On the support of solutions of stochastic differential equations with path-dependent coefficients

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1 On the uppot of olution of tochatic diffeential equation with path-dependent coefficient Rama Cont, Alexande Kalinin To cite thi veion: Rama Cont, Alexande Kalinin. On the uppot of olution of tochatic diffeential equation with path-dependent coefficient <hal v2> HAL Id: hal Submitted on 23 Jun 2018 HAL i a multi-diciplinay open acce achive fo the depoit and diemination of cientific eeach document, whethe they ae publihed o not. The document may come fom teaching and eeach intitution in Fance o aboad, o fom public o pivate eeach cente. L achive ouvete pluidiciplinaie HAL, et detinée au dépôt et à la diffuion de document cientifique de niveau echeche, publié ou non, émanant de établiement d eneignement et de echeche fançai ou étange, de laboatoie public ou pivé.

2 On the uppot of olution of tochatic diffeential equation with path-dependent coefficient Rama Cont Alexande Kalinin June 2018 Abtact Given a tochatic diffeential equation with path-dependent coefficient diven by a multidimenional Wiene poce, we how that the uppot of the law of the olution i given by the image of the Cameon-Matin pace unde the flow of the olution of a ytem of path-dependent odinay) diffeential equation. Ou eult extend the Stoock-Vaadhan uppot theoem fo diffuion pocee to the cae of SDE with path-dependent coefficient. The poof i baed on the Functional Ito calculu. MSC2010 claification: 60H10 ; 28C20 ; 34K50. Keywod: uppot theoem, tochatic diffeential equation, functional equation, emimatingale, Wiene pace, functional Ito calculu. Laboatoie de Pobabilité, Statitique et Modéliation, CNRS-Sobonne Univeité. Rama.Cont@math.cn.f Depatment of Mathematic, Impeial College, United Kingdom. a.kalinin@impeial.ac.uk. 1

3 Content 1 Oveview Suppot theoem fo tochatic diffeential equation Statement of main eult Peliminaie Non-anticipative functional calculu Mild olution to path-dependent ODE Stong olution fo path-dependent SDE Chaacteization of the uppot in Hölde topology Convegence in pobability in Hölde nom Hölde pace fo tochatic pocee A geneal Kolmogoov-Chentov etimate Convegence along a equence of patition Adapted linea intepolation of Bownian motion Auxiliay convegence eult Path-dependent ODE and SDE: poof Poof of Popoition Poof of Popoition Poof of main eult Decompoition into emainde tem Convegence of the fit two emainde Convegence of the thid emainde Poof of Theoem 8 and

4 1 Oveview 1.1 Suppot theoem fo tochatic diffeential equation A tochatic poce may be viewed a a andom vaiable taking value in a pace of path; the topological) uppot of thi andom vaiable then decibe the cloue of) the et of poible ample path and povide inight into the tuctue of ample path of the poce. The natue of the uppot ha been invetigated fo vaiou clae of tochatic pocee, with a focu on tochatic diffeential equation, unde diffeent function pace topologie. Fo diffuion pocee, the uppot unde the unifom nom wa fit decibed by Stoock and Vaadhan 20, 21], a eult known a the Stoock- Vaadhan uppot theoem. An extenion to unbounded coefficient wa given by Gyöngy 14]. The uppot of moe geneal Wiene functional and extenion to SDE in Hilbet pace ae dicued in Aida et al 2] and 1]. Thee eult wee extended to the Hölde topology by Ben Aou et al. 5] and, uing diffeent technique, by Millet and Sanz-Solé 16] ; Bally et al 4] ue imila method to deive a uppot theoem in Hölde nom fo paabolic SPDE. Suppot theoem in p-vaiation topology ae dicued by Ledoux et al 15] uing ough path technique. Suppot theoem in Hölde and p-vaiation topologie ae dicued in 13]. Pakkanen 18] give condition fo a tochatic integal ito have full uppot. In thi wok, we extend ome of thee eult to tochatic diffeential equation with path-dependent coefficient. Let Ω, F, F t ) t 0,T ], P ) be a filteed pobability pace on which thee i a tandad d-dimenional F t ) t 0,T ] - Bownian motion W. Conide the following tochatic diffeential equation dx t = bt, X t ) dt + σt, X t ) dw t fo t, T ], 1.1) whoe coefficient b :, T ] S R m and σ :, T ] S R m d ae nonanticipative i.e. bt, X), σt, X) depend on the path X t = Xt.) of the olution up to t. Unde Lipchitz condition on the coefficient b, σ, thi SDE ha a unique olution X 17,19] whoe ample path lie in ome Hölde pace C α 0, T ], R m ). Ou main eult i a deciption of the uppot of the olution in the Hölde topology: we how that the uppot of the law of the olution i given by the image of the Cameon-Matin pace H 1 unde the flow aociated with a ytem of functional diffeential equation. 3

5 1.2 Statement of main eult Let T > 0 and d, m N. To keep notation imple, we denote by the abolute value function, the Euclidean nom in R d and R m and the Fobeniu nom in R m d. Denote by S := C0, T ], R m ) 1.2) the pace of continuou R m -valued map on 0, T ] equipped with the upemum nom and by C α 0, T ], R m ) the pace of x S that ae α-höldecontinuou on, T ] fo α 0, 1], endowed with the delayed Hölde nom x α, := x + x) xt) up. 1.3),t,T ]: t t α We et C00, T ], R m ) := S and 0, := by convention. Then, unde the aumption tated below, 1.1) admit a unique tong olution whoe ample path lie in Hölde pace C α 0, T ], R m ) fo all α 0, 1/2). We denote H 1 0, T ], R m ) the pace of abolutely continuou function on, T ] whoe deivative ḣ i quae-integable with epect to the Lebegue meaue. We equip thi pace with the nom T 1/2 x H, := x + ẋ) d) ) Then H 1 0, T ], R m ) C 1/2 0, T ], R m ) and evey x H 1 0, T ], R m ) atifie x 1/2, x H,. Uing the concept of hoizontal and vetical diffeentiability fo nonanticipative functional 7, 12], we intoduce in Section 2.1 egulaity aumption on the coefficient σ C 1,2, T ) S, R m d ) and conide the map ρ :, T ) S R m given coodinatewie by d ρ k t, x) := x σ k,l t, x)σt, x)e l, 1.5) l=1 whee {e 1,..., e d } i the canonical bai of R d and x σ k,l :, T ) S R 1 m denote the vetical deivative 6,12] of the k, l)-coodinate function of σ fo any k {1,..., m} and l {1,..., d}. Note that ρ = x σ σ fo m = d = 1. In thi context, the uppot of the unique tong olution to 1.1) may be chaacteized by tudying the following path-dependent odinay diffeential equation diven by an element h H 1 0, T ], R d ): ẋ h t) = b 1/2)ρ)t, x t h) + σt, x t h) ḣt) fo t, T ]. 1.6) Ou main eult may be tated a follow: 4

