Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity

TL;DR

This paper investigates a class of infinite-dimensional backward stochastic differential equations with dissipative operators and locally Lipschitz nonlinearities, providing theoretical results and applications to SPDEs and spin systems.

Contribution

It introduces a framework for BSDEs with unbounded dissipative operators and locally Lipschitz nonlinearities, extending existing theory to more general infinite-dimensional settings.

Findings

Existence and uniqueness results for the class of BSDEs studied.

Application to stochastic partial differential equations.

Application to spin systems.

Abstract

In this paper we study a class of backward stochastic differential equations (BSDEs) of the form dY(t)= -AY(t)dt -f_0(t,Y(t))dt -f_1(t,Y(t),Z(t))dt + Z(t)dW(t) on the interval [0,T], with given final condition at time T, in an infinite dimensional Hilbert space H. The unbounded operator A is sectorial and dissipative and the nonlinearity f_0(t,y) is dissipative and defined for y only taking values in a subspace of H. A typical example is provided by the so-called polynomial nonlinearities. Applications are given to stochastic partial differential equations and spin systems.

Equations

$$d{Y}_t=-A{Y}_{t}dt-f_0(t,Y_t)dt-f_1(t,Y_t,Z_t)dt+Z_{t}d{W}_t,0\le t\le T;Y_T=\xi$$

$$d{Y}_t=-A{Y}_{t}dt-f(t,Y_t,Z_t)dt+Z_{t}d{W}_t,0\le t\le T,Y_T=\xi$$

$$Y_t-{\int }_t^T{e}^{(s-t)A}[f_0(s,Y_s)+f_1(s,Y_s,Z_s)]ds+{\int }_t^T{e}^{(s-t)A}{Z}_{s}d{W}_s=e^{(T-t)A}\xi ,$$

$$\left\{\begin{array}[]{ll}(i)\quad\rho(A)\supseteq S_{\theta,\omega}=\{\lambda\in\mathbb{C}:\lambda\neq\omega,|arg(\lambda-\omega)|<\theta\},\\ (ii)\quad||(\lambda I-A)^{-1}||_{\mathcal{L}(X)}\leq\frac{M}{|\lambda-\omega|}\quad\forall\lambda\in S_{\theta,\omega}\\ \end{array}\right.$$

$$e^{tA}=\frac{1}{2\pi i}{\int }_{\omega +{\gamma }_{r,\eta }}{e}^{t\lambda }(\lambda I-A{)}^{-1}d\lambda ,t>0,$$

$$||t^{-\alpha +\beta }{e}^{tA}|{|}_{L(D_A(\alpha ,p),D_A(\beta ,p))}\le C,0<t\le 1.$$

$$Y_t-{\int }_t^T{e}^{(s-t)A}[f_0(s,Y_s)+f_1(s,Y_s,Z_s)]ds+{\int }_t^T{e}^{(s-t)A}{Z}_{s}d{W}_s=e^{(T-t)A}\xi ,$$

$$Y_t-{\int }_t^T{e}^{(s-t)A}[f_0(s,Y_s)ds+f_1(s)]ds+{\int }_t^T{e}^{(s-t)A}{Z}_{s}d{W}_s=e^{(T-t)A}\xi ,$$

$$<f_0(t,y_1)-f_0(t,y_2),y_1-y_2{>}_H\le \mu |y_1-y_2{|}_H^2;$$

$$\begin{array}[]{l}\displaystyle{|Y_{t}^{n}|_{H}^{2}+\int_{t}^{T}||Z_{s}^{n}||_{\mathcal{L}^{2}(K;H)}^{2}ds}\leq\displaystyle{|J_{n}\xi|_{H}^{2}+2\int_{t}^{T}<J_{n}f_{0}(s,Y_{s}),Y_{s}^{n}>_{H}ds}+\\ \quad\quad+\displaystyle{2\int_{t}^{T}<J_{n}f_{1}(s),Y_{s}^{n}>_{H}ds-2\int_{t}^{T}<Y_{s}^{n},Z_{s}^{n}dW_{s}>_{H}}.\end{array}$$

$$\begin{array}[]{l}\displaystyle{|Y_{t}|_{H}^{2}+\int_{t}^{T}||Z_{s}||_{{\mathcal{L}}^{2}(K,H)}^{2}}\leq\\ \quad\quad\leq\displaystyle{|\xi|_{H}^{2}+2\int_{t}^{T}<f_{0}(s,0),Y_{s}>_{H}+2\int_{t}^{T}<f_{1}(s),Y_{s}>_{H}ds+}\\ \quad\quad\quad\displaystyle{-2\int_{t}^{T}<Y_{s},Z_{s}dW_{s}>_{H}}\\ \quad\quad\leq\displaystyle{|\xi|_{H}^{2}+\int_{t}^{T}|f(s,0)|_{H}^{2}ds+\int_{t}^{T}|f_{1}(s)|_{H}^{2}ds+2\int_{t}^{T}|Y_{s}|_{H}^{2}ds+}\\ \quad\quad\quad\displaystyle{-2\int_{t}^{T}<Y_{s},Z_{s}dW_{s}>_{H}}.\end{array}$$

$$\begin{array}[]{l}\displaystyle{|Y_{t}|_{H}^{2}\leq\mathbb{E}^{\mathcal{F}_{t}}|\xi|_{H}^{2}+2\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}|Y_{s}|_{H}^{2}ds+}\\ \quad\quad\quad\quad\displaystyle{+\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}|f(s,0)|_{H}^{2}ds+\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}|f_{1}(s)|_{H}^{2}ds}\\ \quad\quad\quad\displaystyle{\leq|\xi|_{L^{\infty}(\Omega,H)}^{2}+(S^{2}+C^{2})T+2\int_{t}^{T}\mathbb{E}^{\mathcal{F}_{t}}|Y_{s}|_{H}^{2}ds}.\\ \end{array}$$

