Stochastic Partial Differential Equations driven by Poisson type noise

Transcription

1 Univerität Bielefeld Fakultät für Mathematik Dilomarbeit Stochatic Partial Differential Equation driven by Poion tye noie Reidar Janen 19. Dezember 13 Betreuer Prof. Dr. M. Röckner

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3 Content Introduction 5 1. Stochatic Integration with reect to Poion Procee Poion Random Meaure and Poion Point Procee Stochatic Integration with reect to a Poion Point Proce Proertie of the Poion integral Main Theorem Setting and Aumtion Formulation of the Theorem Proof of the Main Theorem Exitence Finite dimenional equation Contruction of the infinite dimenional Solution Proof of the Main Theorem Uniquene Alication to Examle Semilinear tochatic equation Examle Quai-linear tochatic equation: -Lalacian Examle Aendix 95 A. Sulement B. Inequalitie C. Inequalitie on ilbert ace D. Tool on rocee E. Micellaneou tool F. Some imortant embedding and interolation Bibliograhy 16 3

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5 Introduction In thi diloma thei we are going to how exitence and uniquene of olution to tochatic artial differential equation on a Gelfand trile driven by Poion tye noie with locally monotone coefficient of the form X t = A, X d + B, X dw + f, X, z µ dz, d, X = X, 1 on a finite time horizon, where W i a cylindrical Wiener roce and µ i a comenated Poion random meaure. Stochatic artial differential equation with jum tye noie, uch a Lévy tye ertubation or Poion tye noie, lay an imortant role for modelling real world roblem. In hyical reearch, rik modelling and otion ricing in finance, genetic and even climate baed reearch thee equation emerge to a greater extent, ince a Lévy ertubation or the additional Poion jum term offer more uitable modelling abilitie than tochatic artial differential equation driven by a Wiener roce olely, cf. A4, NØP9, IP9, NR13 or XFL13. Therefore, it hould come to no urrie that in the lat few year there ha been a high reearch interet in mathematic concerning exitence and uniquene of olution to tochatic artial differential equation abbr.: SPDE driven by dicontinuou jum term, eecially driven by Lévy tye noie. For examle one can have a look in LR4, Kno5, P7, R7, A9, NØP9, MPR1, Pr or BL11 and the reference therein to get a rough overview. Thi reearch with reect to jum tye noie extend even to tochatic artial differential equation on earable Banach ace, cf. R6, and ha been recently carried out for multi-valued ma, ee Ste1 and LS14. Earlier, SPDE driven by general dicontinuou martingale had been tudied by Gyöngy and Krylov already, ee GK8, GK8 and Gy. Tyical examle to equation of tye-1 are tochatic Burger equation and the tochatic -Lalace equation. The reult reented in Theorem..1 i baed uon the aer of Brzeźniak, Liu and hu BL11 and tate, that under certain condition, uch a local monontonicity and coercivity, equation 1 ha a unique trong olution in the ene of Defintion.1.1. Due to the Lévy Itô decomoition, the cla of SPDE driven by Lévy tye noie can be reduced to the cla of SPDE, where the tochatic ertubation term i a um of a Wiener roce and a comenated Poion random meaure a in 1, cf. Section D.1 in Ste1 or Section 9 in NØP9. 5

6 Introduction More reciely, a tochatic artial differential equation of the tye X t = A, X d + σ, X dl, where L i a general Lévy roce, can be written in the form of 1. owever, contrary to the reult in BL11, we do not involve big jum in our equation, which would caue the aearance of and additional ummand in 1 driven by a general Poion random meaure. The variational framework wa commenly ued ee e.g. PR7 to how exitence and uniquene to SPDE driven by a cylindrical Wiener roce, i.e. f in 1, under the aumtion that A and B are monotone oerator and that A and B fulfill a coercivity condition. In LR1 and LR14 thi reult wa imroved by auming that the oerator A and B are only locally monotone. In BL11 thi aroach led to exitence and uniquene for 1 under the further aumtion, that alo f i locally monotone. owever, there i no need for f to fulfill a coercivity condition, too. A further, recently ublihed generalization i the ue of a generalized coercivity condition on A and B, cf. LR13, in cae f to handle the tamed 3D-Navier-Stoke equation. Since thi thei i baed uon BL11, we do not coe with a generalized coercivity condition here, but ue ome minor but imortant change which are inired from LR14 to imrove the aumtion made in BL11 and LR1. It i imortant to make note of our diviion of uniquene and exitence of olution to 1 in Theorem..1 deending on the given aumtion, becaue the claimed uniquene and even exitence reult in BL11 doe not follow directly in general from the aumtion made therein, cf. Remark... ence thi work can be undertood a an extenion to PR7 and LR1 with reect to the comenated Poion random meaure-term and cover all the reult therein. Moreover, one hould not loe track of the fact that in cae B thi thei rovide a tool to handle SPDE of ure jum tye ometime called ure Lévy jum tye a well. The intention of thi work i to rove exitence and uniquene of olution to 1 under corrected and weakened aumtion in a comrehenible way and in all detail. For a dicuion on the aumtion, we refer to Remark... Furthermore thi work will rovide ome alication roved in all detail, too. Although thi thei i meant to be elf-contained, the reader i required to have knowledge of tochatic integration in ilbert ace a well a knowledge of cylindrical Wiener rocee. In thi thei we mainly tick to the notation of Section and 3 in PR7. Sobolev embedding are ued frequently in Chater 3, but ummarized in Aendix F. Neverthele, it i recommended to know about functional analyi and weak convergence. Let u mention Alt6 and Bre1 a reference in thi area. Let u briefly outline the tructure of thi thei. In Chater 1 we will introduce Poion oint rocee and Poion random meaure. Briefly we will recall all neceary fundamental of the tochatic integration with reect to Poion oint rocee. Chater contain the main reult of thi thei and it roof. After introducing the variational framework, we define what i meant by a olution to 1 and otulate the 6

7 main aumtion cf. condition A1 A4. They lead to Theorem..1 and Remark.. in which we dicu the difference between our aumtion and the familiar reult in BL11 and other. Afterward we will give a hort outline of the roof and finally rove exitence of olution to 1 and uniquene. In the lat chater of thi thei, Chater 3, Theorem..1 i alied to emilinear and quai-linear tochatic equation driven by Poion tye noie. In the firt cae one can think of a tochatic Burger tye equation. The econd cae will be the -Lalace equation. Following the intention of thi work, the verification of all aumtion made in Chater will be done in all detail. In ection A F of the aendix we will reent all auxiliary reult needed in Chater and 3 for comletene, in articular thoe, that are miing or claimed, but not roved in BL11. I would like to thank my uervior Prof. Dr. Michael Röckner for leading me to the field of tochatic artial differential equation and hi contant uort in the at year. Secial thank are given to Dr. Simon Michel for hi helful comment. Finally, I am very grateful for the uort of my family and my better and wore half, Jule. 7