6 Theoem 1 Suppot theoem fo path-dependent SDE). Let ˆx C0, T ], R m ) and σ C 1,2, T ) S, R m d ) with hoizontal ep. vetical) deivative denoted by t σ ep. x σ). Aume σ and x σ ae bounded and thee ae contant c, η, λ 0 and κ 0, 1) uch that bt, x) c1 + x κ ), bt, x) bt, y) λ x y, t σ k,l t, x) + xx σ k,l t, x) c1 + x η ), σ, y) σt, x) + x σ k,l, y) x σ k,l t, x) λ t 1/2 + y x t ) fo all, t, T ), x, y S, k {1,..., m} and l {1,..., d}. Then: i) Thee i a unique tong olution X to 1.1) atifying X = ˆx) fo all 0, ] a.. Futhe, E X 2p α,] < fo all p 1 and α 0, 1/2). ii) Fo any h H 1 0, T ], R d ) thee i a unique mild olution x h to 1.6) o that x h ) = ˆx) fo all 0, ] and we have x h H 1 0, T ], R m ). In addition, the map H 1 0, T ], R d ) H 1 0, T ], R m ), h x h i Lipchitz continuou on bounded et. iii) Fo each α 0, 1/2), the uppot of the image meaue P X 1 in the delayed Hölde pace C α 0, T ], R m ) i the cloue of the et of all mild olution x h to 1.6), whee h H 1 0, T ], R d ). That i, uppp X 1 ) = {x h h H 1 0, T ], R d )} in C α 0, T ], R m ). 1.7) Thi eult extend peviou eult 2,5,16,20] on the uppot of diffuion pocee to the cae of path-dependent coefficient. In the diffuion cae, we etieve the eult of 5, 16] unde weake aumption on σ. Ou poof adapt the appoach ued by Millet and Sanz-Solé 16] to the path-dependent cae, uing the tool of Functional Ito calculu 6, 12]. We contuct Hölde-continuou appoximation of the olution uing an adapted linea intepolation of Bownian motion and how that thi appoximation convege in pobability to the olution in Hölde nom. A key ingedient i the ue of functional etimate deived in 3] uing the Functional Ito calculu, combined with intepolation eo etimate in Hölde nom fo tochatic pocee. Outline. The emainde of the pape i devoted to the poof of Theoem Suppot Theoem. Section 2 dicue the vaiou building block of the poof. Section 2.1 ecall ome functional calculu concept fom 6] and etablihe eveal eult ueful in ou etting. Section 2.2 give condition fo exitence and uniquene of a mild) olution to the path-dependent ODE 1.6); Section 2.3 give condition fo the exitence of a unique tong olution 5

7 to 1.1); Section 2.4 dicue the intepolation method ued to chaacteize the uppot in Hölde topologie. Section 3 dicue Hölde pace fo tochatic pocee and the notion of convegence in pobability in Hölde nom in moe depth. Section 3.2 deive a vaiation on the Kolmogoov-Chentov theoem with an etimate fo the Hölde nom Lemma 12) and an impoved veion of a tatement fom 16] Popoition 14). Section 3.4 dicue adapted linea intepolation of Bownian motion, impoving ome eult fom 16]. Section 4 ue thee ingedient to pove the exitence and uniquene of mild olution to pathdependent ODE Sec. 4.1) and SDE Sec. 4.2). Finally, Section 5 combine thee ingedient to give a poof of the main eult. 2 Peliminaie We hall denote Id the d d identity matix; fo a matix A we denote by A it tanpoe. 2.1 Non-anticipative functional calculu Let D0, T ], R m ) denote the Banach pace of all R m -valued càdlàg map on 0, T ] equipped with the upemum nom and ecall the following notion fom 6, 8]. A functional F :, T ] D0, T ], R m ) R i non-anticipative if F t, x) = F t, x t ) fo all t, T ] and x D0, T ], R m ), whee x t i path x topped at time t: x t ) = x t) fo each 0, T ]. F i called boundedne-peeving if fo each n N thee i c n 0 uch that F t, x) c n fo evey t, T ] and x D0, T ], R m ) atifying x n. In othe wod, F i ought to be bounded on bounded et. We notice that the following peudometic on, T ] D0, T ], R m ) given by d t, x),, y)) := t 1/2 + x t y 2.1) i complete and if F i d -continuou, then it i non-anticipative. A obeved in 10], Lipchitz continuity with epect to d allow fo a Hölde moothne of degee 1/2 in the time vaiable. 6

8 Let u ecall the definition of the hoizontal and vetical deivative. A functional F :, T ) D0, T ], R m ) R i called hoizontally diffeentiable if fo each t, T ) and x D0, T ], R m ), the function 0, T t) R, h F t + h, x t ) i diffeentiable at 0. In thi cae, it deivative thee, will be denoted by t F t, x). We ay that F i vetically diffeentiable if fo all t, T ) and x D0, T ], R m ), the function R m R, h F t, x + h1t,t ]) i diffeentiable at 0. In thi cae, it deivative thee will be epeented by x F t, x). We call F patially vetically diffeentiable if fo all k {1,..., m}, t, T ) and x D0, T ], R m ), the function R R, h F t, x + hê k 1t,T ]) i diffeentiable at 0, whee {ê 1,..., ê m } i the canonical bai of R m. In thi cae, it deivative thee will be denoted by xi F t, x). By calculu, if F i vetically diffeentiable, then it i patially vetically diffeentiable and x F = x1 F,..., xm F ). F i twice vetically diffeentiable if it i vetically diffeentiable and the ame hold fo x F. In thi cae, we et xx F := x x F ) and xk x l F := xk xl F ) fo all k, l {1,..., m}. It follow fom Schwaz Lemma that if F i twice vetically diffeentiable and xx F i d -continuou, then xx F i ymmetic: xk x l F = xl x k F fo each k, l {1,..., m}. A functional G :, T ) D0, T ], R m ) R i aid to be of cla C 1,2 if it i once hoizontally and twice vetically diffeentiable uch that G itelf and the deivative t G, x G and xx G ae boundedne-peeving and d - continuou. By C 1,2, T ) S) 2.2) we denote the pace of functional F :, T ) S R m d that admit an extenion G :, T ) D0, T ], R m ) R of cla C 1,2, whee S i given by 1.2). Then it follow fom 6, Theoem 5.4.1] that t F := t G, x F := x G and xx F := xx G on, T ) S ae independent of the choice of the extenion G. Note that 2.2) allow u to ue the functional Itô fomula 9] in the poof of Popoition 34, which give one of the main agument to etablih 1.7). To conclude, we wite C, T ) S, R m d ) fo the linea pace of all map F :, T ) S R m d atifying F k,l C, T ) S) fo each k {1,..., m} and l {1,..., d}. 7

9 2.2 Mild olution to path-dependent ODE We how in thi ection a unique mild olution to the ODE 1.6), which belong to the delayed Cameon-Matin pace H 1 0, T ], R m ). To thi end, let u conide the geneal path-dependent odinay diffeential equation ẋt) = F t, x t ) fo t, T ], 2.3) whee F :, T ] S R m denote a non-anticipative poduct meauable map. Then fo each h H 1 0, T ], R d ) the choice F = b 1/2)ρ+σḣ, whee ρ i given by 1.5), yield the uppot chaacteizing ODE 1.6). A fo x S the map, T ] R m, t F t, x t ) may fail to be continuou, one may in geneal not expect to deive claical olution. So, we ecall the concept of a mild olution to 2.3), which i a path x S atifying T F, x ) d < and xt) = x) + F, x ) d fo all t, T ]. By definition, a mild olution x i abolutely continuou on, T ] and it become a claical olution if and only if the Boel meauable map, T ] R m, F, x ) i continuou. Let u intoduce the following egulaity condition, which ae atified unde the aumption of Theoem 1 fo the choice of F mentioned befoe. O.i) Thee exit a meauable function c F c F ) 2 d < and T F t, x) c F t) 1 + x + :, T ] 0, ) atifying T ẋ) d fo all t, T ) and x S that i abolutely continuou on, T ]. O.ii) Fo each n N thee i a meauable function λ F,n :, T ] 0, ) uch that T λ F,n) 2 d < and F t, x) F t, y) λ F,n t) x y H, fo all t, T ) and x, y H 1 0, T ], R m ) o that x H, y H, n. Unde the above gowth condition and Lipchitz moothne on bounded et, we obtain a unique mild olution that can be appoximated by a Picad iteation in the delayed Cameon-Matin nom H, given by 1.4). ) 8

