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ON STOCHASTIC EVOLUTION EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS XICHENG ZHANG Department of Mathematics, Huazhong University of Science and Technology, 8 Wuhan, Hubei 430074, P.R.China 0 Fakulta¨t fu¨r Mathematik, Universita¨t Bielefeld 0 Postfach 100131, D-33501 Bielefeld, Germany 2 Email: [email protected] n a J Abstract. Inthispaper,westudytheexistenceanduniquenessofsolutionsforseveral 0 classes of stochastic evolution equations with non-Lipschitz coefficients, that is, back- 1 ward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat ] R a large class of quasi-linear stochastic equations, which includes the reaction diffusion and porous medium equations. P . h t a m Contents [ 2 1. Introduction 1 v 2. Framework and Preliminaries 3 0 3. Stochastic Evolution Equations in Banach Spaces 8 6 2 4. Backward Stochastic Evolution Equations 17 3 5. Stochastic Functional Integral Evolution Equations 27 0 7 6. Applications 33 0 6.1. Stochastic Porous Medium Equations 33 / h 6.2. Stochastic Reaction Diffusion Equations 35 t a References 37 m : v i X 1. Introduction r a Let O be a bounded open subset of Rd. Consider the following stochastic porous medium equation with Dirichlet boundary condition: du = |w |·∆(|u |p−2u )dt+dw , t t t t t u (x) = 0, x ∈ ∂O,t > 0 (1) t  u = φ ∈ Lp(O),  0 where p > 2, ∆ is the usual Laplace operator, and {w ,t > 0} is a one dimensional  t standard Brownian motion. This is a degenerate non-linear stochastic partial differential equation. Notice that the degeneracy may be caused by w = 0 and u = 0. In the t t deterministic case, it is well known that porous medium equations can be written as abstract monotone operator equations(cf. [34] [29]). Thus, in the stochastic case, it can Keywords andphrases. StochasticReactionDiffusionEquation,StochasticPorousMediumEquation, Stochastic EvolutionEquation,BackwardStochastic EvolutionEquation, Stochastic Functional Integral Evolution Equation. 1 fall into a class of stochastic evolution equations studied by Krylov-Rozovskii [17]. More discussions about the stochastic porous medium equation are referred to [8] [26] [23]. On the other hand, let us consider the following stochastic reaction diffusion equation: du = |w |·(∆u −|u |p−2 ·u )dt+ |w |·u dw , t t t t t t t t u (x) = 0, x ∈ ∂O,t > 0 (2)  t p u = φ ∈ L2(O),  0 where p > 2. Usually, one wants to find an adapted process u such that for almost all w  u(w) ∈ L2([0,T],H1(O))∩Lp([0,T]×O)∩C([0,T],L2(O)), · 0 and (2) holds in the generalized sense, where H1(O) is the usual Sobolev space. 0 However, from the well known results, it seems that one cannot solve Eq.(1) and Eq.(2) because of the presence of |w | in front of the Laplace operator. One of the main purposes t in this paper is to extend the well known results in [17] [11] so that we can solve Eq.(1) and Eq.