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9 1. Stochatic Integration with reect to Poion Procee In thi chater we will introduce the Poion random meaure and the Poion oint roce. Afterward we will etablih the fundament on tochatic integration with reect to a Poion oint roce. Our main reference are Kno5 and IW Poion Random Meaure and Poion Point Procee Let Ω, F, P be a comlete robability ace. Let S, S be a meaurable ace and let M N S denote the et of N = N {+ }-valued meaure on S, S. We write B M N S to denote the mallet σ-field on M N S uch that all maing j B : M N S µ µ B N for B S are meaurable, i.e. B M N S = σ M N S µ µ B B S Definition. A ma µ: Ω S N i called N-valued random meaure, if i µ ω, M N S for each ω Ω and ii µ, B i an N-valued random variable on Ω, F, P for all B S. For imlicity of notation we will write µ B intead of µ, B Definition. An N-valued random meaure µ i called Poion random meaure if the following condition hold. i For all B S with E µ B <, µ B : Ω N i a Poion ditributed random variable with arameter E µ B, i.e. P µ B = n = e EµB for all n N. If Eµ B =, then µ B = P -a.. E µ Bn n! ii For any airwie dijoint B 1,..., B n S, n N, the random variable µ B 1,..., µ B n are indeendent. Now let, be another meaurable ace Definition. A oint function on i a maing : D,, where the domain D of i countable. 9

10 Chater 1. Stochatic Integration with reect to Poion Procee Remark. Each oint function on induce a meaure µ dt, dz on,, B,. Let P D denote the ower et of D and let : D,, t t, t. Let ν be the counting meaure on D, P D defined by ν A = #A for all A P D. Now let u define the meaure µ A B := ν 1 A B for all A B, and B. Then we have µ A B = # {t D t A, t B}. Notation. For given t, and B we will et A =, t and write Let µ t, B = µ, t B. P = {: D, D i countable} be the ace of all oint function on and define B P := σ P µ t, B t >, B Definition. i A random variable : Ω, F P, B P i called oint roce on and Ω, F, P. ii Let θ t be the hift oerator given by θ t :,,, + t. A oint roce i called tationary if for every t > the roce and the hifted roce θ t have the ame robability law. iii A oint roce i called σ-finite if there exit a equence B n n N with B n a n and E µ t, B n < for all t > and n N. iv A Poion oint roce i a oint roce on if there exit a Poion random meaure ν on,, B, and a P -zero et N F uch that for all ω N C and all A B,, B µ ω A B = ν ω, A B Prooition. Let be a σ-finite Poion oint roce on and Ω, F, P. Then i tationary if and only if there exit a σ-finite meaure m on, uch that E µ dt, dz = dt m dz. ere, dt denote the Lebegue-meaure on,, B,. In thi cae the meaure m i uniquely determined and we call it the characteritic meaure of µ. Proof. See Kno5, Prooition.1. 1

11 1.. Stochatic Integration with reect to a Poion Point Proce Definition. Let F t, t be a filtration on Ω, F, P and a oint roce on and Ω, F, P. i The roce i called F t -adated, if for every t and B, µ t, B i F t -meaurable. ii The roce i called an F t -Poion oint roce, if it i F t -adated and σ-finite, uch that {µ t, t + h B h >, B } i indeendent of F t for all t. Further, we define the et Γ µ := {B E µ t, B < for all t > } Definition. Let F t be a right-continuou filtration on Ω, F, P. Let be a Poion oint roce on and Ω, F, P. The roce i aid to be of cla QL or quai-leftcontinuou with reect F t, if it i F t -adated and σ-finite and for all B there exit a roce ν t, B : Ω R, t, uch that the following condition hold: 1 If B Γ µ, the roce ν t, B, t, i a continuou F t -adated increaing roce with ν, B = P.-a.. For all t and for P -a.e. ω Ω, ν ω t, i a σ-finite meaure on,. 3 If B Γ µ, then i an F t -martingale. µ t, B := µ t, B ν t, B, t, In thi cae we call ν the comenator of µ and µ i called comenated Poion random meaure of µ Prooition. Let F t, t, be a right-continuou filtration on Ω, F, P. Let m be a σ-finite meaure on, and let be a tationary F t -Poion oint roce on with characteritic meaure m. Then i quai-left-continuou with reect to F t and with comenator ν t, B = t m B, t, B. 1.. Stochatic Integration with reect to a Poion Point Proce In thi ection we want to contruct the tochatic integral with reect to comenated Poion random meaure, where the random meaure i induced by a tationary Poion oint roce. 11

12 Chater 1. Stochatic Integration with reect to Poion Procee Let Ω, F, P be a comlete robability ace with normal filtration F t, t, and let, be another meaurable ace with a σ-finite meaure m. We fix a tationary F t -Poion oint roce on a defined in the reviou ection with characteritic meaure m. Since i tationary and by Prooition 1.1.9, i quaileft-continuou with reect to F t and the comenator ν of the induced meaure µ i given by ν = dt m. The comenated Poion random meaure i given by µ = µ ν = µ dt m. We will denote thee meaure imly by µ and µ, ince i fixed throughout the whole ection. Further let,, be a earable ilbert ace, T, and et Γ = {B m B < }. Let M T F t. be the ace of all càdlàg quare integrable martingale in with reect to 1..1 Definition. An -valued roce Φ t : Ω, t, T, i an elementary roce, if there exit a artition = t < t 1 < < t k = T, k N and for m {, k 1} there exit airwie dijoint B1 m,..., Bm n m Γ, n m N, and function Φ m i L Ω, F tm, P ;, i n m, uch that the following hold: Φ = k 1 n m m= i=1 Φ m i I tm,t m+1 B m i. The linear ace of all elementary rocee i denoted by E. The tochatic integral with reect to µ can now be defined for an elementary roce Φ E and t, T by Int Φ t := Φ, z µ dt, dz :=,t k 1 n m m= i=1 Φ m i µ t m+1 t, B m i µ t m t, B m i. Then Int Φ i linear in Φ E and P -a.. well defined. We et for Φ E. Φ T := E,t Φ, z m dz d 1.. Prooition. For Φ E we have Int Φ M T, Int Φ = P -a.. For all t, T E Int Φ t = E Φ, z m dz d 1..1,t 1

13 1.. Stochatic Integration with reect to a Poion Point Proce hold. In other word, Int: E, T M T, M i an iometry with T Proof. See Kno5, Prooition.. Int Φ M T = Φ T. U to thi oint, T i only a eminorm on E. Thu let u conider the ace of equivalence clae of elementary rocee with reect to T and let u denote it again by E for imlicity of notation. E i dene in the comletion E T of E with reect to T and hence there exit a unique iometric extenion of Int to E T and the iometry in 1..1 alo hold for each roce in E T. The following rooition will characterize E T. But firt we need to define the redictable σ-algebra on, T Ω by P T := σ g :, T Ω R g i F t -adated and left-continuou = σ {, t F B t T, F F, B } {{} F B F F, B } and et N µ T, ; := Φ:, T Ω Φ i P T /B -meaurable and Φ T = E,T 1..3 Prooition. In the ituation above we have and E T = N µ T, ; Φ, z m dz d 1 N µ T, ; = L, T Ω, P T, dt P m;. Proof. See Kno5, Prooition Proertie of the Poion integral <. We will now collect ome imortant roertie of the tochatic integral with reect to a comenated Poion random meaure Prooition. Let Φ N µ T, ; and let τ be an F t -toing time with P τ T = 1. Then I,τ Φ N µ T, ; and I,τ Φ, z µ d, dz = Φ, z µ d, dz P -a..,t for all t, T.,t τ 13