10 Popoition 2. Let O.i) and O.ii) hold and ˆx C0, T ], R m ), then the ODE 2.3) admit a unique mild olution y F atifying y F ) = ˆx) fo all 0, ] and it hold that y F H 1 0, T ], R m ). Moeove, the equence x n ) n N0 in H 1 0, T ], R m ), ecuively defined via x 0 t) := ˆx t) and x n+1 t) := x 0 t) + t F, x n) d 2.4) fo all n N 0, convege in the delayed Cameon-Matin nom H, to y F. 2.3 Stong olution fo path-dependent SDE We tun to the deivation of a unique tong olution to 1.1), fo which a.e. path lie in C α 0, T ], R m ) fo any α 0, 1/2). We conide the tochatic diffeential equation with path-dependent coefficient dx t = Bt, X t ) dt + Σt, X t ) dw t fo t, T ], 2.5) whee B :, T ] S R m and Σ :, T ] S R m d ae two non-anticipative poduct meauable map. A tong olution to 2.5) i an F t ) t 0,T ] -adapted ight-continuou poce X : 0, T ] Ω R m with a.. continuou path atifying T B, X ) d + X t = X + T B, X ) d + Σ, X ) 2 d < Σ, X ) dw a.. and fo all t, T ] a.. Remak 3. The fact that we do not have to aume the uual condition i claified in Section 3.1 and iepective how B, y) and Σ, y) ae extended fo, T ] and any ight-continuou map y : 0, T ] R m that i not continuou, the above integal emain unchanged up to inditinguihability. We now tate the aumption on the coefficient, valid in the etting of Theoem 1 fo the choice B = b and Σ = σ. S.i) Thee ae a meauable function c B :, T ] 0, ) and a contant c Σ 0 uch that T c B) 2 d < and fo all t, x), T ) S, Bt, x) c B t)1 + x ) and Σt, x) c Σ 1 + x ). 9

11 S.ii) Thee ae α 0 0, 1/2), a meauable function λ B :, T ] 0, ) and a contant λ Σ 0 uch that T λ B) 2 d < and Bt, x) Bt, y) λ B t) x y α0,, Σt, x) Σt, y) λ Σ x y α0, fo all t, T ) and x, y C α 0 0, T ], R m ), whee α0, equal if α 0 = 0 and othewie i given by 1.3) when α i eplaced by α 0. Remak 4. If condition S.ii) hold, then it i alo tue if α 0 i eplaced by any α α 0, 1/2). Thu, it i tonget in the cae that α 0 = 0. Fo p 1 and α 0, 1] we let C α,p0, T ], R m ) denote the pace of all F t ) t 0,T ] -adapted ight-continuou pocee X : 0, T ] Ω R m atifying E X p α,] <, equipped with the intinic eminom C α,p0, T ], R m ) 0, ), X E X p α,]) 1/p, 2.6) which i complete, by Popoition 11. Moeove, if a equence n X) n N in thi linea pace convege with epect to the above eminom, then it alo convege in the delayed Hölde nom α, in pobability. Finally, we et C 0, T ], R m ) := p 1 C 0,p0, T ], R m ) and let C 1/2, 0, T ], R m ) denote the inteection of the pace C α,p0, T ], R m ) ove all p 1 and α 0, 1/2), which yield a completely peudometizable topological pace. Popoition 5. Aume S.i)-S.ii) and let ˆX C 0, T ], R m ). Then up to inditinguihability thee i a unique tong olution X to 2.5) uch that X = ˆX fo all 0, ] a.. we have that X C, 1/2 0, T ], R m ). Futhemoe, the equence n X) n N0 in 0, T ], R m ), ecuively given by 0 X t := ˆX t and C 1/2, n+1x t = 0 X t + t B, n X ) d + t Σ, n X ) dw 2.7) fo all n N 0, convege in the eminom 2.6) to X fo each p 2 and α 0, 1/2). In paticula, lim n P n X X α, ε) = 0 fo all ε > 0. Remak 6. Pathwie uniquene i hown in Lemma 30, equiing only the following Lipchitz condition on bounded et, which follow fom S.ii) in the tonget cae α 0 = 0: 10

12 S.iii) Fo each n N thee i a meauable function λ n :, T ] 0, ) atifying T λ n) 2 d < and Bt, x) Bt, y) + Σt, x) Σt, y) λ n t) x y fo all t, T ] and x, y S with x y n. 2.4 Chaacteization of the uppot in Hölde topology Section 2.2 and 2.3 povide the main agument to pove the fit two aetion of Theoem 1. Let u now decibe how we hall chaacteize the uppot 1.7). Fo n N let T n be a patition of, T ] that we wite in the fom T n = {t 0,n,..., t kn,n } fo ome k n N and t 0,n,..., t kn,n, T ] o that = t 0,n < < t kn,n = T and we denote it meh by T n = max i {0,...,kn 1}t i+1,n t i,n ). We aume that lim n T n = 0 and that the equence of patition i well-balanced in the ene of 11], that i, thee i c T 1 uch that T n c T inf i {0,...,k n 1} t i+1,n t i,n ) fo all n N. 2.8) Moeove, fo n N we define an F t ) t 0,T ] -adapted ight-continuou poce nw : 0, T ] Ω R d by etting n W t := W t fo t 0, t 1,n ), nw t := W ti 1,n + t t i,n ) W t i,n W ti 1,n t i+1,n t i,n 2.9) fo t t i,n, t i+1,n ) with i {1,..., k n 1} and n W T := W tkn 1,n. Then nw can be egaded a adapted linea intepolation of the d-dimenional F t ) t 0,T ] -Bownian motion W on, T ] and almot each of it path belong to H 1 0, T ], R d ). Thu, let u uppoe that the aumption and the fit two claim of Theoem 1 hold. To etablih uppp X 1 ) {x h h H 1 0, T ], R d )} in C α 0, T ], R m ) fo α 0, 1/2), we will jutify in Section 5.4 that it uffice to check that lim P x nw X α, ε) = 0 fo all ε > ) n By definition of a mild olution to 2.3), we ee fo each n N that x nw i a tong olution to the degeneate path-dependent SDE d n Y t = b 1/2)ρ)t, n Y t ) + σt, n Y t ) n Ẇ t ) dt fo t, T ] 11

13 with initial condition n Y = ˆx a.. Fo each h H 1 0, T ], R d ) and n N, we intoduce an a.. continuou local matingale h,n Z : 0, T ] Ω 0, ) by equiing that h,n Z = 1 a.. and h,nz t = exp ḣ) n Ẇ dw 1 ) t 2 ḣ) nẇ 2 d 2.11) fo all t, T ] a.. In fact, h,n Z i a matingale, a claified in Lemma 39. Hence, P h,n : F 0, 1] given by P h,n A) := E h,n Z T 1A] i a pobability meaue equivalent to P. By uing thi fact, we will how that the convee incluion in 1.7) follow once we have poven that lim P h,n X x h α, ε) = 0 fo all ε > ) n By Gianov theoem, the poce h,n W : 0, T ] Ω R d defined via h,nw t := W t t ḣ) n Ẇ d i a d-dimenional F t ) t 0,T ] -Bownian motion unde P h,n and X i a tong olution to the path-dependent SDE d n Y t = bt, n Y t ) + σt, n Y t )ḣt) nẇt) ) dt + σt, n Y t ) d h,n W t 2.13) fo t, T ] unde P h,n with initial condition n Y = ˆx a.. Hence, to pove 2.10) and 2.12) at the ame time, we conide the following geneal famewok. Namely, we let B :, T ] S R m and B H :, T ] S R m d be two non-anticipative poduct meauable map and B C 1,2, T ) S, R m d ). Then fo each n N we intoduce the path-dependent SDE d n Y t = Bt, n Y t ) + B H t, n Y t )ḣt) + Bt, ny t ) n Ẇ t ) dt + Σt, n Y t ) dw t fo t, T ], 2.14) whee Σ :, T ] S R m d i a non-anticipative poduct meauable map, a conideed in Section 2.3. In addition, we intoduce the path-dependent SDE dz t = B + R)t, Z t ) + B H t, Z t )ḣt)) dt + B + Σ)t, Z t ) dw t 2.15) fo t, T ], whee we equie the non-anticipative poduct meauable map R :, T ) S R m given coodinatewie by d R k, y) := x B k,l, y)1/2)b + Σ), y)e l. 2.16) l=1 12