(2) in the generalized sense for almost all path w . t In the present paper, we shall work on the framework of evolution triple. This is crucial for treating a wide class of quasi-linear stochastic partial differential equations(including reaction diffusion equations and porous medium equations). We now recall some well- known results in this direction. In [20] [21], Pardoux considered linear stochastic partial differential equations(SPDEs) using the monotonicity method. In [17], basing on their established Itô’s formula, Krylov and Rozovskii proved a more general result under some monotonicity or dissipative conditions. This classic work was later extended in several aspects: to stochastic evolution equations(SEEs) driven by general (discontinuous) mar- tingales in [10], to SEEs with coercivity constants depending on t in [11], to SEEs related to some Orlicz spaces in [26]. All these works are based on Galerkin’s approximation. It should be remarked that the semigroup method is another main tool in the theory of semi-linear SPDEs (cf. [9] [6] [7] [16] [13] [35] [36]etc.). In order to solve Eq.(1), we need to deal with SEEs with random coercivity coefficients. This is our first goal, and will be done in Section 3 after some preliminaries of Section 2. Here, some stopping times techniques will be used. The second aim is to prove the existence and uniqueness of solutions to backward stochastic evolution equations. Since Pardoux and Peng in [22] proved the existence and uniqueness of solutions to nonlinear backward stochastic differential equations(BSDEs), the theory of BSDEs has already been developed extensively. It is well known that BSDEs can be applied to the studies of stochastic controls, mathematics finances, deterministic PDEs, etc.. Meanwhile, backward SPDEs have also been studied in [14] [25] etc.. In these works, the authors mainly concentrated on semilinear BSPDEs. The second aim in this paper is to prove the existence and uniqueness of solutions to BSEEs with non-Lipschitz coefficients in the framework of evolution triple. Thus, it can be used to deal with a large class of quasi linear BSPDE. We remark that Mao in [18] has already studied the BSDEs with non-Lipschitz coefficients, and the authors in [1] also investigated the BSDEs with monotone and arbitrary growth coefficients. This is the content of Section 4. The third aim is to study the stochastic functional integral evolution equations with non-Lipschitz coefficients, which in particular includes a class of stochastic Volterra type evolution equations. Stochastic Volterra equations driven by Brownian motion were first studied by Berger-Mizel [3]. Later, Protter [24] proved the existence and uniqueness of stochastic Volterra equations driven by general semimartingales. Recently, Wang in [32] studied the the existence and uniqueness of stochastic Volterra equations with singular kernels and non-Lipschitz coefficients. About the stochastic functional differential equations, Mohammend’s book [19] is one of the main references. In [30], using the evolution 2 semigroup approach, theauthorsstudied theexistence, uniqueness andasymptotic behav- ior of mild solutions to