14 Chater 1. Stochatic Integration with reect to Poion Procee Proof. See Kno5, Prooition Prooition. Let Φ N µ T, ; and et X t := Φ, z µ d, dz, t, T. Then X i cádlág and X t = X t P -a.. for all t, T.,t Proof. See Kno5, Prooition Prooition. Let Φ N µ T, ;, be another ilbert ace and let L L ;. Then L Φ N µ T, ; and L,t for all t, T. Φ, z µ d, dz = Proof. See Kno5, Prooition 3.7.,t L Φ, z µ d, dz 1..7 Prooition. Let Φ N µ T, ;. Then for all t, T E,t Φ, z µ d, dz Proof. See Ste1, Prooition.1. = E Let X t denote the quare bracket of an -valued roce X t Prooition. Let Φ N µ T, ; and X t :=,t,t Φ, z m dz d Φ, z µ d, dz, t. P -a... Then X t =,t Φ, z µ d, dz. Proof. Ste1, Corollary.3 14

15 . Main Theorem In thi chater we will formulate and rove the main theorem of thi work. Our main reference i BL Setting and Aumtion Let,, be a earable real ilbert ace identified with it dual ace by the Riez iomorhim. Let be a real reflexive Banach ace with dual ace, uch that i continuouly embedded into, i.e. there exit C > with v C v for all v, and uch that i dene in. We call,, a Gelfand trile. It follow that continuouly and denly cf. ei9, Prooition 3.13 and alo continuouly and denly. If, denote the duality between and, then we have u, v = u, v for all u, v. Note that i earable ince continuouly and denly and hence thi i true for. Let Ω, F, P be a robability ace with normal filtration F t, t. Let,, m be a meaurable ace with a σ-finite meaure m. A in Section 1. we fix a tationary F t - Poion oint roce on and Ω, F, P. The comenated Poion random meaure i given by µ t, B = µ t, B tm B, t, B, where µ = µ and µ = µ. Let U be another earable ilbert ace and let W t t be a U-valued cylindrical Wiener roce on the robability ace Ω, F t, P. Let < T < be fixed. We conider the tochatic artial differential equation of the following tye dx t =A t, X t dt + B t, X t dw t + f t, X t, z µ dt, dz, X =X,

16 Chater. Main Theorem where X i an F -meaurable random variable. We conider the oerator A:, T Ω, B :, T Ω L U,, f :, T Ω, where L U;, L denote the ace of ilbert-schmidt oerator from U to. For imlicity we write A t, v for the maing ω A t, ω, v and analogouly for B and f. The oerator A and B are both aumed to be rogreively meaurable, i.e. for all t, T thee ma retricted to, t Ω are B, t F t B -meaurable where B denote the Borel-σ-algebra. f i aumed to be a P B -meaurable function, where P i the redictable σ-algebra which i generated by all left-continuou and F-adated real-valued rocee on, T Ω. We aume that there exit contant α > 1, β, θ >, K >, a non-negative, F-adated roce F t t,t uch that F L 1, T Ω, dt P ; R and a meaurable, hemicontinuou function ϱ:,, which i locally bounded in. Furthermore we aume that thee contant and function fulfill the following condition for all v, v 1, v, ω Ω and all t, T : A1 emicontinuity. The ma A t, v 1 + v, v i continuou in R. A Local monotonicity. A t, v 1 A t, v, v 1 v + B t, v 1 B t, v L + f t, v 1, z f t, v, z m dz F t + ϱ v v 1 v. A3 Coercivity. A t, v, v + B t, v L + θ v α F t + K v. A4 Growth. α α 1 A t, v F t + K v α 1 + v β..1.1 Definition Solution. A olution to.1.1 i an F t -adated, -valued, càdlàg roce X t t,t, if for it dt P -equivalent cla X the following condition hold: 16

17 .. Formulation of the Theorem i P -a.. we have X L α, T Ω, dt P ; L, T Ω, dt P ;. ii The following equality hold P -a.. for all t, T : X t = X + + t t A, X d + t f, X, z µ d, dz. The integrability of all occuring integral i required. B, X dw.1. Remark. Although A i an -valued roce by definition, we will ee in Prooition.3.1 that the -valued Bochner integral with reect to dt will become -valued. Our main aim in thi chater will be to etablih exitence and uniquene of trong olution to.1.1 in the ene of Definition Formulation of the Theorem Suoe all condition and aumtion from Section.1 hold. Let C BDG > be the generic contant from the Burkholder-Davi-Gundy inequality D.5 i in cae = 1 and define Γ := Γ θ, β, C BDG β + := θ β + β β + C BDG + β β Then Γ > becaue θ, C BDG >, β and β + β+ = β + β+1 4 β+1 > 3 β+1. We can now formulate the main theorem of thi work...1 Theorem. Suoe that condition A1 - A4 are atified and that F L β+, T Ω, dt P. Suoe there exit contant γ < θ β+ β + β β+1 β and C > uch that B t, v L + f t, v, z m dz C 1 + F t + v + γ v α, B1 and let ϱ be uch that f t, v, z β+ m dz C 1 + F β+ t + v β+ ϱ v C 1 + v α 1 + v β + γ v β v α, B B3 for every t T, ω Ω and v. Then equation.1.1 ha a olution X t t,t for every initial value X L β Ω, F, P ;, where β β +. Furthermore, 17

18 Chater. Main Theorem i there exit a contant C = C, γ, θ, C, K, T > uch that u E X t β+ C T 1 + E X β+ + E t,t F β+ t dt. ii if γ < Γ, then there exit a contant Ĉ = Ĉ, γ, θ, C, C BDG, K, T > uch that E u X t β+ t,t Ĉ 1 + E and the olution X = X t t,t i unique. X β+ T + E F β+ t dt Let u do a hort dicuion on thi theorem and it aumtion in contrat to the familiar reult in BL11, LR1... Remark. i Although tated otherwie in BL11, Theorem 1., the claimed reult for uniquene of the olution X = X t t,t in the cae when..1 ii doe not hold, i.e. γ Γ, cannot be achieved. We will ee in the roof of uniquene in Section.4 that we need Corollary.3.13 iii, which only hold for γ < Γ. ii Contrary to the tatement in BL11, Theorem 1., we do not tate in..1 i and ii that T E X t α X t β dt i bounded. Although we will ee that thi term i bounded in cae of the finite dimenional equation, tyical convergence argument are inufficient to how that the term above i alo bounded in the infinite dimenional cae. iii The given bound on γ in Theorem..1 i of a technical origin, a one will ee in the roof of Lemma.3.5 ii. One can ue γ < θ β + + β+ 1 a a better looking, but maller bound in Theorem..1, becaue β + β β+1 β + 1 β + + β+1 β + 4 = β + + β + β+ imlie that γ < θ β + + β+ 1 β + = θ θ β + and o the Theorem hold. β + β β+1 β + 1 β + + β + β