$$|x{|}_{H_{\alpha }}\le c|x{|}_{D_A(\theta ,\infty )}^{\frac{\alpha }{\theta }}|x{|}_H^{1-\frac{\alpha }{\theta }},x\in D_A(\theta ,\infty ).$$

$$\underset{t\in [a,T]}{\sup }||Y_t|{|}_{H_{\alpha }}\le R,\underset{t\in [a,T]}{\sup }|Y_t{|}_H\le K.$$

$$|Y_t{|}_{L^{\infty }(\Omega ,D_A(\theta ,\infty ))}\le C_2\frac{1}{(T-t{)}^{\theta -\alpha }},a\le t<T$$

$$\begin{array}[]{l}\displaystyle{||Y_{t}||_{D_{A}(\theta,\infty)}\leq\mathbb{E}^{\mathcal{F}_{t}}||e^{(T-t)A}\xi||_{D_{A}(\theta,\infty)}}\\ \quad\quad\quad\quad\quad\displaystyle{+\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}||e^{(s-t)A}[f_{0}(s,Y_{s})+f_{1}(s)]||_{D_{A}(\theta,\infty)}ds,\quad a\leq t\leq T}.\\ \end{array}$$

$$\begin{array}[]{l}\displaystyle{\mathbb{E}^{\mathcal{F}_{t}}||e^{(T-t)A}\xi||_{D_{A}(\theta,\infty)}}\leq\\ \quad\leq\displaystyle{\mathbb{E}^{\mathcal{F}_{t}}||e^{(T-t)A}||_{L(D_{A}(\alpha,\infty),D_{A}(\theta,\infty))}||\xi||_{L^{\infty}(\Omega,D_{A}(\alpha,\infty))}}\\ \quad\leq\displaystyle{\frac{C_{0}}{(T-t)^{\theta-\alpha}}||\xi||_{L^{\infty}(\Omega,H_{\alpha})}},\end{array}$$

$$\begin{array}[]{l}\displaystyle{\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}||e^{(s-t)A}[f_{0}(s,Y_{s})+f_{1}(s)]||_{D_{A}(\theta,\infty)}ds}\leq\\ \leq\displaystyle{\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}||e^{(s-t)A}||_{L(H,D_{A}(\theta,\infty))}|f_{0}(s,Y_{s})+f_{1}(s)|_{H}ds}\leq\\ \quad\leq\displaystyle{\mathbb{E}^{\mathcal{F}_{t}}\big{(}\int_{t}^{T}\frac{C_{1}}{(s-t)^{\theta}}[|f_{0}(s,Y_{s})|_{H}+|f_{1}(s)|_{H}]ds\big{)}}\leq\\ \quad\leq\displaystyle{\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}\frac{C_{1}}{(s-t)^{\theta}}[S(1+||Y_{s}||_{H_{\alpha}}^{\gamma})+C]ds}.\\ \end{array}$$

$$\begin{array}[]{l}\quad\quad\displaystyle{\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}\frac{C_{1}}{(s-t)^{\theta}}\left[S(1+c|Y_{s}|_{H}^{\gamma(1-\alpha)/\theta}||Y_{s}||_{D_{A}(\theta,\infty)}^{\gamma\alpha/\theta})+C\right]ds}=\\ \quad\quad=\displaystyle{\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}\frac{C_{1}}{(s-t)^{\theta}}\left[S(1+c|Y_{s}|_{H}^{\gamma(1-\alpha)/\theta}||Y_{s}||_{D_{A}(\theta,\infty)})+C\right]ds,}\\ \end{array}$$

$$\begin{array}[]{l}\quad\quad\displaystyle{\int_{t}^{T}\frac{C_{1}}{(s-t)^{\theta}}S(1+cK^{\gamma(1-\alpha)/\theta}\mathbb{E}^{\mathcal{F}_{t}}||Y_{s}||_{D_{A}(\theta,\infty)}+C)ds}\\ \quad\leq\displaystyle{\int_{t}^{T}\frac{C_{1}}{(s-t)^{\theta}}(C+S)ds+\int_{t}^{T}\frac{C_{1}}{(s-t)^{\theta}}ScK^{\gamma(1-\alpha)/\theta}\mathbb{E}^{\mathcal{F}_{t}}||Y_{s}||_{D_{A}(\theta,\infty)}}ds.\\ \end{array}$$

$$\begin{array}[]{l}\displaystyle{||Y_{t}||_{D_{A}(\theta,\infty)}\leq\frac{C_{0}}{(T-t)^{\theta-\alpha}}||\xi||_{L^{\infty}(\Omega,H_{\alpha})}+\int_{t}^{T}\frac{C_{1}}{(s-t)^{\theta}}(C+S)ds}\\ \quad\quad\quad\quad\quad\displaystyle{+\int_{t}^{T}\frac{C_{1}}{(s-t)^{\theta}}ScK^{\gamma(1-\alpha)/\theta}\mathbb{E}^{\mathcal{F}_{t}}||Y_{s}||_{D_{A}(\theta,\infty)}ds},\\ \end{array}$$

$$\begin{array}[]{l}\displaystyle{||Y_{t}||_{D_{A}(\theta,\infty)}}\leq\\ \quad\leq\displaystyle{\frac{K_{1}}{(T-t)^{\theta-\alpha}}||\xi||_{L^{\infty}(\Omega,H_{\alpha})}+\mathbb{E}^{\mathcal{F}_{t}}[\sup_{s\in[a,T]}(1+||Y_{s}||_{H_{\alpha}}^{\gamma})\int_{t}^{T}\frac{ds}{(s-t)^{\theta}}]}\\ \quad\leq\displaystyle{\frac{K_{1}}{(T-t)^{\theta-\alpha}}||\xi||_{L^{\infty}(\Omega,H_{\alpha})}+(1+R^{\gamma})\int_{t}^{T}\frac{ds}{(s-t)^{\theta}}}.\\ \end{array}$$

