 d E { γ}f, X ) {2 γ}f 2, X 2 )... {n γ}f n, X n ) T n 2 {n γ}f n, X n ) T n {n γ}f n, X n ) d n d n... d 2 d E E { γ}f, X ) {2 γ}f 2, X 2 )... T n {n γ}f n, X n ) d n F n d n... d 2 d. 39

44 4 SOME HELPFUL LEMMAS Since all he erm excep for he la inegral are meaurable wih repec o F n hae T γ E exp f, X ) d we = n= n= T T T α T n 2 E { γ}f, X ) {2 γ}f 2, X 2 )... {n γ}f n, X n )E T T n 3 {n 2 γ}f n 2, X n 2 ) So, by ieraion we ge E exp T γ n f n, X n ) d n F n d n... d 2 d E { γ}f, X ) {2 γ}f 2, X 2 )... T γ T n 2 {n γ}f n, X n ) d n d n 2... d 2 d. f, X ) d n= α n = α. Propoiion 4.6. Le Aumpion 2.2, 2.3 be fulfilled and X ), X 2) o 5) uch ha 6) hold. For A n) := σ n), X ) ) σ n), X 2) Y,n) Y 2,n) 2 ) 2,n) {Y Y 2,n) } and ε from Propoiion 4.4, here exi a conan C > uch ha ] E [e An) T τ R τε C uniformly for all n N. Proof. Conidering σ n) we find ha: xi σ n) = xi x U n)) σ + x U n) xi σ + xi σ. be wo oluion We ue ha σ i bounded and x σ L pt ), ha x U n) i uniformly bounded by 2 and 2 xu n) i euibounded in L pt ) ee Lemma 4.) o deduce ha x σ n) L p T ) C uniformly in n. 4

45 4 SOME HELPFUL LEMMAS Addiionally, σ n) i coninuou, ince x U n) i Hölder coninuou. Then by Lemma A.6 here exi a euence of coninuou funcion u m ) m, which are differeniable wih repec o x in he ordinary ene, uch ha and u m σ n) uniformly on [, T ] B R x u m L p T ) x σ n) L p T ) m N. Then we hae wih Lemma 4. i) T τ R τ ε σ n), X ) ) σ n), X 2) E exp Y,n) Y 2,n) 2 E exp 4 T τ R τ ε ) 2 {Y,n) Y 2,n) σ n), X ) ) σ n), X 2) ) 2 X ) X 2) 2 ) {X X 2) d } d }. By uniform conergence, we receie ha T τ R τ ε σ n), X ) ) σ n), X 2) ) 2 E exp Y,n) Y 2,n) 2 d,n) {Y Y 2,n) } T τ R τ ε lim E u m, X ) ) u m, X 2) ) 2 exp 4 m X ) X 2) 2 d ) {X X 2) } T τ R τ ε 2 = lim E m exp x u m, X λ )X ) X 2) ) dλ 4 X ) X 2) 2 d ) {X X 2) } lim m E exp 4 T τ R τ ε x u m, X λ ) 2 dλ d. An applicaion of Fubini Theorem and Jenen ineualiy yield T τ R τ ε σ n), X ) ) σ n), X 2) ) 2 E exp Y,n) Y 2,n) 2,n) {Y lim m E exp 4 T τ R τ ε x u m, X λ ) 2 d dλ. Y 2,n) d } 4