19 .. Formulation of the Theorem iv We only know that C BDG i generic. Since no information about it calculation can be found in Kal97, we cannot calculate Γ exlicity here. But from LS89 we know that C BDG 3, ince we need it for = 1. v Condition B i weaker than condition 1.3 in BL11, Theorem 1., becaue we allow f t, v, z β+ m dz alo to be bounded by v β v α. A a conequence we cannot chooe γ < to be arbitrary if β =, but we can till chooe γ < θ if β = cf. Remark.3.6 ii and our bound on γ become maller. θ β Anyway, the claimed bound of γ < in BL11, Theorem 1. i not ufficient for β, 1 to how exitence with the method ued therein. The reaon lie in the miing analogue of our Lemma.3.5 i in BL11, which ha not been worked out there. We can ee in the roof that we alway need γ < θ to aly Gronwall inequality, which i e.g. obviouly not true for β = 1 4 then we could chooe θ < γ < θ. Another reaon for weakening condition B here and therefore loing a higher bound on γ i the fact, that all the examle in BL11 do not even involve γ, i.e. γ = there. In Chater 3, we are able to ue thi bound to claim more general condition. vi We ue a weaker local monotonicity condition A here. BL11 or LR1 the bound i given by For condition in K + ϱ v v 1 v in A. ere, we allow F t t,t to be art of the bound intead. Thi generalization i inired from LR14, which i not ublihed yet. vii In a ecial cae, namely for β =, α < 1 5θ 16K 1 and 5θ < 16K, we can deduce B1 from A3 and A4. Indeed, both A3 and A4 imly B t, v L θ v α + F t + K v + F t + K v α 1 + v β α 1 α. By Young inequality we ee that F t + K v α 1 + v β α 1 α Therefore B t, v L ence, ince β =, B t, v L α + F t α + F t + v α α + α 1 α α α α 1 K α F t 1 + v β + K v α 1 + v β. 1 + v β 1 + v β θ + K v. 4 α 1 + v α 4 α 1 K θ + K v α α. 19

20 Chater. Main Theorem So, if θ + γ 4K α 1 α and if C max { } α 1 α ; K; α, then A3 and A4 give u a tronger etimate than B1: B t, v L C 1 + F t + v + γ v α. Furthermore, ince β =, we can dro B1 comletely then, ince B cover the etimate for f. It remain to how that there exit uch a γ with 4K α 1 α θ γ < θ 4. By ome calculation, thi i true for 5θ < 16K and α < 1 5θ 1. 16K The roof of Theorem..1 i lit into an exitence and a uniquene art. The exitence art i baed on the o called Galerkin aroximation. Firt we will conider equation.1.1 in a finite dimenional ace with dimenion n N. Then a olution to thi finite dimenional equation can be found, but intead of roving thi fact, we refer to the literature. owever, we will ee that thi olution fulfill ome ariori etimate under our aumtion and thi will lead to Lemma.3.7 below. There we will ee that each integrand of.1.1 in the finite dimenional cae convergence weakly a n. Thee limiting rocee will be ued to contruct a olution to.1.1 in the general cae. Section.3. deal with an Itô formula for thi roce and finally we will ee that the integrand of our contructed roce are almot everywhere equal to thoe given in.1.1. ence a olution will be contructed, ince all regularity etimate and integrability condition will follow from condition A1 A4 and B1 B3. Section.4 deal with the matter of uniquene. Contrary to the exitence art, thi one i quite eaier. owever, a already mentioned in the introduction, we will ee that the tronger condition on γ in Theorem..1 ii i mandatory to obtain uniquene of a olution. Let u tart with the roof of Theorem..1. For the ret of thi chater we et := β +. Notation. For any given q > 1 we denote by q it dual uch that 1 q + 1 q = 1, i.e. q = q q 1. We aume that for the initial value X from Theorem..1 we have X L β+ Ω, F, P ; without lo of generality. Thi follow from the generalized ölder inequality..3. Proof of the Main Theorem Exitence The roof of exitence i baed on the Galerkin aroximation and therefore, we will firt conider a finite dimenional verion of equation.1.1. Let n N be arbitrary. We will now aume that {e 1, e,... } i an orthonormal bai of, which exit ince i dene and continuou and uch that San e 1, e,... i dene in. Define the finite dimenional ace n := San e 1, e,..., e n

21 .3. Proof of the Main Theorem Exitence and the rojection P n : n, v P n v := n i=1 v, e i e i. For u, v n and t, T we obtain P na t, u, v = P n A t, u, v = A t, u, v. Now let {g 1, g,... } be an orthonormal bai of U and P n the orthogonal rojection onto San g 1, g,..., g n in U. Set W n t := n i= Finite dimenional equation W t, g i U g i = P n W t. The finite dimenional verion of equation.1.1 in n can now be written a dy t =P n A t, Y t dt + P n B t, Y t dw n t + P n f t, Y t, z µ dt, dz, Y =P n X,.3.1 where t, T and X L β+ Ω, F t, P ; i the ame initial value a in Theorem Prooition. Suoe condition A1 A4, B1 B3 hold. Then equation.3.1 ha a trong olution, i.e. there exit an F t -adated, n -valued, càdlàg roce uch that we have X n t =P n X + + t P -a.. for all t, T. t P n A t, X n d + P n f Proof. See ABW1, Theorem 3.1., X n, z t µ d, dz. P n B, X n dw n X n t t,t Remark. The reult in Prooition.3.1 can alo be retrieved from GK8, Theorem 1. The next Lemma i an imortant auxiliary reult and alo known a the Itô formula. 1

22 Chater. Main Theorem.3.3 Lemma Itô formula. Let q < and, for fixed n N, let X t t,t the tochatic roce given in.3.. Then t X t q = X q + q q X q 4 P n B, X P n X d + q t X A, X, X + P n B, X P n d L t t +q X q X, P n B, X dw + X q X, P n f, X, z µ d, dz + t X + P n f, X, z q X q q X q X, P n f, X, z µ d, dz P -a.. for all t, T. Proof. Aly IW81, Theorem 5.1 to the function x x retricted to n..3.4 Remark. Itô formula or Itô lemma for o called Itô-Lévy rocee in finite dimenion i a well known reult in the literature. It can alo be found in Mé8, Theorem 7.1 for general emimartingale. Without claiming to give a full lit, let u further mention A9, Theorem 4.4.7, NØP9, Theorem 9.5 and ABW1, Equation.16. Now let u do ome a riori etimate on X n t olution to.1.1. Recall that we et := β +. t,t before we begin to contruct a.3.5 Lemma. Suoe condition A1 A4 and B1 B3 hold and that F L, T Ω, dt P. Let, n N, be a olution to.3.1 given by Prooiton.3.1. X n t t,t i There exit a contant C 1 = C 1, γ, θ, C, K, T, X L Ω;, F L > Ω,T uch that for all n N. X n u E t t,t T + E X n t ii There exit a contant C = C, γ, θ, C, K, T > uch that for all n N. X n u E t t,t T + E X n t α dt C X n t α dt C E X T + E F t dt.3.4

23 .3. Proof of the Main Theorem Exitence iii If γ < Γ, then there exit a contant C 3 = C 3, γ, θ, C, C BDG, K, T > uch that E u X n t t,t T + E X n t X n t α dt C 3 E X T + E F t dt.3.5 for all n N..3.6 Remark. i In the roof of.3.5 ii and iii and alo Lemma.3.7 ii to iv we heavily ue Young inequality with q =. At firt ight thi i not oible in cae that =, i.e. β =. But we ue it alway in the ame ituation, namely ξ ζ ξ + ζ for ξ, ζ R, and thi inequality hold true even if =. ii One may alo note that.3.5 i and ii are identical if =. But we can ee in te i.6 and ii.7 of the roof that.3.5 i allow u to ue a higher bound, namely γ < θ intead of γ < 1 4θ a in the roof of ii. Thi i becaue in the roof of i we ue the econd art from Lemma C.1 and not the firt art. Proof. Firt we need to introduce a toing time τ n R for given n N and R >, defined by } τ n R {t = inf X n t > R T. By Theorem D.4 we know that τ n R i a toing time. Furthermore we have lim R τ n R = T P -a.. Since X n t take value in the finite dimenional ace n and becaue continuouly, we have X n t R, X n t CR, for all t, τ n R, n N..3.6 Notation. To avoid notational comlexity we et t := t t, n, R := t τ n R for t, T. Ste i.1. Let u aly Itô formula.3.3 to the roce X n t and for q = in there. 3