stochastic semilinear functional differential equations in Hilbert spaces. In our proof of Section 5, the main tool is the usual Picard iteration. As above, the results in Section 5 can be also used to deal with a class of quasi linear stochastic functional partial differential equations. Lastly, in Section 6 we shall present two applications for our abstract results: stochastic porous medium equations and stochastic reaction diffusion equations. In particular, Eq.(1)andEq.(2)willbetwospecialcases. Itisworthy tosaythatthetwoexamples given in Section 6 have stochastic non-linear second order terms. Moreover, we may also consider the corresponding backward and functional stochastic partial differential equations with a slight modification. 2. Framework and Preliminaries In this section we present a general setting in which we can deal with a large class of non-linear stochastic partialdifferential equations, andalsorecallthepowerful Itôformula and a nonlinear Gronwall type inequality (Bihari’s inequality) for treating non-Lipschitz equations. LetXbeareflexive andseparableBanachspace, whichisdensely injectedinaseparable Hilbert space H. Identifying H with its dual we get X ⊂ H ≃ H∗ ⊂ X∗, where the star ‘∗’ denotes the dual spaces. Assume that the norm in X is given by kxkX := kxk1,X +kxk2,X, x ∈ X. Denote by Xi, i = 1,2 the completions of X with respect to the norms k·ki,X =: k·kXi. Then X = X ∩X . Let us also assume that both spaces are reflexive and embedded in 1 2 H. Thus, we get two triples: X ⊂ H ≃ H∗ ⊂ X∗, X ⊂ H ≃ H∗ ⊂ X∗. 1 1 2 2 Noticing that X∗ and X∗ can be thought as subspaces of X∗, one may define a Banach 1 2 space Y := X∗ + X∗ ⊂ X∗ as follows: f ∈ Y if and only if f = f +f , f ∈ X∗,i = 1,2 1 2 1 2 i i and the norm of f is defined by kfkY = inf (kf1kX∗ +kf2kX∗). f=f1+f2 1 2 In the following, the dual pairs of (X,X∗) and (X ,X∗),i = 1,2 are denoted respectively i i by [·,·]X, [·,·]X , i = 1,2. i Then, for any x ∈ X and f = f +f ∈ Y ⊂ X∗, 1 2 [x,f]X = [x,f1]X1 +[x,f2]X2. We remark that if f ∈ H and x ∈ X, then [x,f]X = [x,f]X1 = [x,f]X2 = hx,fiH, where h·,·i stands for the inner product in H. H Let (Ω,F,(F ) ,P) be a complete separable filtration probability space, and Q a t t>0 nonnegative definite and symmetric bounded linear operator on another Hilbert space U. A cylindrical Q-Wiener process {W(t),t > 0} defined on (Ω,F,P) is given and assumed to be adapted to (F ) (cf. [9]). In the following we shall only consider the case of Q ≡ I t t>0 for simplicity. Let L (U,H) denote the Hilbert space consisting of all the Hilbert-Schmidt 2 3 operators from U to H, where the norm is denoted by k·k , and the inner product L2(U,H) by h·,·,i . L2(U,H) Fix T > 0. Let M be the total of progressively measurable subsets of [0,T]×Ω. The following Itô formula is taken from Gyöngy-Krylov [12]. Theorem 2.1. Let X be an F -measurable H-valued random variable. Let 0 0 Y : [0,T]×Ω → X∗ ∈ M/B(X∗), i = 1,2, i i i and M an H-valued continuous locally square integrable martingale starting form zero. Let λ ,λ be two M/B(R)-measurable real valued processes such that for (dt×dP)-almost all 1 2 (t,ω), λ (t,ω),λ (t,ω) > 0. Assume that for some q ,q > 1 and for almost all ω, 1 2 1 2 λi(·,ω) ∈ L1([0,T],dt), Yi(·,ω)·λi−q1i(·,ω) ∈ Lqiq−i1([0,T],dt;X∗), i = 1,2. Define an X∗-valued process by t t X(t) := X + Y (s)ds+ Y (s)ds+M(t). 