46 4 SOME HELPFUL LEMMAS Now, chooe µ > o mall ha p, 2d + ) + µ) which exi ince p, > 2d + ). Then we hae for β > wih Young ineualiy T τ R τ ε σ n), X ) ) σ n), X 2) ) 2 E exp Y,n) Y 2,n) 2 d,n) {Y Y 2,n) } lim E exp 4β T τ R τ ε x u m, X λ ) 2 d dλ m β lim E exp T τ R τ +µ ε β x u m, X λ ) 2 d + µ ) +µ 4 µ dλ. m µ + + µ β And wih Hölder ineualiy T τ R τ ε σ n), X ) ) σ n), X 2) E exp Y,n) Y 2,n) 2 ) +µ ) 4 µ exp µ + µ lim m β E exp T τ R τ ε β +µ + µ T µ ) 2 {Y,n) Y 2,n) d } +µ x u m, X λ ) 2+µ) d dλ. Furhermore, we hae wih Propoiion 4.4 for all T on he e { τ R τ ε } T τ R τ ε E β +µ + µ T µ +µ x u m, X λ ) 2+µ) d F Cd, p,, µ, T, c σ, c σ, b L p T )) β+µ + µ T µ +µ x u m 2+µ) = Cd, p,, µ, T, c σ, c σ, b L p T ))β +µ x u m 2+µ) L pt ) Cd, p,, µ, T, c σ, c σ, b L p T ))β +µ x σ n) 2+µ) L pt ) =: α. 2+µ) L p T ) 2+µ) Since x σ n) L p T ) i euibounded, we can chooe β o mall ha hi i le han for all n N. Then we hae by Lemma 4.5 and ineualiy 2) ha ] ) +µ ) µ E [e An) T τ R τε 4 µ exp + µ β α C, 2) 42

47 4 SOME HELPFUL LEMMAS where C doe no depend on n Conergence of he ranformed drif In he following we proe ha E T b n), X ) ) b n), X 2) ) 2 d conerge o for n. The proof i much impler han in [FF] ince we are able o apply Krylo eimae. The price o pay i ha we hae o aume p, > 2d + ). Lemma 4.7. Le c), c3), c4) of Aumpion 2.2 and Aumpion 2.3 be fulfilled and X ), X 2) be wo oluion of 5) uch ha condiion 6) hold. Then we hae T lim b n), X ) ) b n), X 2) ) 2 d =. n E Proof. Young ineualiy, Lemma 3. wih = p, r = and an applicaion of Lemma on he ariing T n+ b) 2 erm yield E T 2E b n), X ) ) b n), X 2) T b n), X ) ) ) 2 d 2 d + 2E T Cd, p,, T, c σ, c σ, b L p T )) b n) 2 L pt ) b n), X 2) = Cd, p,, T, c σ, c σ, b L p T )) T n+ b) 2 L pt ) Cd, p,, T, c σ, c σ, b L p T )) 2 2n+) b 2 L pt ) n. ) 2 d 4.4. Bounded fir and econd momen In hi ecion we how ha X and X 2 are inegrable which i he la neceary ool o proe pahwie uniuene. Furhermore, we alo obain he finiene of E[up [,T ] X ] and herefore, here i no exploion for our SDE. All we need i Lemma 3. and he ineualiy of Burkholder, Dai and Gundy. 43

48 4 SOME HELPFUL LEMMAS Lemma 4.8. Le c), c3) and c4) of Aumpion 2.2 be fulfilled. If X i a oluion o SDE 5) uch ha condiion 6) hold, we hae [ ] E up X [,T ] < and up E [ X 2] <. [,T ] Proof. We hae [ ] E up X = E up [,T ] [,T ] x + b, X ) d + σ, X ) dw x + E up b, X ) d + E up σ, X ) dw [,T ] [,T ] T x + E b, X ) d + E up σ, X ) dw. [,T ] Then applicaion of Lemma 3. o he fir expecaion erm and of he ineualiy of Burkholder, Dai and Gundy ee e.g. [RY5] Corollary IV.4.2) o he econd yield [ ] T 2 E up X x + C b L p T ) + CE σ, X ) 2 d. [,T ] Since σ i bounded and b L pt ), hi i finie. Furhermore, up E [ X 2] [,T ] = up E [,T ] 2 x + b, X ) d + σ, X ) dw 2 b, X ) d + σ, X ) dw 2 2 E b, X ) d + 4 up E σ, X ) dw. [,T ] 2 x up E [,T ] 2 x up [,T ] We apply Hölder ineualiy o he fir expecaion and he mulidimenional Iô Iomery o he econd one o receie up [,T ] E [ X 2] 2 x 2 + 4T E T b, X ) 2 d + 4 up E [,T ] σ, X ) 2 d. 44

49 4 SOME HELPFUL LEMMAS Again, we ue Lemma 3., c4) and c) o obain up E [ X 2] 2 x 2 + 4T C b 2 L pt ) + 4T c2 σ <. [,T ] 45