24 Chater. Main Theorem Then P -a.. for all t, T we have X n t X = n t =: t t t X n, P nb t X n A, X n, X n, X n + Pn B dw n X n, P nf, X n, z X n + P nf, X n, z X n, P nf, X n, z µ d, dz X n µ d, dz, X n Pn d µ d, dz + 1 t + t + 3 t + 4 t 5 t. Ste i.. Alying A3 to 1 t yield to 1 t = A3 t t A F d + K, X n, X n + Pn B, X n Pn d t X n t d θ X n α Ste i.3. We come to t. Let u note that we have, for all t, T and v, B t, v L B t, v L + f t, v, z d. L L.3.7 B1 m dz C 1 + F t + v + γ v α. The tochatic integral t X n, P nb, X n dw n real-valued, local martingale, ince a a càdlàg, F t -adated roce X n t E t P n B, X n L d becaue from.3.6 we ee that.3.8 <, E t X n C 1 + F t + < and X n L 1, T Ω, dt P ; R. ence we deduce E t =..3.8 i well-defined a a contiuou, X n α + γ i redictable and X n α d < and we have F Ste i.4. Let u how that E 3 t =. Let Φ, z := X n, P nf, X n, z, then the roce Φ, i redictable, ince f i redictable. From Prooition 1.. 4

25 .3. Proof of the Main Theorem Exitence and 1..3 we deduce that 3 t i a martingale. Indeed, for all t, T and v, we have by the Cauchy-Schwarz inequality and condition B1 E E B1 E t X n t X n t X n, P nf C, X n, z P n f, X n, z 1 + F + X n m dz d m dz d + γ X n α d <, becaue by.3.6 the - and -norm of X n are bounded and F L 1, T Ω, dt P ; R. Ste i.5. A the lat te of rearation we want to etimate E 4 t 5 t. Prooition 1..7 allow u to change the integrator from µ d, dz to m dz d and then we aly Lemma C.1: E 4 t 5 t E 4 t 5 t E 4 t 5 t t E X n + P nf, X n, z X n X n, P nf t 1..7 = E C.1 = E X n X n, P nf t, X n, z µ d, dz + P nf, X n, z X n, X n, z f, X n, z m dz d m dz d. Therefore, by B1, we know that, for all t, T, E 4 t 5 t E t B1 t CT + C E F d + C E f, X n, z t X n d + γ E m dz d t X n α d. Ste i.6. The reult i. to i.5 combined and ued in the toed verion of.3.7 5

26 Chater. Main Theorem in exectation deliver, for all t, T, X X n E n = E t + E 1 t + E t + E 3 t + E 4 t 5 t }{{}}{{} = = T t E X C E F d + C + K E X n d t + CT + γ θ E X n α d..3.9 ere we ued that, ince X n i n -valued, we have X n X and that, becaue F i non-negative and hence the integral with reect to d i increaing in time, we get t E F T d E F d. If > then we oberve by ölder inequality E X < and E X E 1 = E X T T 1 + C E F d 1 + C T E F d <. So we et Θ := E X C T T E d F + CT. We alo have Θ <. In the cae = we ee immediately that Θ = E X T + E F d + CT < by aumtion. Since γ < θ θ < θ by aumtion, we have θ γ > and we bring the lat ummand in.3.9 to the left hand ide and get by Fubini theorem with ϕ t = E X n t ϕ t + θ γ ψ t Θ + X n, ψ t = E t ψ. Therefore we can aly Gronwall lemma B.5 on α C + K ϕ d. Furthermore we have C + K > and and we get ϕ t + θ γ ψ t Θ + t C + K ϕ + θ γ ψ d ϕ t + θ γ ψ t Θe C+Kt Θe C+KT. Then ϕ t Θe C+KT and hence u,t ϕ ΘeC+KT. ψ t 1 ϕ t + θ γ ψ t Θ θ γ θ γ ec+kt 6

27 .3. Proof of the Main Theorem Exitence alo hold true. Reubtitution give u X n u E,t e C+KT θ γ X n + E t α E X T T E F d + CT := C 1 for all t, T and n N and with C 1 = C 1, γ, θ, C, K, T, X L Ω;, F L < Ω,T. The right hand ide i indeendent of t, R, n and the toing time τ n R. Ste i.7. Now, we aly the monotone convergence theorem..3.1 hold for T, T and we have τ n R T a R P -a.. Then we have X n u E X n + E T,T X = lim u E n R = lim R.3.1 C 1,T τ n R u,t τ n R X E n α + E lim R Xn T τ n R + E X n T τ n R α α for all n N. We come to the roof of.3.5 ii. Ste ii.1. We aly Itô formula.3.3 to the roce X n t with q =. Then P -a.. for 7

28 Chater. Main Theorem all t, T we have X n t X = n t =: t X n t X n t X n t X n 4 A X n, P nb X n + P nf t X n X n, X n P n B, X n, X n Pn, X n + Pn B dw n X n, P nf, X n, z, X n, z X n, P nf X n, X n, z µ d, dz X n, X n µ d, dz µ d, dz d Pn d L + I 1 t + I t + I 3 t + I 4 t + I 5 t I 6 t Ste ii.. We ue.3.8 for I 1 t. I 1 t =.3.8 = C +C t X n t X n t X n t X n t X n 4 4 P n B P n B C, X n, X n 1 + F + F d + C d. Pn Pn X n t X n X n d X n + γ d X n α t d + γ d X n X n α d The firt ummand in the lat line can be litted by Young inequality with q = into t X n d+ C t F d and the lat ummand into T C + t X n d. Then we obtain I 1 t γ t X n X n α d + C + t X n d + C t F d + T C. 8

29 .3. Proof of the Main Theorem Exitence Ste ii.3. We aly A3 to I t. I t = A3 = t X n t X n t X n A F + K F d + K, X n X n, X n + Pn B, X n Pn d L X θ n d t X n α t d θ X n X n α A in the te before we lit the firt ummand of the lat line by Young inequality and have the following reult I t θ t X n X n α d + K + t X n d + Ste ii.4. Let u how that Φ, z := X n X n, P nf, X n, z N µ t, ; R, then I 4 t = t Φ, z µ d, dz i a real-valued martingale by Prooition 1.. and 1..3 and we get E I 4 t =E Φ, z µ d, dz =. Since f i redictable, the t roce Φ, i redictable. It remain to how that Φ t <. By condition B1 we get for all t, T and v f t, v, z and the Cauchy-Schwarz inequality deliver ince Φ t = X n C. S. t E E = E.3.1 E <, and t F d. d. B1 m dz C 1 + F t + v + γ v α.3.1 X n t X n 4 X n t X n 1 t X n 1 C X n X n, P nf, X n, z P n f, X n, z P n f, X n, z 1 + F + X n m dz d + γ m dz d m dz d X n α d are bounded by.3.6 and F L, T Ω, dt P. Ste ii.5. Since, a a càdlàg, F t -adated roce, X n t i redictable, the tochatic integral t X n X n, P nb, X n dw n i well-defined a a continuou, real- 9