0 1 2 Z0 Z0 If there exists a (dt×dP)-version X˜ of X such that for almost all ω, 1 X˜(·,ω)·λqi(·,ω) ∈ Lqi([0,T],dt;X ), i = 1,2, i i then for almost all ω, (i) [0,T] ∋ t 7→ X(t,ω) ∈ H is continuous; (ii) for all t ∈ [0,T] t kX(t,ω)k2H = kX0(ω)k2H +2 [X˜(s,ω),(Y1+Y2)(s,ω)]Xds Z0 t +2 hX(s),dM(s)i (ω)+hMi(t,ω), H Z0 where h·i denotes the quadratic variation of H-valued martingale. Proof. Set N (t) := tλq1i(s)ds and Y˜(t) := Y (t)·λ−q1i(s),i = 1,2. Then i 0 i i i i t t X(t)R= X + Y˜ (s)dN (s)+ Y˜ (s)dN (s)+M(t). 0 1 1 2 2 Z0 Z0 By the assumptions and Hölder’s inequality, we have for almost all ω, Y˜(·,ω) ∈ L1([0,T],dN ;X∗), i = 1,2, i i i X˜(·,ω) ∈ ∩ L1([0,T],dN ;X ). i=1,2 i i Moreover, by Hölder’s inequality we have for i = 1,2 and almost all ω T kY˜i(t,ω)kX∗i ·kX˜(t,ω)kXidNi(t) Z0 T −1 1 = kYi(t,ω)kX∗iλi qi(t)·kX˜(t,ω)kXiλiqi(t)dt Z0 qi−1 T qi − 1 qi 6 kY (t,ω)kqi−1λ qi−1(t)dt i X∗ i i (cid:18)Z0 (cid:19) 1 T qi × kX˜(t,ω)kqiλ (t)dt < +∞. X i i (cid:18)Z0 (cid:19) 4 Thus, we can prove this Theorem along the same lines as in the proof of [12, Theorem 2] (see also [17] [28] [23]). We omit the details. (cid:3) We now recall the following Bihari inequality(cf. [4]). A multi-parameter version with jump was proved in [38]. Lemma 2.2. Let ρ : R+ → R+ be a continuous and non-decreasing function. Let g(s) and λ(s) be two strictly positive functions on R+ such that for some g > 0 0 t g(t) 6 g + λ(s)·ρ(g(s))ds, t > 0. 0 Z0 If λ is locally integrable, then t g(t) 6 G−1 G(g )+ λ(s)ds , 0 (cid:18) Z0 (cid:19) where G(x) := x 1 dy is well defined for some x > 0, and G−1 is the inverse function x0 ρ(y) 0 of G. R In particular, if g = 0 and for some ε > 0 0 ε 1 dx = +∞, (3) ρ(x) Z0 then g(t) ≡ 0. Remark 2.3. The typical concave functions satisfying (3) are given by ρ (x), k = k 1,2,··· , c ·x·Πk logjx−1, x 6 η ρ (x) := 0 j=1 (4) k c ·η ·Πk logjη−1 +c ·ρ′(η−)·(x−η), x > η, (cid:26) 0 j=1 0 k where logjx−1 := loglog···logx−1 and c > 0, 0 < η < 1/ek. 0 In the sequel, we use the following convention: c ,c ,··· will denote positive constants 0 1 whose values may change in different occasions. Moreover, the following Young inequality will be used frequently: Let a,b > 0 and α,β > 1 satisfying 1 + 1 = 1, then for any ε > 0 α β bβ ab 6 εaα + . (5) (αε)β/αβ For simplicity of notations, we also write A := ([0,T]×Ω,B([0,T])×F,dt×dP) and A := ([0,T]×Ω,M,dt×dP). a We now introduce three evolution operators used in the present paper: A : [0,T]×Ω×X → X∗ ∈ M×B(X )/B(X∗), i = 1,2, i i i i i B : [0,T]×Ω×X → L (U,H) ∈ M×B(X)/B(L (U,H)). 2 2 In the following, for the sake of simplicity, we write A = A +A ∈ Y ⊂ X∗. 1 2 Assume that (H1) (Hemicontinuity) For any (t,ω) ∈ [0,T]×Ω and x,y,z ∈ X, the mapping [0,1] ∋ ε 7→ [x,A(t,ω,y+εz)]X is continuous. 