14 In Theoem 8 below, we in paticula how that wheneve n Y and Z ae tong olution to 2.14) and 2.15), epectively, uch that n Y = Z = ˆx a.. fo each n N, then lim P ny Z α, ε) = 0 fo all ε > ) n Then the choice B = b 1/2)ρ, B H = 0, B = σ and Σ = 0 yield 2.10), ince R = 1/2)ρ in thi cae. Moeove, by chooing B = b, B H = σ, B = σ and Σ = σ intead, 2.12) follow. Since thee ae the two deied eult, we conide the following egulaity condition: C.i) B C 1,2, T ) S, R m d ) and thee ae c, η 0 and κ 0, 1) uch that fo all t, x), T ) S, Bt, x) + B H t, x) c1 + x κ ), m ) d 1/2 t Bt, x) + xx B k,l t, x) 2 c1 + x η ), k=1 l=1 m ) d 1/2 Bt, x) + x B k,l t, x) 2 + Σt, x) c. k=1 l=1 C.ii) B i Lipchitz continuou in x S, unifomly in t, T ), and B H, B, x B and Σ ae Lipchitz continuou with epect to d given by 2.1). C.iii) Thee i a meauable function b :, T ] R o that T and B, y) = Σ, y)b) fo all, y), T ) S. b) 2 d < Remak 7. Condition C.iii) allow u to pefom a change of meaue to get a unique tong olution to 2.14). Howeve, when deiving 2.17) in Section 5.1, 5.2 and 5.3 we meely aume that C.i) and C.ii) hold. Theoem 8. Let C.i)-C.iii) hold and h H 1 0, T ], R d ). Aume that ˆX C 0, T ], R m ) and n ˆX)n N i a equence in C 0, T ], R m ) atifying up E ˆX n 2p] < fo each p 1. n N Then the following thee aetion hold: i) Fo any n N thee i a unique tong olution n Y to 2.14) atifying ny = ˆX n a.. Moeove, up n N E n Y 2p α,] < fo all p 1 and α 0, 1/2). ii) Thee i a unique tong olution Z to 2.15) uch that Z = ˆX a.. and we have E Z 2p α,] < fo all p 1 and α 0, 1/2). 13

15 iii) Let α 0, 1/2) and lim n E ˆX n ˆX 2 ]/ T n 2α = 0, then it follow that lim E max ny tj,n Z tj,n 2] / T n 2α = ) n j {0,...,k n} In paticula, 2.17) hold, that i, n Y ) n N convege in the delayed Hölde nom α, in pobability to Z. 3 Convegence in pobability in Hölde nom 3.1 Hölde pace fo tochatic pocee Fo α 0, 1] let C α 0, T ], R m ) denote the linea pace of all R m -valued adapted ight-continuou pocee X atifying X C α 0, T ], R m ) a.., endowed with the peudometic C α 0, T ], R m ) C α 0, T ], R m ) 0, ), X, Y ) E X Y α, 1]. 3.1) We notice that a equence n X) n N in thi peudometic pace convege to ome X C α 0, T ], R m ) if and only if it convege to thi poce in the delayed Hölde nom α, in pobability. Put diffeently, n X X α, ) n N convege to zeo in pobability. Futhe, n X) n N i Cauchy if and only if it i Cauchy in the nom α, in pobability in the ene that lim up k n N: n k P n X k X α, ε) = 0 fo all ε > 0. Next, we et C 0, T ], R m ) := C 0 0, T ], R m ), which i the linea pace of all R m -valued adapted ight-continuou a.. continuou pocee. Depite the fact that we do not aume the uual condition, C 0, T ], R m ) i complete, which yield the following eult. Lemma 9. C α 0, T ], R m ) endowed with the metic 3.1) i complete. Poof. Let n X) n N be a Cauchy equence in C α 0, T ], R m ). By Lemma in 21], thee i X C 0, T ], R m ) to which n X) n N convege unifomly in pobability. Fo given ε, η > 0 thee i n 0 N uch that P up,t,t ]: t n X k X ) n X t k X t ) ε ) < η t α 2 2 fo all k, n N with k n n 0. We fix l N and et δ l := T )/l, then thee exit k l N uch that k l n 0 and P kl X X ε/4)δ α l ) < η/2. 14

16 Hence, P ) n X X ) n X t X t ) up > ε,t,t ]: t δ l t α < η fo all n N with n n 0. By the continuity of meaue, n X X α, ) n N convege in pobability to zeo. In paticula, X α, < a.. Fo p 1 we ecall that C,p0, α T ], R m ) denote the pace of all pocee X C α 0, T ], R m ) atifying E X p α,] <, endowed with the eminom 2.6). A equence n X) n N in thi pace i called p-fold unifomly integable if n X α, ) n N atifie thi popety in the uual ene. Lemma 10. Any equence n X) n N in C α,p0, T ], R m ) that i Cauchy with epect to the eminom 2.6) i p-fold unifomly integable. Poof. Let ε > 0, then thee i n 0 N o that E k X n X p α,] < ε/2 p fo all k, n N with k n n 0. A the andom vaiable Y := max n {1,...,n0 } n X α, i p-fold integable, we obtain that ]) up E n X p 1/p α, 1 A EY p 1A ]) 1/p + ε 1/p /2 n N fo all A F. Fit, by chooing A = Ω, thi give up n N E n X p α,] <. Secondly, by etting δ := ε/2 p, it follow that up n N E n X p α, 1 A] < ε fo all A F with EY p 1A] < δ. We conclude with the following convegence chaacteization. Popoition 11. A equence n X) n N in C,p0, α T ], R m ) convege in the eminom 2.6) if and only if it i p-fold unifomly integable and thee i X C α 0, T ], R m ) uch that lim P nx X α, ε) = 0 fo all ε > 0. n In the latte cae, we have E X p α,] < and lim n E n X X p α,] = 0. Moeove, C α,p0, T ], R m ) equipped with 2.6) i complete. Poof. By Lemma 9 and 10, it uffice to how the if-diection of the fit claim. To thi end, let ν n ) n N be a tictly inceaing equence in N uch that νn X X α, ) n N convege to zeo a.., then E X p α,] lim inf n E νn X p α, ] up E n X α,] p <, n N 15

17 by Fatou Lemma. Now let ε > 0, then thee exit δ > 0 and n 0 N uch that up n N E n X p α, 1 A] < ε/3) p fo each A F with P A) < δ and P n X X α, ε/3) < δ fo all n N with n n 0. Thu, E n X X p α, ]) 1/p E n X X p α, 1 { nx X α, ε/3} ]) 1/p + ε/3 < ε fo evey uch n N, ince imila eaoning a above give E X p α, 1 A] up n N E n X p α, 1 A] fo all A F. Thi complete the poof. 3.2 A geneal Kolmogoov-Chentov etimate In thi ection we eviit the poof of the Kolmogoov-Chentov Theoem to allow fo pocee that ae meely ight-continuou and to obtain a quantitative etimate of the Hölde nom. Let k α,q,p := 2 q+2p 2 q/2p) α 1) 2p 3.2) fo p 1/2, q > 0 and α 0, q/2p)) and note that lim α q/2p) k α,q,p =. Then we have the following eult. Popoition 12. Let H be a et of R m -valued ight-continuou pocee and, t, T ] be uch that < t. Aume that thee ae c 0 0, p 1/2 and q > 0 uch that up E U u U v 2p] c 0 u v 1+q 3.3) U H fo all u, v, t]. Then fo each α 0, q/2p)) it hold that up E U H up u,v,t]: u v ) 2p ] U u U v k u v α α,q,p c 0 t ) 1+q 2αp. Poof. Fo given n N 0 let D n be the n-th dyadic patition of, t], whoe point ae d i,n := + i2 n t ), whee i {0,..., 2 n }. We define n := {u, v) D n D n u v 2 n t )}, then it i eadily een that thee ae 2 n tuple u, v) n atifying u < v. Fo fixed U H we et V n := up u,v) n U u U v, then 3.3) give E ] Vn 2p E U u U v 2p] 2 nq c 0 t ) 1+q. 3.4) u,v) n: u<v 16