30 Chater. Main Theorem valued, local martingale if Φ :=P n B Φ t =.3.8 E <, X n t E P n B, X n t C 1 + F t + N W n, t. L d X n + γ X n α d a in Ste ii.4. ence we have E I 3 t =. Ste ii.6. Now let u come to I 5 t and I 6 t. Firt, we want to etimate E I 5 t I 6 t by Lemma C.1. By Prooition 1..7 we can relace µ d, dz by m dz d in the integral and then there exit a contant C 4 = C 4 uch that E I 5 t I 6 t E 1..7 = E t X n t C.1 C 4 E X n t + P nf, X n, z X n X n X n X n, P nf X n X n, P nf X n, X n, z µ d, dz + P nf, X n, z X n + Pn f, X n, z, X n, z m dz d m dz d. Continuing in the lat row of the equation above and uing B we now obtain E I 5 t I 6 t C 4 E t C 4 C + 1 E + γc 4 E X n 1 + C + F t X n t X n X n + X n t d + C C 4 E + γ F α d + C C 4 T. X n d X n α d Ste ii.7. We aly the exectation to the toed verion of.3.11 and ue our 3

31 .3. Proof of the Main Theorem Exitence reult from the Ste ii. to ii.6 on it. X n E = E X t + E I1 t + E I t + E I 3 t }{{} = where C 5 := + E I 5 t I 6 t }{{} EI 5t I 6t E I 5t I 6t E X + + C 4 γ θ E + C 5 E t X n C + C 6 := C + C C d + C 6 E t F t X n X n d + K + + C 4 C + 1 ere we ued that ince X n i n -valued we have econd ummand to the left ide we get X n E t + θ + C 4 γ E E X + C5 E + C C 4 + C t X n T, X n t X n + E I 4 t }{{} = α d + C C 4 + C T, and X. Bringing the d + C 6 E X n t α d where + C 4 γ < X θ by aumtion on γ. For imlicity we et ϕ t := E n t ψ t := E X n X n α d and C 7 = C 7, γ, θ = θ + C 4 γ >. t We want to aly Gronwall inequality, therefore by Fubini theorem the inequality above can be written a ϕ t + C 7 ψ t Θ + t C 5 ϕ d, where Θ := E X T + C6 E F d + C C 4 + C T. Note that C 5 > becaue K >. Since ψ i non-negative, we focu on ϕ t + C 7 ψ t Θ + t C 5 ϕ d Θ + t F d C 5 ϕ + C 7 ψ d, 31

32 Chater. Main Theorem and Gronwall inequality B.5 give u ϕ t + C 7 ψ t Θe C 5t Θe C 5T. Non-negativity and C 7 > give u ϕ t ϕ t +C 7 ψ t Θe C5T, therefore u r,t ϕ r Θe C 5T, and ψ t 1 C 7 ϕ t + C 7 ψ t 1 C 7 Θe C 5T. Altogether we have with reubtituting Θ for all t, T and n N u ϕ r + ψ t r,t 1 + 1C7 e C 5T E T X + C6 E F d + C C 4 + C T 1 + 1C7 1 + C C C 4 + C where the contant C := C, γ, θ, C, K, T := T e C 5T E X T + E F d + 1, C7 1 + C C C 4 + C T e C 5T i indeendent of R, t, n and the toing time τ n R. Ste ii.8. In thi lat te we aly the monotone convergence theorem. Since.3.13 hold for T, T and τ n R X n u E r r,t = lim R = lim R u r,t τ n R r u,t τ n R T a R P -a.. we get T + E X E n r X E n r X n X n n T τ R + E lim R.3.13 C E X T + E F d n T τ R + E + 1 α d X n X n X n X n α d α d for all n N. Now let u rove.3.5 iii. Ste iii.1. Again, we aly Itô formula.3.3 to the roce X n t ee Next, we ue the reult from Ste ii. and ii.3 to calculate the occuring term I 1 t and 3

33 .3. Proof of the Main Theorem Exitence I t in the toed verion of Then we aly the abolute value and the triangle inequality to get X n t X n t + θ X n t + γ + t C + 1 F d + C t X n t X n t X n, P nb X n α d X n X n α + K +, X n dw n X n, P nf, X n, z X n + P nf, X n, z X n X n X n, P nf, X n, z d t X n µ d, dz d + T C µ d, dz for t, T. On.3.14 we aly the uremum over, τ n R t =, t. The Lebegueintegral tay unchanged, becaue all integrand are non-negative alo F wa choen to be non-negative and hence the integral are increaing in time. So we have u X r n r,t + θ t X n X n α X n t +C 1 + C 8 X n t X n d X n α d + C 9 t F d + J 1 t + J t + J 3 t + T C, with C 8 = γ, C 9 = C + 1, C 1 = C + + d.3.15 K + 33

34 Chater. Main Theorem and J 1 t = u J t = u J 3 t = u r X n r,t r X n r,t r r,t X n, P nb, X n dw n X n, P nf, X n, z X n + P nf, X n, z X n X n, P nf, X n X n, z, µ d, dz, µ d, dz. We want to etimate J 1, J and J 3 from above by the Lebegue-integral, which already aeared, in exectation in the next three te. Ste iii.. Let u etimate E J 1 t. Since J 1 without the uremum and abolute value i a real-valued, local martingale by Ste ii.5, we may aly the Burkholder-Davi- Gundy inequality D.5 i and then condition B1 in the form of.3.8. Remember that C BDG > i the generic contant from D.5 i. Then E J 1 t = E u D.5 i r,t C BDG E r X n C BDG E u.3.8 C BDG E u X n t X n, P n B B, X n, X n X r n t X n r,t X r n t X n r,t L B dw n 1 d, X n C d L F + X n + γ X n α d 1. Let ε > arbitrary. We want to aly Young inequality with q = = q to the right 34

35 .3. Proof of the Main Theorem Exitence hand ide of the equation above which i equal to Y oung E ε u CBDG ε E u =ε E u X r n r,t t ε X r n r,t X n 1 + C 11 E C + C C 11 E X r n r,t t X n X n + γc 11 E 1 + F + X n t X n t X n C α d + C C 11 E + γ X n α d 1 + F + X n d + C C 11 E t X n 1 + γ X n α d t X n d F d.3.16 where C 11 =C 11 ε, C BDG = C BDG 4ε. Again with Young inequality alied with q = q = X on the n -term we arrive at, E J 1 t ε E u + C C 11 X r n r,t t E F d + C 11 C + γc 11 E 1 + E t X n X n t X n d α d + C C 11 T for all t, T. Ste iii.3. Now we come to E J t. By Ste ii.4, J without the uremum and abolute value i a martingale and again, we may aly the Burkholder-Davi-Gundy inequality D.5 i with the already given, generic contant C BDG >, cf. Ste iii.. 35