5 (H2) (Weak monotonicity) There exists 0 6 λ ∈ L1(A) such that for all x,y ∈ X and 0 (t,ω) ∈ [0,T]×Ω 2[x−y,A(t,ω,x)−A(t,ω,y)]X+kB(t,ω,x)−B(t,ω,y)k2L2(U,H) 6 λ (t,ω)·kx−yk2. 0 H (H3) (Weak coercivity) There exist q ,q > 2,c > 0 and positive functions λ ,λ ,λ ,ξ ∈ 1 2 1 1 2 3 L1(A) satisfying that for almost all (t,ω) 0 6 λ (t,ω) < c ·(λ (t,ω)∧λ (t,ω)) (6) 0 1 1 2 and (t,ω) 7→ λi(t,ω)·eqi2−2R0tλ0(s,ω)ds ∈ L1(A), i = 1,2, (7) where λ is same as in (H2), and such that for all x ∈ X and (t,ω) ∈ [0,T]×Ω 0 2[x,A(t,ω,x)]X +kB(t,ω,x)k2L2(U,H) 6 − λ (t,ω)·kxkqi +λ (t,ω)·kxk2 +ξ(t,ω). i X 3 H i iX=1,2(cid:16) (cid:17) (H4) (Boundedness) There exist cAi > 0 and 0 6 ηi ∈ Lqiq−i1(A), i = 1,2 such that for all x ∈ X and (t,ω) ∈ [0,T]×Ω 1 kAi(t,ω,x)kX∗i 6 ηi(t,ω)·λiqi(t,ω)+cAi ·λi(t,ω)·kxkXqii−1, i = 1,2, where q and q are same as in (H3). 1 2 In order to emphasize λ ,ξ and q ,η below, we shall say that i i i (A,B) satisfies H (λ ,λ ,λ ,λ ,ξ,η ,η ,q ,q ). 0 1 2 3 1 2 1 2 If there are no special declarations, we always suppose that q > 2,c ,c > 0, η ∈ i 1 Ai i Lqiq−i1(A), i = 1,2 and λi,ξ ∈ L1(A),i = 0,1,2,3 are strictly positive functions, and (6)-(7) hold. Remark 2.4. By (H3), (H4) and Young’s inequality (5), it follows that for any x ∈ X and (t,ω) ∈ [0,T]×Ω 1 kB(t,ω,x)k2L2(U,H) 6 2 cAi ·λi(t,ω)·kxkqXii +ηi(t,ω)·λiqi(t,ω)·kxkXi iX=1,2(cid:16) (cid:17) +λ (t,ω)·kxk2 +ξ(t,ω) 3 H qi 6 c ·λ (t,ω)·kxkqi +ηqi−1(t,ω) B i Xi i iX=1,2(cid:16) (cid:17) +λ (t,ω)·kxk2 +ξ(t,ω), 3 H where c > 1 only depends on c and q , i = 1,2. B Ai i The following lemma is well known(cf. [17]). Lemma 2.5. Let (A,0) satisfy H (0,λ ,λ ,λ ,ξ,η ,η ,q ,q ), and 0 6 τ 6 T a bounded 1 2 3 1 2 1 2 random variable. Let X and Y (i = 1,2) be respectively X and X∗-valued measurable i i processes with 1 1 (·)·λqi−1 ·X ∈ Lqi−1(A;X ), 1 (·)·Y ∈ L1(A;X∗), i = 1,2. [0,τ] i i [0,τ] i i 6 Assume that for any X-valued measurable process Φ satisfying 1 1 (·)·λqi−1 ·Φ ∈ Lqi−1(A;X ), i = 1,2, [0,τ] i i it holds τ E [X(s)−Φ(s),Y(s)−A(s,Φ(s))]Xds 6 0, (8) (cid:18)Z0 (cid:19) where Y = Y +Y ∈ Y ⊂ X∗. 1 2 Then Y(t,ω) = A(t,ω,X(t,ω)) for almost all (t,ω) ∈ {(t,ω) : t ∈ [0,τ(ω)]}. Proof. For any ε ∈ (0,1) and X-valued bounded measurable process φ, letting Φ = X−εφ in (8) and dividing both sides by ε, we get τ E [φ(s),Y(s)−A(s,X(s)−εφ(s))]Xds 6 0. (cid:18)Z0 (cid:19) By (H4) and the assumptions, we have 1[0,τ](·)· kY(·)kX∗ + sup kA(·,X(·)−εφ(·))kX∗ ∈ L1(A). ε∈(0,1) (cid:16) (cid:17) Hence, by (H1) and the dominated convergence theorem τ E [φ(s),Y(s)−A(s,X(s))]Xds 6 0. (cid:18)Z0 (cid:19) By changing φ to −φ and the arbitrariness of φ, we conclude that Y = A(·,X). (cid:3) The following lemma is simple and will be used in Section 4. A short proof is provided here for the reader’s convenience. Lemma 2.6. Let (S,S) be a measurable space. Let X : Rd × S → Rd be a measurable field. Assume that for every s ∈ S, Rd ∋ x 7→ X(x,s) ∈ Rd is a homeomorphism. Then, the inverse (x,s) 7→ X−1(x,s) is also a measurable