18 We now et D := n N 0 D n and let u, v D atify 0 < v u < 2 n t ) fo ome n N 0. Then fo each k N 0 thee ae unique i k, j k {1,..., 2 k } uch that d ik 1,k u < d ik,k, and d jk 1,k v < d jk,k fo v < t and d jk,k = t fo v = t, epectively. Since d ik,k) k N0 and d jk,k) k N0 ae two deceaing equence conveging to u and v, epectively, two telecoping um yield that U u U v = U din,n U djn,n + U U dik+1,k+1 d + ik U U,k djk+1,k+1 d. jk,k k=n k=n We note that eithe i n = j n o intead n 1, j n 2 and i n = j n 1, a 0 < v u < 2 n t ). In both cae, d in,n, d jn,n) n. Moeove, d ik,k, d ik+1,k+1), d jk,k, d jk+1,k+1) k+1 fo all k N 0, by contuction. So, U u U v V n + 2 V k+1 2 V k. 3.5) k=n k=n Next, pick u, v D uch that 0 < v u < t, then thee i a unique n N 0 uch that 2 n 1 t ) v u < 2 n t ). By 3.5), v u) α U u U v 2 1+α t ) α 2 αk V k becaue 2 αn 2 αk fo all k N 0 with k n. Clealy, u, v D atify v u = t if and only if u = and v = t. In thi cae, U U t /t ) α t ) α Z 0. Thu, we have hown that U u U v up u,v,t]: u v u v α 21+α t ) α k=n 2 αk V k, 3.6) k=0 a D i a countable dene et in, t] containing t and U i ight-continuou. Hence, 3.6), the tiangle inequality, monotone convegence and 3.4) yield that E up u,v,t]: u v ) 2p ]) 1 2p U u U v 2 1+α t ) α u v α k=0 2 1+α c 1 2p 0 t ) 1+q 2p α 2 αk E V 2p ]) 1 2p k k=0 2 α q 2p )k. The powe eie on the ight-hand ide convege abolutely to the invee of 1 2 α q/2p), ince α < q/2p). Fo thi eaon, the popoition follow. 17

19 3.3 Convegence along a equence of patition In thi ection, we tate a ufficient citeion fo a equence of pocee to convege in pobability in the delayed Hölde nom α, whee α 0, 1]. Fo thi pupoe, we equie the following etimate. Lemma 13. Let T be a patition of, T ] of the fom T = {t 0,..., t k } fo ome k N and t 0,..., t k, T ] uch that = t 0 < < t k = T. Then x) xt) up 2 max,t,t ]: t t α fo each map x :, T ] R m. up j {0,...,k 1} u,v t j,t j+1 ]: u v xt i ) xt j ) + max i,j {0,...,k}: i j t i t j α xu) xv) u v α Poof. Fit, aume that i, j {0,..., k 1} ae uch that i < j 1 and t i, t i+1 ), then x) xt j ) t α x) xt i+1) t i+1 α + xt i+1) xt j ) t i+1 t j α, becaue t > t i+1 t i+1 t j. Now uppoe that i, j {0,..., k 1} atify i < j, t i, t i+1 ) and t t j, t j+1 ). In thi cae, x) xt) t α x) xt i+1) t i+1 α + xt i+1) xt j ) t i+1 t j α + xt j) xt) t j t α, ince t > t i+1 t i+1 t j t j t. Now the aetion follow. Thi yield the befoe mentioned citeion. Popoition 14. Let n U) n N be a equence of R m -valued ight-continuou pocee and aume thee ae p, q > 0 o that fo each β 0, q/2p)) thee i c β 0 atifying P max up j {0,...,k n 1} u,v t j,n,t j+1,n ] ) n U u n U v λ c u v β β λ 2p 3.7) fo all n N and λ > 0. If n U ) n N and max j {0...,kn} n U tj,n / T n α ) n N convege in pobability to zeo, then o doe the equence n U α, ) n N fo all α 0, q/2p)). 18

20 Poof. Let n N and fix β α, q/2p)), then it follow that n U max up j {0,...,k n 1} t t j,n,t j+1,n ) n U t n U tj,n t t j,n β T n β + max j {0,...,k n 1} nu tj,n, ince n U t n U t n U tj,n / t t j,n β ) T n β + n U tj,n fo all j {0,..., k n 1} and t t j,n, t j+1,n ). Conequently, fom 3.7) we obtain that P n U ε ) c β 2/ε) 2p T n 2βp + P max nu tj,n ε/2 ) j {0,...,k n} fo all ε > 0, which diectly entail that n U ) n N convege in pobability to zeo. Next, fo fixed n N Lemma 13 give u that n U n U t up 2 max,t,t ]: t t α up j {0,...,k n 1} u,v t j,n,t j+1,n ]: u v n U ti,n n U tj,n + max i,j {0,...,k n}: i j t i,n t j,n. α n U u n U v u v α By uing the fact that u v β α T n β α and t i,n t j,n T n /c T fo all i, j {0,..., k n } with i j and u, v t j,n, t j+1,n ], we ee that P up,t,t ]: t ) n U n U t ε c t α β 4/ε) 2p T n β α)2p + P 2c α T max nu tj,n / T n α ε/2 j {0,...,k n} fo any ε > 0. A the tem on the ight-hand ide convege to zeo a n, the aetion i hown. Remak 15. If p 1/2 and thee i c 0 0 uch that E n U u n U v 2p ] c 0 u v 1+q fo all n N and u, v, T ], then Popoition 12 and Chebyhev inequality aue that condition 3.7) i atified. 3.4 Adapted linea intepolation of Bownian motion We tudy the equence n W ) n N of adapted linea intepolation of W given by 2.9) and fo which a.e. path lie in H 1 0, T ], R d ). To thi end, we intoduce the following notation. Fo given n N and t, T ), let i {0,..., k n 1} be uch that t t i,n, t i+1,n ), then we et t n := t i 1) 0,n, t n := t i,n and t n := t i+1,n. 3.8) ) 19

21 That i, t n i the pedeceo of t n with epect to T n, unle i = 0, and t n i the ucceo of t n. We alo et T n := t kn 1,n, T n := T and T n := T. In addition, we ue the following abbeviation: t i,n := t i,n t i 1,n ) and W ti,n := W ti,n W ti 1,n 3.9) fo each i {1,..., k n 1}. Afte thee pepaation, let u begin with a geneal integal epeentation. Lemma 16. Let n N and, t, T ] be uch that < t. Then each R m d -valued pogeively meauable quae-integable poce Y atifie Y un d n W u = t t i+1,n i,n t i 1,n Y un dw u a.., wheneve i {1,..., k n 1} i uch that, t t i,n, t i+1,n ], and Y un d n W u = t i+1,n + t i+1,n j 1,n t i,n i,n t i 1,n Y un dw u Y un dw u + t t j,n t j+1,n j,n t j 1,n Y un dw u a.., if i, j {1,..., k n 1} atify i < j, t i,n, t i+1,n ] and t t j,n, t j+1,n ]. In paticula, fo all j {1,..., k n } we have j,n Y un d n W u = j 1,n Y un dw u a.. Poof. A n W u = 0 fo each u, t 1,n ], the econd claim follow fom the fit, by chooing = t 1,n and t = t j,n fo j {1,..., k n }. To check the fit claim, uppoe initially that, t t i,n, t i+1,n ] fo ome i {1,..., k n 1}, then Y un d n W u = t t i+1,n Y ti 1,n W ti,n = t t i+1,n i,n t i 1,n Y un dw u a.. Now aume intead that thee ae i, j {1,..., k n 1} uch that i < j, t i,n, t i+1,n ] and t t j,n, t j+1,n ]. Then the a.. decompoition Y un d n W u = i+1,n Y un d n W u + j 1 h=i+1 h+1,n t h,n Y un d n W u + and the cae conideed above imply the aeted epeentation. t j,n Y un d n W u 20