$$v(t)={\int }_t^T{e}^{(s-t)A}\varphi (s)ds,a\le t\le T.$$

$$||v|{|}_{C^{1-\alpha }([a,T];D_A(\alpha ,1))}\le G_0||\varphi |{|}_{L^{\infty }((a,T);H)}.$$

$$||v|{|}_{C^{1-\alpha }([a,T];H_{\alpha })}\le G||\varphi |{|}_{L^{\infty }((a,T);H)}.$$

$$|f_0(t,y_1)-f_0(t,y_2){|}_H\le L_R||y_1-y_2|{|}_{H_{\alpha }}0\le t\le T,||y_i|{|}_{H_{\alpha }}\le R$$

$$\mathbb{K}=\{Y\in L^2(\Omega ;C([T-\delta ,T],H_{\alpha })):\underset{t\in [T-\delta ,T]}{\sup }||Y_t|{|}_{H_{\alpha }}\le R,a.s.\}.$$

$$Y_t-{\int }_t^T{e}^{(s-t)A}[f_0(s,U_s)ds+f_1(s)]ds+{\int }_t^T{e}^{(s-t)A}{Z}_{s}d{W}_s=e^{(T-t)A}\xi$$

$$|f_0(t,U_t)+f_1(t){|}_H\le S(1+||U_t|{|}_{H_{\alpha }}^{\gamma })+C\le S(1+R^{\gamma })+C,$$

$$Y_{t}={\mathbb{E}}^{\mathcal{F}_{t}}\big{(}e^{(T-t)A}\xi+\int_{t}^{T}e^{(s-t)A}[f_{0}(s,U_{s})+f_{1}(s)]ds\big{)}.$$

Full text

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in an infinite dimensional Hilbert space $H$, where the unbounded operator $A$ is sectorial and dissipative and the nonlinearity $f_{0}(t,y)$ is dissipative and defined for $y$ only taking values in a subspace of $H$. A typical example is provided by the so-called polynomial nonlinearities. Applications are given to stochastic partial differential equations and spin systems.

Key words Backward stochastic differential equations, stochastic evolution equations.

MSC classification. Primary: 60H15 Secondary: 35R60

1 Introduction

Let $H,K$ be real separable Hilbert spaces with norms $|\cdot|_{H}$ and $|\cdot|_{K}$. Let $W$ be a cylindrical Wiener process in $K$ defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $\{\mathcal{F}_{t}\}_{t\in[0,T]}$ denote its natural augmented filtration. Let $\mathcal{L}^{2}(K,H)$ be the Hilbert space of Hilbert-Schmidt operators from $K$ to $H$.

We are interested in solving the following backward stochastic differential equation

[TABLE]

where $\xi$ is a random variable with values in $H$, $f(t,Y_{t},Z_{t})=f_{0}(t,Y_{t})+f_{1}(t,Y_{t},Z_{t})$ and $f_{0},f_{1}$ are given functions, and the operator $A$ is an unbounded operator with domain $D(A)$ contained in $H$. The unknowns are the processes $\{Y_{t}\}_{t\in[0,T]}$ and $\{Z_{t}\}_{t\in[0,T]}$, which are required to be adapted with respect to the filtration of the Wiener process and take values in $H$, $\mathcal{L}^{2}(K,H)$ respectively.

In finite dimensional framework such type of equations has been solved by Pardoux and Peng [12] in the nonlinear case. They proved an existence and uniqueness result for the solution of the equation (1) when $A=0$, the coefficient $f(t,y,z)$ is Lipschitz continuous in both variables $y$ and $z$, and the data $\xi$ and the process $\{f(t,0,0)\}_{t\in[0,T]}$ are square integrable. Since this first result, many papers were devoted to existence and uniqueness results under weaker assumptions. In finite dimension, when $A=0$, the Lipschitz condition on the coefficient $f$ with respect to the variable $y$ is replaced by a monotonicity assumption; moreover, more general growth conditions in the variable $y$ are formulated. Let us mention the contribution of Briand and Carmona [1], for a study of polynomial growth in $L^{p}$ with $p>2$, and the work of Pardoux [11] for an arbitrary growth. In [13] Pardoux and Rascanu deal with a BSDE involving the subdifferential of a convex function; in particular, one coefficient is not everywhere defined for $y$ in ${\mathbb{R}}^{k}$.

In other works the existence of the solution is proved when the data, $\xi$ and the process $\{f(t,0,0)\}_{t\in[0,T]}$, are in $L^{p}$ for $p\in(1,2)$. El Karoui, Peng and Quenez [4] treat the case when $f$ is Lipschitz continuous; in [2] this result is generalized to the case of a monotone coefficient $f$ (both for equations on a fixed and on a random time interval) and is studied even the case $p=1$.

In the infinite-dimensional framework Hu and Peng [6], and Oksendal and Zhang [10] give an existence and uniqueness result for the equation with an operator $A$, infinitesimal generator of a strongly continous semigroup and the coefficient $f$ Lipschitz in $y$ and $z$. Pardoux and Rascanu [14] replace the operator $A$ with the subdifferential of a convex function and assume that $f$ is dissipative, everywhere defined and continuous with respect to $y$, Lipschitz with respect to $z$ and with linear growth in $y$ and $z$.