field, i.e.: for each x ∈ Rd, X−1(x,·) is S/B(Rd)-measurable. Proof. Fix x ∈ Rd. It suffices to prove that for any bounded open set U ⊂ Rd S := {s : X−1(x,s) ∈ U¯} ∈ S, (9) 1 where U¯ denotes the closure of U in Rd. Let Q be the set of rational points in Rd. Then S = ∩∞ ∪ {s : kX(y,s)−xk < 1/k} =: S . (10) 1 k=1 y∈Q∩U Rd 2 In fact, if s ∈ S , then there is a y ∈ U¯ such that x = X(y,s). Since U is open 1 and X(·,s) is continuous, there exists a sequence y ∈ U ∩ Q such that y → y and n n X(y ,s) → X(y,s) = x. So, s ∈ S . On the other hand, if s ∈ S , then there is a n 2 2 sequence y ∈ U ∩Q such that lim kX(y ,s)−xk = 0, and so y → X−1(x,s) ∈ U¯. n n→∞ n Rd n (9) now follows from (10). (cid:3) 7 3. Stochastic Evolution Equations in Banach Spaces In this section, we consider the following stochastic evolution equation: dX(t) = A(t,X(t))dt+B(t,X(t))dW(t), (11) X(0) = X ∈ H, 0 (cid:26) where (A,B) satisfies H (λ ,λ ,λ ,λ ,ξ,η ,η ,q ,q ). Here and after, one should keep in 0 1 2 3 1 2 1 2 mind that A = A +A ∈ Y ⊂ X∗, where A ∈ X∗,A ∈ X∗. 1 2 1 1 2 2 Set t H(t,ω) := λ (s,ω)ds, (12) 3 Z0 and define θ (ω) := inf{s ∈ [0,T] : H(s,ω) > t}. t Then t 7→ θ are continuous stopping times, and θ ↑ T as t ↑ ∞. Here, inf{∅} = T by t t convention. Set for each m ∈ N µm(dt×dω) := 1 (dt×dP), {t6θm(ω)} and define completed measurable spaces Mm := ([0,T]×Ω,B([0,T])×F)µm and Mm := ([0,T]×Ω,M)µm. a We introduce the following stochastic Banach spaces for later use: for each m ∈ N Km := Lqiq−i1(Mm,λ−qi1−1(t,ω)·µm(dt×dω);X∗), i = 1,2, 1,i i i Km := Lqi(Mm,λ (t,ω)·µm(dt×dω);X ), i = 1,2, 2,i i i Km := L2(Mm,µm(dt×dω);L (U,H)), 3 a 2 Km := L2(Mm,µm(dt×dω);H), 4 Km := L2(Mm,λ (t,ω)·µm(dt×dω);H), 5 3 where the norms are defined in a natural manner, and denoted by k·kK, where K stands for the above spaces. For instance, θm 1/qi kXkKm := E kX(t)kqXi ·λi(t)dt , i = 1,2. 2,i i (cid:20) (cid:18)Z0 (cid:19)(cid:21) Remark 3.1. If λ is non-random, then for some m sufficiently large, θ ≡ T. In this 3 m case, we shall omit the superscript ‘m’ of Km. We need the following lemma. Lemma 3.2. (i) Km,i,j = 1,2 and Km,Km,Km are separable and reflexive Banach i,j 3 4 5 spaces. (ii) For any Y ∈ Km1,i, we have E 0θm kY(t)kX∗idt 6 c0 ·kYkKm1,i, where i = 1 or 2. (iii) Let {Y ,n ∈ N} weakly conver(cid:16)ge to Y in Km,(cid:17)then for any X ∈ Km n R 1,i 2,i θm θm lim E [X(t),Yn(t)]Xidt = E [X(t),Y(t)]Xidt , n→∞ (cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19) where i = 1 or 2. 8 (iv) Let {X ,n ∈ N} weakly converge to X in Km, then for any Y ∈ Km n 2,i 1,i θm θm lim E [Xn(t),Y(t)]Xidt = E [X(t),Y(t)]Xidt , n→∞ (cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19) where i = 1 or 2. Moreover, if {X ,n ∈ N} also weakly converges to X¯ in Km, then n 5 X¯(t,ω) = X(t,ω) for µm-almost all (t,ω). (v) Define a linear operator from Km to Km as 3 4 ·∧θm J(G) := G(s)dW(s), (13) Z0 then J is a continuous linear operator. In particular, J is continuous with respect to the weak topologies. Proof. (i). It follows from the separabilities and reflexivities of X ,X∗,i = 1,2, and i i H,L (U,H). 