22 Let u ecall that fo each p 1 thee i a contant w p > 0 depending only on p uch that fo evey R m d -valued pogeively meauable poce Y it hold that v 2p ] ) t p ] E up Y u dw u w p E Y u 2 du M) v,t] fo all, t, T ] with t. In fact, w p = 2 p p 3p /p 1/2) p and the dimenion m and d do not alte w p. We deive a coeponding eult fo the equence n W ) n N of adapted linea intepolation of W. Popoition 17. Fo each p 1 thee i a contant ŵ p > 0 uch that evey R m d -valued pogeively meauable poce Y atifie v 2p ] E up Y un d n W u ŵ p max E Y tj,n 2p] t ) p 3.10) j {0,...,k n}: t j,n n,t n ] v,t] fo each n N and, t, T ] with t. Poof. Fit, if t t 1,n, then v Y u n d n W u = 0 fo each v, t] a.. Fo < t 1,n and t t 1,n we have v Y un d n W u = v t 1,n Y un d n W u fo all v t 1,n, t] a.. Thu, let u ue Lemma 16 and aume that, t t i,n, t i+1,n ] fo ome i {1,..., k n 1}, then M) yield that v 2p ] t )2p E up Y un d n W u = t i+1,n ) E ti,n 2p ] Y 2p un dw u t i 1,n v,t] w p c p TE Y ti 1,n 2p] t ) p, whee c T i the contant appeaing in 2.8). Next, uppoe that thee ae i, j {1,..., k n 1} uch that i < j, t i,n, t i+1,n ] and t t j,n, t j+1,n ]. Then v max Y v,t] un d n W u t i+1,n i,n Y t i+1,n un dw u t i 1,n h,n + max Y h {i+1,...,j 1} un dw u + t t j,n tj,n Y un dw u a.. t j 1,n t i,n t j+1,n Thi i due to Lemma 16, which aet that the poce, t] Ω R m, v, ω) v Y u n ω) d n W u ω) i piecewie linea. Hence, we obtain that v 2p ] E up Y un d n W u ŵ p max E Y th,n 2p] t ) p h {i 1,...,j 1} v,t] fo ŵ p := 3 2p 1 w p c p T, which yield the claim. 21

23 Lemma 18. Fo each p, q 1 thee exit a contant ŵ p,q > 0 atifying ) t p ] E n Ẇ u q du ŵ p,q T n qp/2 t ) p 3.11) fo all, t, T ] with t and n N. Poof. Clealy, if t t 1,n, then nẇu q du = 0. Fo < t 1,n and t t 1,n we have n Ẇ u q du = n Ẇ u q du. t 1,n So, let now, t t i,n, t i+1,n ] fo ome i {1,..., k n 1} and Z be an R d -valued andom vecto uch that Z N 0, Id), then ) t p ] E n Ẇ u q du = E Z qp] t i,n ) qp/2 t i+1,n ) qp t )p c T n qp/2 t ) p, whee ŵ p,q := E Z qp ]c qp T and c T i the contant in 2.8). Next, let intead i, j {1,..., k n 1} be uch that i < j, t i,n, t i+1,n ] and t t j,n, t j+1,n ]. Then p ]) 1/p j ) E n Ẇ u du) q t th+1,n p ]) 1/p E n Ẇ u q du h=i t h,n ŵp,q 1/p T n q/2 t ), by what we have jut hown. Theefoe, the claim hold. 3.5 Auxiliay convegence eult Lemma 19. Let n U) n N be a equence of non-negative meauable pocee fo which thee ae p > 1 and c p > 0 uch that E n U 2p ] c p T n 2p fo each, T ) and n N. Then thee i c 1 > 0 atifying ) T 2 ] E nu n Ẇ d c 1 T n fo all n N. Poof. Let q > 1 be uch that 1/p+1/q = 1, then Lemma 18 give a contant ŵ q,2 > 0 that i independent of n uch that ) T q ] E n Ẇ 2 d ŵ q,2 T n q T ) q. 22

24 Thu, we define c 1,1 := ŵ 1/q q,2 T ) and c 1 := c 1/p p T )c 1,1, then it follow fom Cauchy-Schwaz and Hölde inequalitie that ) T 2 ] T p ]) 1/p E nu n Ẇ d E nu d) 2 c 1,1 T n 1 c 1 T n. Lemma 20. Let n N and fo evey ight-continuou map x : 0, T ] R m et L n x)t) := x t) fo all t 0, t 1,n ), L n x)t) := xt i 1,n ) + t t i,n ) xt i,n) xt i 1,n ) t i+1,n fo t t i,n, t i+1,n ) with i {1,..., k n 1} and L n x)t ) := xt kn 1,n). Then it hold that L n x) t x max j {1,...,kn 1}: t j,n t xt j,n ) and L n x) t x t 2 fo each t t 1,n, T ]. max up j {0,...,k n 1}: t j,n t t j,n,t j+1,n ] x t ) xt j 1) 0,n ) Poof. Fix t 1,n, t] and let i {1,..., k n 1} be o that t i,n, t i+1,n ], then L n x)) xt i 1,n ) xt i,n ), ince L n i linea on t i,n, t i+1,n ]. In addition, we immediately obtain that L n x)) x) x) xt i 1,n ) + xt i,n ) xt i 1,n ) and the aetion follow. 2 up v t i,n,t i+1,n ] x t v) xt i 1,n ) Lemma 21. Let n U) n N denote a equence of R m -valued ight-continuou pocee fo which thee ae c 0 0, p 1/2 and q > 0 uch that E n U n U t 2p] c 0 t 1+q fo all n N and, t, T ]. Then fo evey α 0, q/2p)), lim E n U L n n U) 2p] / T n 2αp = 0. n Poof. Fix β α, q/2p)). Fom Lemma 20 we obtain that n U L n n U) 2 1+β T n β 23 n U n U t up,t,t ]: t t β

25 fo evey n N. Conequently, Popoition 12 implie that the contant c β := 2 21+β)p k β,q,p c 0 T ) 1+q 2βp atifie E n U L n n U) 2p] c β T n 2βp fo all n N, whee k β,q,p i given by 3.2) when α i eplaced by β. Now the claim follow, ince α < β. Lemma 22. Let G :, T ] S R m be d -Lipchitz continuou, n U) n N be a equence in C 0, T ], R m ) and c 0 0 be uch that Gt, x) c x ) and E n U 2] + E n U n U t 2] / t c E n U 2]) fo all n N,, t, T ] with t and x S. Then thee i c 1 > 0 atifying fo each n N, ) tj,n E max G j {0,...,k n} n, n U n 2 ] n ) 1 d n c 1 T n 1 + E n U 2]). Poof. We aume that E n U 2 ] <, a othewie thee i nothing to how. By decompoing the integal, we can ewite that j,n G n, n U n ) n n d = j 1,n G n, n U n ) d fo each j {1,..., k n }. Thu, let λ 0 be a Lipchitz contant fo G, then tj 1,n 2 ] E max G n, n U n ) G n, n U n ) d j {1,...,k n} c 1,1 T n 1 + E n U 2]) with c 1,1 := 2λ 2 T ) c 0 ). In addition, we etimate that j,n 2 ] E max G j {1,...,k n} n, n U n ) d c t j 1,n 2,2 T n E n U 2]), whee c 2,2 := 2c c 0 ). So, the contant c 1 := 2c 1,1 + c 1,2 T )) yield the claim. Popoition 23. Let G :, T ] S R m d be d -Lipchitz continuou and n U) n N be a equence in C 0, T ], R m ). Suppoe thee ae c 0 0 and p > 1 uch that Gt, x) c x ) and ] E n U 2p] + E n U n U t 2p c t p 0 24

26 fo each n N,, t, T ] uch that t and x S. Then fo evey α 0, 1/2 1/2p)), lim T n 2α E n j,n max j {0,...,k n} 2 ] G n, n U n ) dn W W ) = 0. Poof. We fix n N and once again decompoe the integal to get that j,n G n, n U n ) dn W = j 1,n G n, n U n ) dw a.. fo all j {1,..., k n }. Let λ 0 denote a Lipchitz contant fo G, then E j 1,n max j {0,...,k n} G n, n U n ) G n, n U n ) dw 2 ] T 2w 1 λ 2 n n ) + E n U n n U ] n 2 d c 1 T n with c 1 := 2w 1 λ 2 T )1+c 1/p 0 ), whee w 1 i the contant atifying M) fo p = 1. Next, we let n M C 0, T ], R m ) be a quae-integable matingale atifying nm t = G n, n U n ) dw fo all t, T ] a.., then n M tj,n n M tj 1,n each j {1,..., k n }. Futhemoe, = j,n t j 1,n G n, n U n ) dw a.. fo E n M n M t 2p] 2 2p 1 w p c 2p 0 t ) p 1 + E n U 2p]) c 2 t ) p fo all, t, T ] with t, whee c 2 := 2 2p 1 w p c 2p c 0 ). Thu, let β α, 1/2 1/2p)), then it follow fom Popoition 12 that E up,t,t ]: t ) 2p ] n M n M t k t β β,p 1,p c 2 T ) 1 2β)p, ince p > 1, by aumption. Finally, we et c β := k β,p 1,p c 2 ) 1/p T ) 1 2β, then j,n 2 ] E max G j {1,...,k n} n, n U n ) dw c t j 1,n β T n 2β, by Hölde inequality, and the aetion follow. 25