Special results deal with stochastic backward partial differential equations (BSPDEs): we recall in particular the works of Ma and Yong [8] and [9]. Earlier, Peng [16] studied a backward stochastic partial differential equation and regarded the classical Hamilton-Jacobi-Bellman equation of optimal stochastic control as special case of this problem.

Our work extends these results in a special direction. We consider an operator $A$ which is the generator of an analytic contraction semigroup on $H$ and a coefficient $f(t,y,z)$ of the form $f_{0}(t,y)+f_{1}(t,y,z)$. The coefficient $f_{1}(t,y,z)$ is assumed to be bounded and Lipschitz with respect to $y$ and $z$. The term $f_{0}(t,y)$ is defined for $y$ only taking values in a suitable subspace $H_{\alpha}$ of $H$ and it satisfies the following growth condition for some $1<\gamma<1/\alpha$, $S\geq 0$, $\mathbb{P}$-a.s.

$\quad\quad\quad\quad|f_{0}(t,y)|_{H}\leq S(1+||y||_{H_{\alpha}}^{\gamma})\quad\forall t\in[0,T],\quad\forall y\in H_{\alpha}.$

Following [6], we understand the equation (1) in the following integral form

[TABLE]

requiring, in particular, that $Y$ takes values in $H_{\alpha}$. This requires generally that the final condition also takes values in the smaller space $H_{\alpha}$. We take as $H_{\alpha}$ a real interpolation space which belongs to the class $J_{\alpha}$ between $H$ and the domain of an operator $A$ (see Section 2). Moreover $f_{0}(t,\cdot)$ is assumed to be locally Lipschitz from $H_{\alpha}$ into $H$ and dissipative in $H$. We prove (Theorem 5) that if $\xi$ takes its values in the closure of $D(A)$ in $H_{\alpha}$ and is such that $||\xi||_{H_{\alpha}}$ is essentially bounded, then equation (2) has a unique mild solution, i.e. there exists a unique pair of progressively measurable processes $Y:\Omega\times[0,T]\rightarrow H_{\alpha}$, $Z:\Omega\times[0,T]\rightarrow{\mathcal{L}}^{2}(K;H)$, satisfying $\mathbb{P}$-a.s. equality (2) for every $t$ in $[0,T]$ and such that $\mathbb{E}\sup_{t\in[0,T]}||Y_{t}||_{H_{\alpha}}^{2}+\mathbb{E}\int_{0}^{T}||Z_{t}||_{\mathcal{L}^{2}(K,H)}^{2}dt<\infty.$

This result extends former results concerning the deterministic case to the stochastic framework: see [7], where previous works of Fujita - Kato [5], Pazy [15] and others are collected. In these papers similar assumptions are made on the coefficients $f_{0}$, $f_{1}$ and on the operator $A$.

The plan of the paper is as follows. In Section 2 some notations and definitions are fixed. In Section 3 existence and uniqueness of the solution of a simplified equation are proved, where $f_{1}$ is a bounded progressively measurable process which does not depend on $y$ and $z$. In Section 4, applying the previous result, a fixed point argument is used in order to prove our main result on existence and uniqueness of a mild solution of (2). Section 5 is devoted to applications.

2 Notations and setting

The letters $K$ and $H$ will always denote two real separable Hilbert spaces. Scalar product is denoted by $\langle\cdot,\cdot\rangle$; $\mathcal{L}^{2}(K;H)$ is the separable Hilbert space of Hilbert-Schmidt operators from $K$ to $H$ endowed with the Hilbert-Schmidt norm. $W=\{W_{t}\}_{t\in[0,T]}$ is a cylindrical Wiener process with values in $K$, defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$. $\{\mathcal{F}_{t}\}_{t\in[0,T]}$ is the natural filtration of $W$, augmented with the family of $\mathbb{P}$-null sets of $\mathcal{F}$.

Next we define several classes of stochastic processes with values in a Banach space $X$.

• •

$L^{2}(\Omega\times[0,T];X)$ denotes the space of measurable $X$-valued processes $Y$ such that $\displaystyle{\left[\mathbb{E}\int_{0}^{T}|Y_{\tau}|^{2}\,d\tau\right]^{1/2}}$ is finite, identified up to modification.

• •

$L^{2}(\Omega;C([0,T];X))$ denotes the space of continuous $X$-valued processes $Y$ such that $\displaystyle{\left[\mathbb{E}\sup_{\tau\in[0,T]}|Y_{\tau}|^{2}\right]^{1/2}}$ is finite, identified up to indistinguishability.

• •

$C^{\alpha}([0,T];X)$ denotes the space of $\alpha$-Hölderian functions on $[0,T]$ with values in $X$ such that $[f]_{\alpha}=\displaystyle{\sup_{0\leq x<y\leq T}\frac{|f(x)-f(y)|}{(y-x)^{\alpha}}<\infty}$.

Now we need to recall several preliminaries on semigroup and interpolation spaces. We refer the reader to [7] for the proofs and other related results.