36 Chater. Main Theorem Then condition B1 in form of.3.1 give u, for an arbitrary ε >, D.5 i E J t = E u C BDG E.3.1 t C BDG E u C BDG E u = E ε u CBDG r,t r X n X r n t r,t X n f X r n t X n r,t X r n r,t t ε X n 1 C, X n X n, P nf, X n, z, z X n 1 + F + f C 1 m dz d, X n 1 + F + X n, z + γ X n µ d, dz m dz d X n α + γ d 1 X n α 1. d 1. Since thi i exactly the ame ituation a in.3.16, we directly conclude E J t = ε E u + C C 11 X r n r,t t E F d + C C γc 11 E E t X n t X n X n d α d + C C 11 T for all t, T. 36

37 .3. Proof of the Main Theorem Exitence Ste iii.4. For the term E J 3 t we have by Prooition 1..7 and Lemma C.1 E J 3 t t E t 1..7 = E + P nf, X n, z X n X n X n C.1 t C 1 E X n X n X n, P nf + P n f X n, X n, P n f, X n, z, X n, z X n, z µ d, dz m dz d X n + f, X, z m dz d with C 1 = C 1 = cf. roof of Lemma C.1. Now we aly B to the right hand ide and arrive at E J 3 t C 1 + C E + γc 1 E t X n t X n X n d + C C 1 E t F α d + C C 1 T for all t, T. Ste iii.5. We want to aly Gronwall inequality. But firt, let u combine the reult from Ste iii. to iii.4 and aly them to.3.15 in exectation. Then for all t, T and ε > E u X r n r,t + θ E t X n X n t F E X + C9 + 4C C 11 + C C 1 E + C 1 + C 11 C C 8 + γc 11 + γc 1 E + C 1 + C t X n X n α d d E + T d C C 1 + C C 11 + C t X n d α d + ε E u X r n r,t

38 Chater. Main Theorem We ued X n X here. We chooe ε = 1 3 and define C 13 := C 13, θ, γ, C BDG := θ C 8 + γc 11 + γc 1 = θ γ + C BDG ε = θ γ + 3 CBDG Then C 13 >, ince we aumed γ < Γ. Furthermore we et C 14 := C 14 C BDG, := + 3CBDG 3 + C1, CBDG C 15 := C 15, K, C, C BDG := C C 16 := C 16, T, C, C BDG := T C 1 + C 1 and C C C C BDG + C Now bringing the lat two ummand to the left hand ide in.3.17 yield to 1 3 E u X r n t + C 13 E X n X n α d r,t E T X + C14 E F d + C 15 E. t X n d + C By the definition of our toing time τ n R the right hand ide i finite. Since X n u r, X r n for all, T and ince the integral are iotone we have by Fubini theorem t E X n t t X d n = E d E u X r n d. For imlicity we define ϕ t := E t ψ t := E u X r n r,t X n Then we can rewrite.3.18 in the following way: ϕ t + 3C 13 ψ t 3Θ + X n t r, α d. 3C 15 ϕ d 38

39 .3. Proof of the Main Theorem Exitence for all t, T, where Θ :=E X T + C14 E F d + C 16. Note that C 15 > becaue K >. Since ψ i non-negative we have ϕ t + 3C 13 ψ t 3Θ + t 3C 15 ϕ d 3Θ + and we can aly Gronwall inequality B.5 to get t ϕ t + 3C 13 ψ t 3Θe 3C 15t 3Θe 3C 15T. 3C 15 ϕ + 3C 13 ψ d The non-negativity of ϕ and ψ and the fact that C 13 > give u ϕ t ϕ t + 3C 13 ψ t 3Θe 3C 15T and ψ t 1 3C 13 ϕ t + 3C 13 ψ t 1 C 13 Θe 3C 15T. ence we have for all t, T and n N ϕ t + ψ t C 13 e 3C 15T E T X + C14 E F d + C C C 16 e 3C 15T E X T + E F d + 1, C where the contant C 3 := C 3, γ, θ, C, C BDG, K, T := C C C 16 e 3C 15T i indeendent of R, t, n and the toing time τ n R. Ste iii.6. Finally we aly the monotone convergence theorem. Since.3.19 hold for T, T and τ n R E u X r n r,t R = E lim = lim R E T a R P -a.. we get r r u,t τ n R u,t τ n R T + E X r n X r n X n T τ C 3 E X T + E F d for all n N. + E + 1 X n n R X n T τ n R α d X n α X n X n d α d 39

40 Chater. Main Theorem Notation. To imlify the ued ace in the following, we introduce the abbreviation L α = L α, T Ω, dt P ;, L α = L α α 1, T Ω, dt P ;, L = L, T Ω, dt P, L U;, M = M T P, dt P m;..3.7 Lemma. Suoe condition A1 A4 and B1 B3 hold and that F L, T Ω, dt P. For each n N let be a olution to.3.1. Then there exit X n t t,t a ubequence n k k N and element X L α L, T ; L Ω;, Y L α, L, g M uch that the following hold: i X n k X weakly in L α and weakly tar in L, T ; L Ω; a k. ii P nk A, X n k Y weakly in L α a k. iii P nk B, X n k weakly in L and P nk B, X n k weakly in L, T ; L Ω; a k. iv P nk f, X n k, g weakly in M and P nk f, X n k, z weakly in L, T ; L Ω; a k. dw n k µ d, dz Proof. Part i. By Lemma.3.5 i we know that T u E n N X n t α dt <, dw g, z µ d, dz i.e. the equence X n n N i bounded in Lα. Since 1 < α <, L α i reflexive and hence there exit a weakly convergent ubequence X n k k N and an element X L α uch that X nk X weakly a k. Furthermore Lemma.3.5 ii tell u that X n u u E k <. k N t,t So the equence X n k k N i bounded in L, T ; L Ω;. We can identify L, T ; L Ω; = L, 1 T ; L 1 Ω; and by the Banach-Alaoglu theorem t 4

41 .3. Proof of the Main Theorem Exitence E.1 there exit another weakly tar convergent ubequence X n k and an element k N X L, T ; L Ω; uch that X n k X weakly tar a k. But we alo have X n k X weakly a k, o we conclude X = X. Part ii. Alo the ace L α i reflexive ince 1 < α < and o we only have to how that the equence P n k A, X n k, where k N n k k N i the ubequence from the lat te, i bounded in L α. Then there exit another ubequence n k k N, and an element Y L α uch that P n k A, X n k Y weakly a k. We have by A4 and Young inequality remember = β + T α u E A t, X n k α 1 t k N T Y oung A4 u E n N T = u E k N u k N F t + K dt k Xn t F t + K k Xn t T E F t + K k Xn t + K k Xn t α α α α 1 + k Xn t + F t X n k t + F t + k β Xn t dt. β dt + K k Xn t k Xn t α k Xn t β dt.3. We ue Lemma.3.5 i to get T K u E k N k Xn t α dt <. Lemma.3.5 ii give u T K u E k Xn t k N α k Xn t β dt < and with Fubini theorem T u E k Xn t k N dt = T u k N = u k N T T E X n k t dt X n k u E,T u E t,t X n k t dt.3.4 <. 41