50

51 5 PATHWISE UNIQUENESS 5. Pahwie uniuene Thi ecion i deoed o he proof of Theorem 2.7. We ar on a mall ineral [, T ] which i gien by Lemma 4. and how pahwie uniuene up o hi T ju by eimaing he difference of wo oluion wih he ame iniial alue. In he econd par we hen conclude ha i i poible o exend hi o arbirarily large T. 5.. On mall ineral To proe pahwie uniuene we how ha he expecaion of he difference of wo oluion wih he ame iniial alue i zero. Afer he preparaion in he preiou ecion hi can be done eaily, when we pa oer from X o Y n), which i gien by our ranformaion. Uing Iô formula, Lemma 4. and Propoiion 4.6 we find ha he expecaion of he difference of wo oluion i bounded by a erm depending on n. By aking he limi n and applying Lemma 4.7 and Lemma 4.8 we hen finally conclude ha pahwie uniuene hold. Proof of Theorem 2.7 for mall T. In he following, we denoe by x i he i-h enry of a ecor x. Le Aumpion 2.2 and 2.3 be fulfilled and X ), X 2) be wo oluion o 5), uch ha condiion 6) hold. Furhermore, le T := T from Lemma 4. and gien by 4) in Secion 3.2. By Iô formula we hen hae Y i,n) d Y,n) Y 2,n) 2 = = 2 d i= = Y,n) d i= i= d i= ) i Y 2,n) d Y,n) 2 d Y,n) Y,n) d i= Y,n) j= ) ) i 2 Y 2,n) ) i Y 2,n) ) i i Y 2,n) b n), X ) ) b n), X 2) )) d ) i ) ) i Y 2,n) σ n), X ) ) σ n), X 2) ) dw d m ) σ n), X ) ) σ n), X 2) ) ij d i= Y,n) Y,n) Y 2,n), d m i= j= dw j ) 2 ) i i Y 2,n) b n), X ) ) b n), X 2) )) d ) σ n), X ) ) σ n), X 2) ) dw ) 2 σ n), X ) ) σ n), X 2) ) ij d, 47

52 5 PATHWISE UNIQUENESS and herefore d Y,n) Y 2,n) 2 = d i= Y,n) Y,n) Y 2,n), ) i i Y 2,n) b n), X ) ) b n), X 2) )) d σ n), X ) ) σ n), X 2) σ n), X ) ) σ n), X 2) ) 2 d. ) ) dw Applying he ineualiy of Cauchy and Schwarz yield d Y,n) Y 2,n) 2 2 Y,n) Y 2,n) b n), X ) ) b n), X 2) ) d ) + 2 Y,n) Y 2,n), σ n), X ) ) σ 2), X 2) ) dw + σ n), X ) ) σ n), X 2) ) 2 d. 22) Moreoer, wih A n) = σ n), X ) ) σ n), X 2) Y,n) from Propoiion 4.6, we hae d e An) Y,n) Y 2,n) 2) = e An) d + d Y 2,n) 2 ) 2 {Y,n) Y 2,n) Y,n) Y 2,n) 2 + Y,n) [ ] e An). ) Y,n),. ) Y. 2,n) 2 ), } d, Y 2,n) 2 de An) where [, ] denoe he uadraic coariaion which i zero due o he monooniciy of e An). So, we deduce ha d e An) Y,n) Y 2,n) 2) = e An) d Y,n) Y 2,n) 2 Y,n) Y 2,n) 2 e An) da n). Now, we ue ineualiy 22) o conclude ha d e An) Y,n) Y 2,n) 2) 2e An) Y,n) Y 2,n) b n), X ) ) b n), X 2) ) d ) + 2e An) Y,n) Y 2,n), σ n), X ) ) σ n), X 2) ) dw + e An) σ n), X ) ) σ n), X 2) ) 2 d e An) Y,n) Y 2,n) 2 da n). 48