A linear operator $A$ in a Banach space $X$, with domain $D(A)\subset X$, is called sectorial if there are constants $\omega\in\mathbb{R}$, $\theta\in(\pi/2,\pi)$, $M>0$ such that

[TABLE]

where $\rho(A)$ is the resolvent set of $A$. For every $t>0$, (3) allows us to define a linear bounded operator $e^{tA}$ in $X$, by means of the Dunford integral

[TABLE]

where, $r>0,\eta\in(\pi/2,\pi)$ and $\gamma_{r,\eta}$ is the curve $\{\lambda\in\mathbb{C}:|arg\lambda|=\eta,|\lambda|\geq r\}\cup\{\lambda\in\mathbb{C}:|arg\lambda|\leq\eta,|\lambda|=r\}$, oriented counterclockwise. We also set $e^{0A}x=x,\forall x\in X.$ Since the function $\lambda\mapsto e^{t\lambda}R(\lambda,A)$ is holomorphic in $S_{\theta,\omega}$, the definition of $e^{tA}$ is independent of the choice of $r$ and $\eta$. If $A$ is sectorial, the function $[0,+\infty)\rightarrow L(X)$, $t\mapsto e^{tA}$, with $e^{tA}$ defined by (4) is called analytic semigroup generated by A in $X$. We note that for every $x\in X$ the function $t\mapsto e^{tA}x$ is analytic (and hence continuous) for $t>0$. $e^{tA}$ is a strongly continuous semigroup if and only if $D(A)$ is dense in $X$; in particular this holds if $X$ is a reflexive space.

We need to introduce suitable classes of subspaces of $X$.

Definition 1**.**

Let $(\alpha,p)$ be two numbers such that $0<\alpha<1$, $1\leq p\leq\infty$ or $(\alpha,p)=(1,\infty)$. Then we denote with $D_{A}(\alpha,p)$ the space

$\quad\quad\quad\quad D_{A}(\alpha,p)=\{x\in X:t\mapsto v(t)=||t^{1-\alpha-1/p}Ae^{tA}x||\in L^{p}(0,1)\}$

where $||x||_{D_{A}(\alpha,p)}=||x||_{X}+[x]_{\alpha}=||x||_{X}+||v||_{L^{p}(0,1)}.$

(We set as usual $1/\infty=0$).

We recall here some estimates for the function $t\mapsto e^{tA}$ when $t\rightarrow 0$, which we will use in the sequel. For convenience, in the next proposition we set $D_{A}(0,p)=X,\quad p\in[1,\infty].$

Proposition 1**.**

Let $(\alpha,p)$, $(\beta,p)\in(0,1)\times[1,+\infty]\cup\{(1,\infty)\}$, $\alpha\leq\beta$. Then there exists $C=C(p;\alpha,\beta)$ such that

[TABLE]

Definition 2**.**

Let $0\leq\alpha\leq 1$ and let $D,X$ be Banach spaces, $D\subset X$. A Banach space $Y$ such that $D\subset Y\subset X$ is said to belong to the class $J_{\alpha}$ between $X$ and $D$ if there is a constant $C$ such that $||x||_{Y}\leq C||x||_{X}^{1-\alpha}||x||_{D}^{\alpha},\quad\forall x\in D.$ In this case we write $Y\in J_{\alpha}(X,D)$.

Now we give the definition of solution to the BSDE:

[TABLE]

Definition 3**.**

A pair of progressively measurable processes $(Y,Z)$ is called mild solution of (5) if it belongs to the space $L^{2}(\Omega;C([0,T];H_{\alpha}))\times L^{2}(\Omega\times[0,T];\mathcal{L}^{2}(K,H))$ and $\mathbb{P}$-a.s.solves the integral equation (5) on the interval $[0,T]$.

We finally state a lemma needed in the sequel. It is a generalization of the well known Gronwall’s lemma. Its proof is given in the Appendix.

Lemma 1**.**

Assume $a,b,\alpha,\beta$ are nonnegative constants, with $\alpha<1$, $\beta>0$ and $0<T<\infty$. For any nonnegative process $U\in L^{1}(\Omega\times[0,T])$, satisfying $\mathbb{P}$-a.s. $U_{t}\leq a(T-t)^{-\alpha}+b\int_{t}^{T}(s-t)^{\beta-1}\mathbb{E}^{\mathcal{F}_{t}}U_{s}ds$ for almost every $t\in[0,T]$, it holds $\mathbb{P}$-a.s. $U_{t}\leq aM(T-t)^{-\alpha},$ for almost every $t\in[0,T]$. $M$ is a constant depending only on $b,\alpha,\beta,T$.

3 A simplified equation

As a preparation for the study of (2), in this section we consider the following simplified version of that equation:

[TABLE]

for all $t\in[0,T]$.

We suppose that the following assumptions hold.

Hypothesis 2**.**

1.

$A:D(A)\subset H\rightarrow H$ is a sectorial operator. We also assume that $A$ is dissipative, i.e. it satisfies $<Ay,y>\leq 0,\forall y\in D(A)$;

2.

for some $0<\alpha<1$ there exists a Banach space $H_{\alpha}$ continuously embedded in $H$ and such that

(i) $\quad D_{A}(\alpha,1)\subset H_{\alpha}\subset D_{A}(\alpha,\infty)$;

(ii) $\quad\mbox{the part of }A\mbox{ in }H_{\alpha}\mbox{ is sectorial in }H_{\alpha}.$

3.

the final condition $\xi$ is an $\mathcal{F}_{T}$-measurable random variable defined on $\Omega$ with values in the closure of $D(A)$ with respect to $H_{\alpha}$-norm. We denote this set $\overline{D(A)}^{H_{\alpha}}$. Moreover $\xi$ belongs to $L^{\infty}(\Omega,\mathcal{F}_{T},\mathbb{P};H_{\alpha})$;

4.

$f_{0}:\Omega\times[0,T]\times H_{\alpha}\rightarrow H$ satisfies:

i) $\{f_{0}(t,y)\}_{t\in[0,T]}$ is progressively measurable $\forall y\in H_{\alpha}$;

ii) there exist constants $S>0$, $1<\gamma<1/\alpha$ such that $\mathbb{P}$-a.s.

$\quad\quad\quad|f_{0}(t,y)|_{H}\leq S(1+||y||_{H_{\alpha}}^{\gamma})\quad t\in[0,T],y\in H_{\alpha};$

iii) for every $R>0$ there is $L_{R}>0$ such that $\mathbb{P}$-a.s.