2 (ii). By Hölder’s inequality we have θm θm E kY(t)kX∗idt = E kY(t)kX∗iλ−21/qi(t)·λ12/qi(t)dt (cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19) T 1/qi 6 kYkKm E(λi(t))dt . 1,i (cid:18)Z0 (cid:19) (iii). It follows from 1 − 1 X(·)·λqi−1(·) ∈ Lqi(Mm,λ qi−1(t,ω)·µm(dt×dω);X ) ⊂ (Km)∗. i i i 1,i (iv). The first conclusion follows from Y(·)·λ−i 1(·) ∈ Lqiq−i1(Mm,λi(t,ω)·µm(dt×dω);X∗i) ⊂ (Km2,i)∗. As for the second conclusion, by the well known Banach-Saks-Kakutani theorem, there exists a subsequence of X (still denoted by X ) such that its Césaro means X˜ strongly n n n converges to X and X¯ in Km and Km respectively. Therefore, there is a subsequence X˜ 2,i 5 nk such that for µm-almost all (t,ω), X˜ (t,ω) → X(t,ω) in X, and X˜ (t,ω) → X¯(t,ω) in nk nk H. Since X is continuously and densely embedded in H, we have X¯(t,ω) = X(t,ω) for µm-almost all (t,ω). (v). It follows from θm t∧θm 2 kJ(G)k2 = E G(s)dW(s) dt Km 4 Z0 (cid:13)Z0 (cid:13)H ! (cid:13) (cid:13) T (cid:13)t∧θm (cid:13) 6 E (cid:13) kG(s)k2 d(cid:13)s dt L2(U,H) Z0 (cid:18)Z0 (cid:19) 6 TkGk2 . Km 3 The proof is complete. (cid:3) Definition 3.3. An H-valued continuous F -adapted process X(t,ω) is called a solution t of Eq.(11) if for almost all ω ∈ Ω, X(·,ω) ∈ ∩ Lqi([0,T],λ (·,ω)dt;X ) i=1,2 i i 9 and for all t ∈ [0,T] t t X(t,ω) = X (ω)+ A(s,ω,X(s,ω))ds+ B(s,X(s))dW(s)(ω), 0 Z0 Z0 where the first integral is understood as an X∗-valued Bochner integral. Remark 3.4. Note that t t t A(s,ω,X(s,ω))ds = A (s,ω,X(s,ω))ds+ A (s,ω,X(s,ω))ds. 1 2 Z0 Z0 Z0 Since X is M/B(H)-measurable, 1Xi(X)·X is M/B(Xi)-measurable by [17, Lemma 2.1] for i = 1,2. The above integrals are meaningful. We have the following estimates for the solutions of Eq.(11). Theorem 3.5. Assume that (H1)-(H4) hold and X ∈ L2(Ω,F ,P;H). Let X(t) be any 0 0 solution of Eq. (11) in the sense of Definition 3.3. Then, we have for any m ∈ N T EkX(θ )k2 + kXkqi +kXk2 6 c EkX k2 + E(ξ(s))ds , m H Km Km m 0 H 2,i 5 i=1,2 (cid:18) Z0 (cid:19) X and qi E sup kX(t)k2 +kXk2 +kB(·,X(·))k2 + kA (·,X(·))kqi−1 H Km Km i Km t∈[0,θm] ! 4 3 i=1,2 1,i X T q1 q2 6 c EkX k2 + E ξ(s)+ηq1−1(s)+ηq2−1(s) ds , m 0 H 1 2 (cid:18) Z0 (cid:18) (cid:19) (cid:19) where c only depends on m, T and c ,q ,i = 1,2. m Ai i Proof. By Itô’s formula (Theorem 2.1) and (H3), we have kX(t)k2 −kX k2 −M(t) H 0 H t = 2[X(s),A(s,X(s))]X +kB(s,X(s))k2L2(U,H) ds Z0 (cid:0) (cid:1) t 6 − λ (s)·kX(s)kqi +λ (s)·kX(s)k2 +ξ(s) ds, (14) i X 3 H i Z0 iX=1,2(cid:16) (cid:17) ! where M(t) is a continuous local martingale given by t M(t) := 2 hX(s),B(s,X(s))dW(s)i . H Z0 For any R > 0, define the stopping time t τR := inf t ∈ [0,T] : kX(t)kH > R, λi(s)·kX(s)kqXids > R,i = 1,2 . (15) i (cid:26) Z0 (cid:27) Then, by Definition 3.3, τ ↑ T a.s. as R ↑ ∞. R By Remark 2.4 and the change of clock(cf. [27]), we know that {M(θ ∧τ ),t > 0} is t R a continuous F -martingale. Indeed, this follows from θt θt∧τR hM(θ ∧τ )i(t) 6 4 kX(s)k2 ·kB(s,X(s))k2 ds 6 c . · R H L2(U,H) R Z0 10