27 4 Path-dependent ODE and SDE: poof In thi ection, we give the poof fo the exitence and uniquene of mild olution to path-dependent ODE in Section 2.2 and the exitence and uniquene of tong olution to path-dependent SDE in Section Poof of Popoition 2 We fit deive a global etimate fo any mild olution. Thi allow u to ue O.ii), the Lipchitz condition on bounded et, to deive exitence and uniquene eult. Lemma 24. Unde O.i), thee i c H > 0 depending only on T uch that any mild olution x to the ODE 2.3) atifie fo all t, T ], x t 2 H, c H e c t ) t H c F ) 2 d x 2 + c F ) 2 d. 4.1) Poof. By etimating x t + ẋ) d fo given t, T ], it follow eadily fom Gonwall inequality that x t + ) ẋ) d e 2 t t c F ) d x + 2 c F ) d. Moeove, fo c 1 := 2 2 T + 1) we have x t 2 H, 2 x 2 + c 1 c F ) x + ẋu) du) 2 d. 4.2) Thu, we et c H := 2e 2 c 1, then fom 2 c F ) d 1 + T ) c F ) 2 d we infe that x t + 2 ẋ) d) c H e c t ) t H c F ) 2 d x 2 + c F ) 2 d. The claim follow fom 4.2), the fundamental theoem of calculu fo Riemann- Stieltje integal and the tanitivity of abolutely continuou meaue. We now how uniquene of mild olution, which implie uniquene fo claical olution. 26

28 Lemma 25. Aume that O.i) and O.ii) hold, then any two mild olution x and y to the ODE 2.3) that atify x = y mut coincide. Poof. By Lemma 24, thee i n N uch that x H, y H, n. Thu, x t y t 2 H, 2T + 1) λ 2 F,n) x y 2 H, d fo all t, T ]. Gonwall inequality implie that x = y. Poof of Popoition 2. A the uniquene claim follow fom Lemma 25, we diectly tun to the exitence aetion. To thi end, let H be the et of all x H 1 0, T ], R m ) atifying x) = ˆx) fo evey 0, ] and the etimate 4.1), whee c H i choen lagely enough o that c H 2 4 T + 1) ) By Lemma 24, a map x S i a mild olution to the ODE 2.3) uch that x) = ˆx) fo all 0, ] if and only if x H and it i a fixed-point of the opeato Ψ : H H 1 0, T ], R m ) given by Ψy)t) := x 0 t) + t F, y ) d. We emak that condition 4.3) aue that Ψ map H into itelf. Indeed, thi follow by ineting 4.1) into the inequality Ψx) t 2 H, c H x c H c F ) x 2 H,) d, valid fo all x H and t, T ]. A x 0 H and x n = Ψx n 1 ) fo each n N, by definition 2.4), we now know that x n ) n N0 i a equence in H. Next, let u chooe l N atifying x H, l fo all x H and et c 1 := 2T + 1). Then we obtain that Ψx) t Ψy) t 2 H, c 1 λ F,l ) 2 x y 2 H, d fo each x, y H and t, T ], which in paticula how that Ψ mut be H, -Lipchitz continuou. Moeove, it follow inductively that x t n+1 x t n 2 H, δ2 ) t n c 1 λ F,l ) 2 d n! fo evey n N 0, whee we have et δ := Ψx 0 ) x 0 H,. Hence, the tiangle inequality give u that x n x k H, δ n 1 i=k ) 1/2 1 ) T i/2 c 1 λ F,l ) 2 d i! 27

29 fo all k, n N 0 with k < n. Now the atio tet yield that the eie i=0 1/i!) 1/2 x i/2 convege abolutely fo all x 0. Hence, we have hown that lim k up n N: n k x n x k H, = 0. A H i cloed with epect to the complete nom H,, thee exit a unique map y F H uch that lim n x n y F H, = 0. Lipchitz continuity of Ψ implie lim n x n+1 Ψy F ) H, = 0. Fo thi eaon, y F = Ψy F ) and the popoition i poven. 4.2 Poof of Popoition 5 Lemma 26. Let X be an R m -valued adapted ight-continuou poce and B R m be cloed, then τ := inf{t 0, T ] {X 0, t]} B } i a topping time atifying τ = inf{t 0, T ] X t B} on {X S}. Poof. Fit, we check that {τ t} = {{X 0, t]} B } fo fixed t 0, T ]. To thi end, it uffice to how that if t < T and ω Ω atifie τω) = t, then {X ω) 0, t]} B. In thi cae, fo each n N thee ae n 0, t + T t)/n) and y n B atifying y n X n ω) < 1/n. So, we chooe a tictly inceaing equence ν n ) n N in N uch that νn ) n N convege to ome 0, t], then it follow that X ω) = lim n X νn ω) B, which yield the intemediate claim. Next, we et B n := {x B x n} fo all n N and ue the notation ditx, C) = inf y C x y fo all x R m and C R m. Let t 0, T ] and D be a countable dene et in 0, t] containing t, then {τ t} = {inf ditx, B n ) = 0} = {ditx, B n ) < 1/k} F t, D n N n N k N D by the above epeentation of {τ t}. Finally, if ω Ω atifie Xω) S, then {X ω) 0, t]} i compact and in paticula cloed. Fo thi eaon, τω) < inf{t 0, T ] X t ω) B} would violate the definition of τω). Example 27. Let X C 0, T ], R m ) and n N, then the above lemma give a topping time τ n uch that τ n = inf{t 0, T ] X t n} a.. Thi enue that X t τn X n a.. fo all t, T ], ince we have that X t τn n a.. on { X n} and τ n = a.. on { X > n}. 28

30 In thi ection, wheneve p 1 and condition S.i) i atified, we et ) T p m p := c B ) 2 d + c 2p Σ w p, whee w p i the contant appeaing in M). Lemma 28. Unde S.i), fo each p > 2 and α 0, 1/2 1/p) thee i c α,p > 0 depending only on α, p and T uch that any tong olution X to 2.5) atifie fo all t, T ]. E X t 2p α,] cα,p e cα,pmpt ) E X 2p] + m p t ) ) 4.4) Poof. Aume that E X 2p ] < and let n N. Then Example 27 yield a topping time τ n uch that X τn X n a.. Fit, X t τn X t τn + B, X v τn ) d + up Σu, X u ) dw u a.. v,t] fo fixed t, T ]. Thu, fom Jenen and Cauchy-Schwaz inequality we obtain that E X t τn 2p] 3 2p 1 E X 2p] + 6 2p 1 m p t ) p E X τn 2p] d. Moeove, a imila computation how that E X τn u X τn v 2p ] 4 2p 1 m p v u) p E X τn 2p] d fo all u, v, t] with u < v. Theefoe, Popoition 12 yield that ) 2p ] Xu τn Xv τn E u v α up u,v,t]: u v k α,p 2,p 4 2p 1 m p t ) p 1 2αp 1 + E X τn 2p] d, whee the contant k α,p 2,p i given by 3.2) fo q = p 2. Thu, E ] X t τn 2 α, /cα,p E X 2p] + m p 1 + E X τn 2p] d fo c α,p := 12 2p k α,p 2,p )T + 1) p 1. By Gonwall inequality and Fatou lemma, E X t α,] 2p lim inf E X t τn α,] 2p cα,p e cα,pmpt ) E X 2p] +m p t ) ), n which i the claim. 29