$\quad\quad\quad|f_{0}(t,y_{1})-f_{0}(t,y_{2})|_{H}\leq L_{R}||y_{1}-y_{2}||_{H_{\alpha}}$

for $t\in[0,T]$ and $y_{i}\in H_{\alpha}$ with $||y_{i}||_{H_{\alpha}}\leq R$;

iv) there exists a number $\mu\in\mathbb{R}$ such that $\mathbb{P}\text{-a.s.}$, $\forall t\in[0,T]$, $y_{1},y_{2}\in H_{\alpha}$,

[TABLE]

5.

$f_{1}:\Omega\times[0,T]\rightarrow H$ is progressively measurable and for some constant $C>0$ it satisfies $\mathbb{P}$-a.s. $|f_{1}(t)|_{H}\leq C,$ for $t\in[0,T]$.

Remark 1**.**

We note that the pair $(Y,Z)$ solves the BSDE (6) with final condition $\xi$ and drift $f=f_{0}+f_{1}$ if and only if the pair $(\bar{Y},\bar{Z}):=(e^{\lambda t}Y_{t},e^{\lambda t}Z_{t})$ is a solution of the same equation with final condition $e^{\lambda T}\xi$ and drift $f^{\prime}(t,y):=f_{0}^{{}^{\prime}}(t,y)+f_{1}^{{}^{\prime}}(t)$ where $f_{0}^{{}^{\prime}}(t,y)=e^{\lambda t}(f_{0}(t,e^{-\lambda t}y)-\lambda y)$, $f_{1}^{{}^{\prime}}(t)=e^{\lambda t}f_{1}(t).$ If we choose $\mu=\lambda$, then $f_{0}^{{}^{\prime}}$ satisfies the same assumption as $f_{0}$, but with (7) replaced by $<f_{0}(t,y_{1})-f_{0}(t,y_{2}),y_{1}-y_{2}>_{H}\leq 0.$ If this last condition holds, then $f_{0}$ is called dissipative. Hence, without loss of generality, we shall assume until the end that $f_{0}$ is dissipative, or equivalently that $\mu=0$ in (7).

3.1 A priori estimates

We prove a basic estimate for the solution in the norm of $H$.

Proposition 2**.**

Suppose that Hypothesis 2 holds; if $(Y,Z)$ is a mild solution of (6) on the interval $[a,T]$, $0\leq a\leq T$, then there exists a constant $C_{1}$, which depends only on $||\xi||_{L^{\infty}(\Omega;H)}$ and on the constants S of 4.ii) and C of 5. such that $\mathbb{P}$-a.s. $\sup_{a\leq t\leq T}||Y_{t}||_{H}\leq C_{1}.$ In particular the constant $C_{1}$ is independent of $a$.

Proof.

Let the pair $(Y,Z)\in L^{2}(\Omega,C([a,T];H_{\alpha})\times L^{2}(\Omega\times[a,T];\mathcal{L}^{2}(K;H))$ satisfy (6). Let us introduce the operators $J_{n}=n(nI-A)^{-1}$, $n>0$. We note that the operators $AJ_{n}$ are the Yosida approximations of $A$ and they are bounded. Moreover $|J_{n}x-x|\rightarrow 0$ as $n\rightarrow\infty$, for every $x\in H$. We set $Y_{t}^{n}=J_{n}Y_{t}$, $Z_{t}^{n}=J_{n}Z_{t}$. It is readily verified that $Y^{n}$ admits the Itô differential

$dY_{t}^{n}=-AY_{t}^{n}dt-J_{n}f(t,Y_{t})dt-J_{n}f_{1}(t)dt+Z_{t}^{n}dW_{t},\mbox{and}\quad Y_{T}^{n}=J_{n}\xi.$

Applying the Ito formula to $|Y_{t}^{n}|_{H}^{2}$, using the dissipativity of $A$, we obtain

[TABLE]

We note that $\int_{t}^{T}<J_{n}f_{0}(s,Y_{s})+J_{n}f_{1}(s),Y_{s}^{n}>_{H}ds\rightarrow\int_{t}^{T}<f_{0}(s,Y_{s})+f_{1}(s),Y_{s}>_{H}ds$ by dominated convergence, as $n\rightarrow\infty$. Moreover by the dominated convergence theorem we have $\int_{t}^{T}||(Z_{s}^{n})^{*}Y_{s}^{n}-Z_{s}^{*}Y_{s}||_{K}^{2}ds\rightarrow 0\quad\mathbb{P}\mbox{-a.s.}$ and it follows that $\int_{t}^{T}<Y_{s}^{n},Z_{s}^{n}dW_{s}>_{H}\rightarrow\int_{t}^{T}<Y_{s},Z_{s}dW_{s}>_{H}$ in probability. If we let $n\rightarrow\infty$ in (8) we obtain

$\displaystyle{|Y_{t}|_{H}^{2}+\int_{t}^{T}||Z_{s}||_{\mathcal{L}^{2}(K;H)}^{2}ds}\leq\displaystyle{|\xi|_{H}^{2}+2\int_{t}^{T}<f_{0}(s,Y_{s})+f_{1}(s),Y_{s}>_{H}ds}$

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\displaystyle{-2\int_{t}^{T}<Y_{s},Z_{s}dW_{s}>_{H}.}$

Recalling (7), that we assume to hold with $\mu=0$, it follows that

[TABLE]

Now, since $\sup_{0\leq t\leq T}|f(t,0)|_{H}^{2}\leq S^{2}$ and since the stochastic integral $\int_{a}^{t}<Y_{s},Z_{s}dW_{s}>_{H}$, $t\in[a,T]$ is a martingale, if we take the conditional expectation given $\mathcal{F}_{t}$ we have

[TABLE]

Since $Y$ belongs to $L^{2}(\Omega;C([a,T];H_{\alpha}))$ and, consequently, $||Y||_{H_{\alpha}}^{2}\in L^{1}(\Omega\times[0,T])$, we can apply Lemma 1 to $|Y|_{H}^{2}$ and conclude that

$\quad\quad\quad\quad\quad|Y_{t}|_{H}^{2}\leq(|\xi|_{L^{\infty}(\Omega,H)}^{2}+[S^{2}+C^{2}]T)(1+2Te^{2T}).$ ∎

Now we will show that the result of Proposition 2, together with the growth condition satisfied by $f_{0}$, yields an a priori estimate on the solution in the $H_{\alpha}$-norm.