42 Chater. Main Theorem Finally, by the aumtion that F L, T Ω, dt P and Young inequality, we ee that T E F t + F T 4 t dt E F t + dt = 4 E T F t T dt + <, hence.3. i finite and o there exit the required ubequence n k k N n k k N. Part iii. A before, ince the ace L i reflexive, too, it i ufficient to how that P n B, X n k i bounded in k k N L. Condition B1, Fubini theorem and Lemma.3.5 i give u T u E P n B t, X n k k t dt k N L B1 T u E C 1 + F t + k Xn t + γ k α Xn t dt k N = u k N.3.3 u k N.3.3 CT + C E T CT + C E T CT + C E F t dt + C T E Xn t F t dt + CT u E t,t T F t dt + C 1 CT + γ <, T ince by ölder inequality E F t dt a ubequence n k k N P uch that n B k element L. k Xn t T dt + γ E k T T E F t dt + γc 1 Xn t k α dt.3.1 <. So there exit, X n k converge weakly in k N L to an Let u come to the econd art of iii: Since P n i the orthogonal rojection onto San {g 1,..., g n } in U, without lo of generality we have that P n B t, X n k k t P n k converge weakly to in L. Furthermore P n B, X n t dw n = P n B, X n t Pn dw hold for all n N. The maing Int W : L L, T Ω;, Φ Int W Φ := Φ dw i linear and continuou, o it reerve weak convergence. ence P n B, X n k k t dw n k = P n B, X n k P k n dw k dw 4

43 .3. Proof of the Main Theorem Exitence weakly a k. Part iv. If we identify M = L Ω, T, P, P dt m;, then we ee that M i reflexive, too. The roof of boundedne of P n f, X n k k, k N i alo done with B1: T E P n f t, X n k k t, z m dz dt u k N B1 u k N T E C 1 + F t + < cf Xn k t + γ Xn k t So there exit a ubequence n k k N n k k N which fulfill i-iv. Eecially there i an element g M that i the weakly limit of P nk f, X n k, a k. The econd art of iv follow identically a in the roof before: Since the maing Int µ : M L, T Ω;, Φ Int µ Φ := Φ, z µ d, dz i linear and continuou, it reerve weak convergence and we obtain that P nk f, X n k, z µ d, dz g, z µ d, dz weakly a k. α dt.3.8 Remark. In the ituation of Lemma.3.7 all the dt P -verion X, Y and are rogreively meaurable, ince the aroximant are rogreively meaurable..3.. Contruction of the infinite dimenional Solution Let u recall what we have achieved o far. By the Galerkin aroximation we conidered the following tochatic artial differential equation in the finite dimenional ace n, n N: dy t =P n A t, Y t dt + P n B t, Y t dw n t + P n f t, Y t, z µ dt, dz, Y =P n X, By Lemma.3.1 thi equation ha a unique trong olution X n t t,t.3. for each n N. Each olution fulfill ome ariori etimate from Lemma.3.5 which allowed u to find limiting element X, Y, and g a in Lemma.3.7 for the equence of olution X n n N. 43

44 Chater. Main Theorem Now we come back to our origin equation dx t =A t, X t dt + B t, X t dw t + X =X, f t, X t, z µ dt, dz, t, T. Let X, Y,, g be a in Lemma.3.7. We can define the following tochatic roce: t t t X t = X + Y d + dw + g, z µ d, dz,.3.3 t, T. In the following we will ee that thi roce i a -valued modification of the -valued roce X and that thi roce i a olution to our equation.1.1 which finihe the roof of uniquene. Notation. For abbreviation we et Y n k := P nk A, X n k, n k := P nk B, X n k, f nk := P nk f, X nk,..3.9 Lemma. The tochatic roce X t t,t defined by.3.3 i a -valued modification of X. Proof. Thi roof i a traightforwarded extenion of the roof of PR7, Theorem 4..4,. 86. We have to how X = X dt P almot everywhere in. Let v n 1 n and ϕ L, T Ω. Uing Lemma.3.7 i and then equation.3. and Fubini theorem we get T E = lim k = lim k X t, ϕ t v T E T dt = lim E X nk t, ϕ t v dt k T t P nk X, ϕ t v dt + E T t + E nk dw nk, ϕ t v t T + E T E P nk X, v ϕ t dt dt f nk, z µ d, dz, ϕ t v T t + E nk dw nk, ϕ t v t T + E Y nk, ϕ t v d dt dt T + E Y nk, f nk, z µ d, dz, ϕ t v dt dt T ϕ t dt v d 44

45 .3. Proof of the Main Theorem Exitence On the right hand ide of the above equation we can now ue our weak convergence reult from Lemma.3.7 ii-iv and thi yield to lim k T T T E P nk X, v ϕ t dt + E Y nk, ϕ t dt v d T t + E nk dw nk, ϕ t v t T + E T = E T = E X + t f nk, z µ d, dz, ϕ t v Y d + X t, ϕ t v dt, what wa to be hown. t dw + dt dt t g, z µ d, dz, ϕ t v dt.3.1 Prooition. The tochatic roce X t t,t defined by.3.3 i i -valued, càdlàg, F t -adated and atifie P -a.. the following Itô-formula: t X t = X + Y, X d + t L d + t g, z µ d, dz + M t for t, T, where M t = t t X, dw + X, g, z µ d, dz i a càdlàg, real-valued, local martingale. ii fulfill E u X t t,t <. Proof. Thi roof i inired by Ste1, Theorem 5.9. Ste i.1. Let u aly GK8, Theorem with h t := t dw + t g, z µ d, dz 45

46 Chater. Main Theorem and =. Then X defined by.3.3 i -valued, càdlàg, F t -adated and we have P -a.. X t = + X + t t Y, X X, dh + h t d for all t, T, where h t denote the quare bracket of h ee Definition D.3 and we already ued that X i a -valued modification of X by Lemma.3.9. Ste i.. Since X t a an F t -adated càdlàg roce i redicatble and ince N W, T, we know that the tochatic integral t X, dw i a real-valued local martingale. Furthermore, ince g N µ T, ;, we obtain that t i a real-valued local martingale if we to it by X, g, z µ d, dz =: N t τ n := inf {t N t > n} T for n N, ince we have lim n τ n = T P -a.. From thi we ee that the tochatic integral t X, dh = t X, dw + t X, g, z µ d, dz i a real-valued, càdlàg, local martingale. Ste i.3. We know that dw t = t L d and by Prooition 1..8 o we conclude h t = Now we want to rove ii. Ste ii.1. We define g, z µ d, dz = t t L d + t t g, z g, z τ R := inf { t Xt > R } T, µ d, dz, µ d, dz. 46

47 .3. Proof of the Main Theorem Exitence which i a toing time due to Theorem D.4 and ue the notation t := t t, R = t τ R for t, T and R >. We have τ R T for R P -a.. Itô formula from.3.1 i alied with ölder inequality yield to T X t X + Y T + + t α α 1 α 1 α T 1 d X α d α T L d + g, z µ d, dz t X, dw + X, g, z µ d, dz. Now we take the uremum over, t and then aly the exectation to both ide: E u X,t T E X + E Y T + E + E u L r,t r α α 1 α 1 α T 1 d X α d α T d + E g, z µ d, dz r X, dw + E u r,t X, g, z µ d, dz..3.4 Ste ii.. We aly the Burkholder-Davi-Gundy inequality D.5 i on the firt ummand in the lat row of.3.4 and Lemma.3.9 E u r,t r X, dw D.5 i r C BDG E X, dw.3.9 = C BDG E t X L 1 t d C BDG E 1. t X L 1 d 47