31 Remak 29. If p 1 and α 0, 1/2) ae uch that α < 1/2 1/p fail, then, unde S.i), we till have that E X t 2p α,] E X t 2q α,]) p/q < fo any q > p uch that α < 1/2 1/q, by Hölde inequality. Lemma 30. Unde S.iii), pathwie uniquene hold fo 2.5). Poof. Let X and X be two weak olution to 2.5) defined on a common filteed pobability pace Ω, F, F t ) t 0,T ], P ) on which thee i a tandad d-dimenional F t ) t 0,T ] -Bownian motion W uch that X = X a.. We fix n N, then it follow fom Example 27 that thee i a topping time τ n uch that τ n = inf{t 0, T ] X t n o X t n} a.. Clealy, τn X t τn X t τn B, X ) B, X ) d v τn + up Σu, X u ) Σu, X u ) d W u v,t] fo given t, T ]. We et c 1 := T ) + w 1 ), whee w 1 i the contant in M) fo p = 1, then Ẽ X t τn X t τn 2] c 1 λ n ) 2 Ẽ X τn X τn 2] d. So, X τn τn = X a.., by Gonwall inequality. A τ n τ n+1 a.. fo all n N and up n N τ n = a.., we get that X t = lim n Xt τn = lim n n Xτ t = X t a.. fo all t, T ]. Right-continuity implie that X = X a.. Poof of Popoition 5. We define H be the et of all X C 0, T ], R m ) atifying X = ˆX fo all 0, ] a.. and the etimate 4.4) fo any p > 2 and α 0, 1/2 1/p), whee the contant c α,p i choen lagely enough o that c α,p 12 2p k α,p 2,p )T + 1) p ) By Lemma 28 and Remak 29, we have that H C, 1/2 0, T ], R m ) and a poce X C 0, T ], R m ) i a olution to 2.5) atifying X = ˆX fo all 0, ] a.. if and only if X H and it i a fixed point of the opeato Ψ : H C 0, T ], R m ) pecified by equiing that ΨY ) t = 0 X t + t B, Y ) d + t Σ, Y ) dw. fo all t 0, T ] a.. We te the fact that, due to Popoition 12 and condition 4.5), fo evey X H, p > 2 and α 0, 1/2 1/p) it follow that E ] ΨX) t 2p α, cα,p E 0 X 2p] + c α,p m p 1 + E X 2p] d 30 a..

32 fo all t, T ]. Thu, ΨH ) H follow fom plugging 4.4) into the above inequality. Since 0 X H and n X = Ψ n 1 X) a.. fo all n N, by 2.7), we have hown that n X) n N0 i a equence in H. Next, chooe p > 2 and α 0, 1/2) uch that α 0 α < 1/2 1/p, whee α 0 i the contant in the Lipchitz condition S.ii). Futhe, we et l p := T λ B) 2 d) p + λ 2p Σ w p, then E ΨX) t ΨY ) t 2p] 2 2p 1 l p t ) p 1 E X Y 2p α 0,] d fo all given X, Y H and t, T ]. Afte applying Popoition 12 and uing x α0, T + 1) α α 0 x α, fo all x C α 0, T ], R m ), we get E ΨX) t ΨY ) t 2p α, ] cα,p l p E X Y 2p α,] d with c α,p := 4 2p k α,p 2,p )T + 1) 2p 1. Hence, Gonwall inequality entail that thee i at mot a unique olution X to 2.5) atifying X = ˆX fo all 0, ] a.. We alo infe fom the above inequality that Ψ i Lipchitz continuou with epect to the eminom 2.6), whee p i eplaced by 2p. In addition, E ] n+1 X t n X t 2p δ 2p α, n! c α,pl p ) n t ) n fo each n N 0, by induction with δ := E Ψ 0 X) 0 X 2p α,]) 1/2p). Hence, the tiangle inequality give ]) E n X k X 2p 1 2p α, δ n 1 i=k ) 1 1 2p cα,p l p ) i i 2p T ) 2p i! fo each k, n N 0 with k < n. The atio tet implie that the eie i=0 1/i!) 1/2p) x i/2p) convege abolutely fo each x 0. So, lim up k n N: n k E n X k X 2p α,] = 0. Due to Popoition 11, becaue H i cloed with epect to the complete eminom 2.6), whee p i eplaced by 2p, thee exit a poce X H that i unique up to inditinguihability uch that lim E n X X α,] 2p = ) n 31

33 Lipchitz continuity of Ψ implie that lim n E n+1 X ΨX) 2p α,] = 0. Fo thi eaon, X = ΨX) a.. Finally, aume p 1 and α 0, 1/2) ae uch that α 0 α < 1/2 1/p fail. If α < α 0, then E ] n X X 2p α, T + 1) α 0 α)2p E ] n X X 2p α 0, fo all n N, which implie 4.6). Fo α 1/2 1/p we take q > p o that α < 1/2 1/q and ue that E n X X 2p α,] E n X X 2q α,]) p/q fo all n N. A thi alo give 4.6), the poof i complete. 5 Poof of main eult 5.1 Decompoition into emainde tem Popoition 31. Let C.i) hold, h H 1 0, T ], R d ) and B be d -Lipchitz continuou. Then fo each p 1 thee i c p > 0 uch that any n N and any tong olution n Y to 2.14) atify E n Y 2p] + E n Y n Y t 2p] / t p c p 1 + E n Y 2p]) 5.1) fo all, t, T ] with t. Poof. We let l N and ue Example 27 to define a topping time τ l,n uch that n Y τ l,n n Y l a.. Futhe, we etimate that τl,n n Y τ l,n n Y t τ l,n Bu, n Y u u ) + B H u, n Y )ḣu) du v τl,n + up Bu, v,t] n Y u ) n Ẇ u du v τl,n + up Σu, n Y u ) dw u a.. v,t] fo fixed, t, T ] with t. Thu, the tiangle inequality and the inequalitie of Cauchy-Schwaz and Jenen yield that E n Y τ l,n n Y t τ l,n 2p]) 1 2p + c p,1 t ) p E n Y u τ l,n 2κp] du E v τl,n up v,t] Bu, n Y u ) n Ẇ u du 32 ) 1 2p 2p ]) 1 2p, 5.2)

34 whee we have et c p,1 := 6c) 2p T ) p + T ḣu) 2 du) p + w p ) and w p i the contant appeaing in M). Since κ < 1, we can pick γ 1, κ 1 ), then Lemma 18 povide a contant c p,2 > 0 uch that 3.11) hold when q and p ae eplaced by 2 and p/1 γκ), epectively. Thi yield that ) t τl,n ) 2p ] E Bu, n Y u ) Bu n, n Y u n ) nẇu du ) t 2λ) 2p /2t ) p 1 u u n ) p du c 1 γκ p,2 T n p t ) p t ) t p ] + 2λ) 2p /2t ) p 1 E n Y u τ l,n n Y u n τ l,n 2p n Ẇ v 2 dv du c p,3 t ) p T n p E n Y u τ l,n n Y u n τ l,n 2p γκ ]) γκ du, by the inequalitie of Cauchy-Schwaz, Jenen and Hölde, whee λ 0 denote a Lipchitz contant fo B and c p,3 := 2 3p λ 2p T ) p c 1 γκ p,2. Note hee that the choice of c p,2 entail that E n Ẇ v 2 dv ) p 1 γκ ]) 1 γκ c 1 γκ p,2 T n p t ) p. Next, let c p,4 > 0 be a contant atifying 3.11) when q and p ae eplaced by 2 and p/γ 1)κ), epectively. Then Cauchy-Schwaz, Jenen and Hölde inequality imply that u τl,n E Bv, n Y v ) n Ẇ v dv u n 2c) 2p γκ 2 1 u u n ) p γκ 1 u ) 2p ] γκ u n 1 dv + 2c) 2p γκ 2 1 u u n ) p γκ 1 u u n E c p,5 T n p γκ ) 1 + E n Y u τ l,n 2p]) 1 γ ) γ 1 γ cp,4 T n p γκ u un ) p n Y v τ l,n 2p γ u γκ u n n Ẇ w 2 dw ) p γκ ] dv fo any given u, T ], whee we have et c p,5 := 4c) 2p)/γκ) c γ 1)/γ p,4, ince by the choice of c p,4 we can utilize that E u u n n Ẇ v 2 dv ) p ]) γ 1 γ 1)κ γ 33 γ 1 γ cp,4 T n p γκ u un ) p γκ.