Let $0<\alpha<1$ and let $\gamma>1$ be given by 4.ii). We fix $\theta=\alpha\gamma$ and consider the Banach space $D_{A}(\theta,\infty)$ introduced in Definition 1. It is easy to check (see [7]) that, if we take $\theta\in(0,1),\theta>\alpha$, then $H_{\alpha}$ contains $D_{A}(\theta,\infty)$ and belongs to the class $J_{\alpha/\theta}$ between $D_{A}(\theta,\infty)$ and $H$, hence the following inequality is satisfied:

[TABLE]

Proposition 3**.**

Suppose that Hypothesis 2 is satisfied. Let $(Y,Z)$ be a mild solution of (6) in $[a,T]$, $a\geq 0$ and assume that there exists two constants $R>0$ and $K>0$, possibly depending on $a$, such that, $\mathbb{P}$-a.s.,

[TABLE]

Then the following inequality holds $\mathbb{P}$-a.s.:

[TABLE]

with $C_{2}$ depending on the operator $A$, $||\xi||_{L^{\infty}(\Omega,H_{\alpha})}$, $\theta$, $\alpha$, $K$, $C$ of 5. and $S$ of 4.ii) of Hypothesis 2.

Proof.

Taking the conditional expectation given $\mathcal{F}_{t}$ in equation (6) we find

\quad\quad Y_{t}=\mathbb{E}^{\mathcal{F}_{t}}\big{(}e^{(T-t)A}\xi+\int_{t}^{T}e^{(s-t)A}[f_{0}(s,Y_{s})+f_{1}(s)]ds\big{)},\quad a\leq t\leq T.

Consequently, we have

[TABLE]

Since $H_{\alpha}\subset D_{A}(\alpha,\infty)$, we have

[TABLE]

with $C_{0}=C_{0}(\alpha,\theta,\infty)$, where in the last inequality we use Proposition 1. Moreover

[TABLE]

In the inequality we used Hypotheses 4.ii) and 5. and Proposition 1. Recalling (9), we conclude that the last term is dominated by

[TABLE]

by choosing $\theta=\alpha\gamma$. By the second inequality in (10) this can be estimated by

[TABLE]

Hence by (13) and (14) it follows

[TABLE]

and (11) follows from Lemma 1. In order to justify the application of Lemma 1, we need to prove that $||Y||_{D_{A}(\theta,\infty)}$ belongs to $L^{1}(\Omega\times[a,T])$. This also follows from(13) and (14) since, for some constant $K_{1}$,

[TABLE]

∎

3.2 Local existence and uniqueness

We prove that, under Hypothesis 2, there exists a unique solution of (6) on an interval $[T-\delta,T]$ with $\delta$ sufficiently small.

To treat the ordinary integral in the left hand side of (6), we need the following result, whose proof can be found in [7], Proposition 4.2.1 and Lemma 7.1.1.

Lemma 3**.**

Let $\phi\in L^{\infty}((a,T);H),\quad 0<a<T$ and set

[TABLE]

If $0<\alpha<1$, then $v\in C^{1-\alpha}([a,T];D_{A}(\alpha,1))$ and there is $G_{0}>0$, not depending on $a$, such that

[TABLE]

Since $D_{A}(\alpha,1)\subset H_{\alpha}$, we also have $v\in C^{1-\alpha}([a,T];H_{\alpha})$ and there is $G>0$, not depending on $a$, such that

[TABLE]

Theorem 4**.**

Let us assume that Hypothesis 2 holds, except possibly 4.iv). Then there exists $\delta>0$ such that the equation (6) has a unique local mild solution $(Y,Z)\in L^{2}(\Omega;C([T-\delta,T];H_{\alpha}))\times L^{2}(\Omega\times[T-\delta,T];{\mathcal{L}}^{2}(K;H))$.

Remark 2**.**

The dissipativity condition 4.iv) only plays a role in obtaining the a priori estimate in $H$ (Proposition 2) and consequently global existence, as we will see later.

Proof.

Let $M_{\alpha}:=\sup_{0\leq t\leq T}||e^{tA}||_{L(H_{\alpha})}$. We fix a positive number $R$ such that $R\geq 2M_{\alpha}||\xi||_{L^{\infty}(\Omega;H_{\alpha})}$. This implies that $\sup_{0\leq t\leq T}||e^{tA}\xi||_{H_{\alpha}}\leq R/2$ $\mathbb{P}$-a.s. Moreover, let $L_{R}$ be such that

[TABLE]

We recall that the space $L^{2}(\Omega;C([T-\delta,T];H_{\alpha}))$ is a Banach space endowed with the norm Y\rightarrow\big{(}\mathbb{E}\sup_{t\in[T-\delta,T]}||Y_{t}||_{H_{\alpha}}^{2}\big{)}^{1/2}. We define

[TABLE]

It easy to check that $\mathbb{K}$ is a closed subset of $L^{2}(\Omega;C([T-\delta,T],H_{\alpha}))$, hence a complete metric space (with the inherited metrics). We look for a local mild solution $(Y,Z)$ in the space $\mathbb{K}$. We define a nonlinear operator $\Gamma:\mathbb{K}\rightarrow\mathbb{K}$ as follows: given $U\in\mathbb{K}$, $Y=\Gamma(U)$ is the first component of the mild solution $(Y,Z)$ of the equation

[TABLE]

for $t\in[T-\delta,T]$. Since $U\in\mathbb{K}$ we have $\mathbb{P}$-a.s.