53 5 PATHWISE UNIQUENESS For he la erm we find ha e An) Y,n) Y 2,n) 2 da n) = e An) = e An) Therefore, we ge d e An) 2e An) Y,n) Y 2,n) 2 σ n), X ) ) σ n), X 2) Y 2,n) 2 2 d. Y,n) σ n), X ) ) σ n), X 2) Y,n) + 2e An) Y,n) Y 2,n) 2) Y 2,n) Y,n) Y 2,n), and hu, [ E e An) Y,n) Y 2,n) 2] [ Y,n) E Y 2,n) 2] + 2E + 2E e An) e An) Y,n) Y,n) Wih he help of Lemma 4., we ge [ E e An) Y,n) Y 2,n) 2] 9 4 x ) x 2) 2 + 3E + 2E ) ) 2 { Y,n) Y 2,n) b n), X ) ) b n), X 2) ) d ) σ n), X ) ) σ n), X 2) ) dw Y 2,n) b n), X ) ) b n), X 2) ) d Y 2,n), σ n), X ) ) σ n), X 2) ) ) dw. } d X ) X 2) b n), X ) ) b n), X 2) ) d 23) e An) Y,n) Y 2,n), σ n), X ) ) σ n), X 2) ) ) dw. 49

54 5 PATHWISE UNIQUENESS Summarizing, for wo oluion wih he ame iniial alue, R >, and ε from Propoiion 4.4, we hae for all T [ ] [ ] X ) E τ R τ ε X 2) τ R τ ε E 2 Y,n) τ R τ ε Y 2,n) τ R τ ε = 2E [e 2 An) τ R τε e 2 An) τ R τε ] 2E [e An) 2 τ R τε E [ e An) τ R τε Wih Propoiion 4.6 and ineualiy 23) we obain [ ] X ) E τ R τ ε X 2) τ R τ ε T C E X ) X 2) b n), X ) ) b n), X 2) ) d + E τ R τ ε e An) Y,n) ] Y,n) τ R τ ε Y 2,n) τ R τ ε ] Y,n) τ R τ ε Y 2,n) 2 2 τ R τ ε. Y 2,n), σ n), X ) ) σ n), X 2) ) ) dw The econd expecaion erm anihe due o he maringale propery of he ochaic inegral which i well defined a σ n) i bounded and Y,n) Y 2,n) 2 i inegrable by Lemma 4.8. Tha lead o [ ] X ) E τ R τ ε X 2) τ R τ ε T CE X ) X 2) b n), X ) ) b n), X 2) ) 2 d n N. And herefore, [ ] X ) E τ R τ ε X 2) τ R τ ε T X ) C lim up n CE T lim up n E X ) X 2) 2 d E T X 2) b n), X ) ) b n) X 2) ) d 4 b n), X ) ) b n), X 2) ) 2 d

55 5 PATHWISE UNIQUENESS Wih Fubini Theorem, we hen obain [ ] X ) E τ R τ ε X 2) τ R τ ε T [ X C ) E X 2) 2] d = lim up n E T 4 b n), X ) ) b n), X 2) ) 2 d ince by Lemma 4.8 he fir erm i bounded and by Lemma 4.7 he econd one i zero. So, we hae X ) = X 2) P-a.. T τ R τ ε. By he definiion of τ ε, he eualiy hold rue for all T τ R. Since we can ake R arbirarily large, we hae Thu, X ) = X 2) P-a.. [, T ]. ) P X ) = X 2) Q [, T ] = and by coninuiy of he oluion ) P X ) = X 2) [, T ] = Exenion from mall o arbirarily large ineral Unil now, we only proed pahwie uniuene up o ome poibly mall T. Now, le T be arbirarily large and ake T from Lemma 4.. Le u horly remind how T wa choen. We aumed ha σ i uch ha for all f L pt ) he euaion u + 2 d d σσ ) ij x 2 i x j u = f, ut, x) = i= j= ha a oluion u W,2,p T ) uch ha u W,2,p T ) C f L pt ), 24) where C i independen of f and increaing in T. In paricular, we can find a uniform upper bound for C for all T T. By Hölder coninuiy of x u we ge up x u, x) Cd, p, ε)t ) ε 2 u W,2,x) [,T ],p T ) 5