[TABLE]

for all $t$ in $[T-\delta,T]$. Hence $f_{0}(\cdot,U_{\cdot})+f_{1}(\cdot)$ belongs to $L^{2}(\Omega\times[T-\delta,T];H)$ and, by a result of Hu and Peng [6], there exists a unique pair $(Y,Z)\in L^{2}(\Omega\times[T-\delta,T];H)\times L^{2}(\Omega\times[T-\delta,T];{\mathcal{L}}^{2}(K;H))$ satisfying (15). Moreover, by taking the conditional expectation given $\mathcal{F}_{t}$, $Y$ has the following representation

[TABLE]

We will show that $\Gamma$ is a contraction for the norm of $L^{2}(\Omega,C([T-\delta,T];H_{\alpha})$ and maps $\mathbb{K}$ into itself, if $\delta$ is sufficiently small; clearly, its unique fixed point is the required solution of the BSDE.

We first check the contraction property. Let $U^{1},U^{2}\in\mathbb{K}$. Then

[TABLE]

Let v(t)=\int_{t}^{T}e^{(s-t)A}\big{(}f_{0}(s,U_{s}^{1})-f_{0}(s,U_{s}^{2})\big{)}ds. Then, noting that $v(T)=0$ and recalling Lemma 3, for $t\in[T-\delta,T]$

[TABLE]

where $\{M_{t},\,t\in[T-\delta,T]\}$ is a martingale. Hence, by Doob’s inequality

[TABLE]

If $\delta\leq{\delta}_{0}={2GL_{R}}^{\frac{-1}{(1-\alpha)}}$, then $\Gamma$ is a contraction with constant $1/2$.

Next we check that $\Gamma$ maps $\mathbb{K}$ into itself. For each $U\in\mathbb{K}$ and $t\in[T-\delta,T]$ with $\delta\leq{\delta}_{0}$ we have

[TABLE]

where in the last inequality we have used the fact that $D_{A}(\alpha,1)\subset H_{\alpha}$. Now, by Proposition 1, and from 4.ii) and 5., it follows that

[TABLE]

Then, since $U\in\mathbb{K}$, we arrive at

[TABLE]

where $C_{\alpha}$ depends on $A$, $\alpha$. Hence, if $\delta\leq{\delta}_{0}$ is such that $C_{\alpha}S\frac{[(1+R^{\gamma})+C]}{1-\alpha}{\delta}^{1-\alpha}$ is less or equal to $R/2$, then $\displaystyle{\sup_{t\in[T-\delta,T]}||{\Gamma(U)}_{t}||_{H_{\alpha}}}\leq R$. Due to Lemma 3, $\mathbb{P}$-a.s. the function $t\mapsto Y_{t}-{\mathbb{E}}^{\mathcal{F}_{t}}e^{(T-t)A}\xi$ belongs to $C[T-\delta,T];H_{\alpha})$; moreover, the map $t\mapsto{\mathbb{E}}^{\mathcal{F}_{t}}e^{(T-t)A}\xi$ belongs to $C[T-\delta,T];H_{\alpha})$, since $\xi$ is a random variable taking values in $\overline{D(A)}^{H_{\alpha}}$. Therefore, $\mathbb{P}$-a.s. $Y_{\cdot}\in C([T-\delta,T];H_{\alpha})$ and $\Gamma$ maps $\mathbb{K}$ into itself and has a unique fixed point in $\mathbb{K}$. ∎

Remark 3**.**

By Lemma 3, using properties of analytic semigroups, it can be proved that for every fixed $\omega$ the range of the map $\Gamma$ is contained in $C^{1-\beta}([T-\delta,T-\epsilon];D_{A}(\beta,1))$ for every $\epsilon\in(0,\delta)$, $\beta\in[0,1]$.

3.3 Global existence

Now we are able to prove a global existence theorem for the solution of the equation (6), using all the results presented above.

Theorem 5**.**

If Hypothesis 2 is satisfied, the equation (6) has a unique mild solution $(Y,Z)\in L^{2}(\Omega;C([0,T],H_{\alpha}))\times L^{2}(\Omega\times[0,T]);\mathcal{L}^{2}(K;H))$.

Proof.

By Theorem 4 equation (6) has a unique mild solution $(Y^{1},Z^{1})\in L^{2}(\Omega;C([T-\delta_{1},T],H_{\alpha}))\times L^{2}(\Omega\times[T-\delta_{1},T]);\mathcal{L}^{2}(K;H))$ on the interval $[T-\delta_{1},T]$, for some $\delta_{1}>0$. By Proposition 2 we know that there exists a constant $C_{1}$ such that $\mathbb{P}$-a.s.

[TABLE]

We recall that the constant $C_{1}$ depends only on $|\xi|_{L^{\infty}(\Omega;H)}$ and on the constants $S$ of 4.ii) and $C$ of 5. and is independent of $\delta_{1}$. Moreover, by Proposition 3, there exists a constant $C_{2}$ such that $\mathbb{P}$-a.s.

[TABLE]

with $C_{2}$ depending on the operator $A$, $||\xi||_{L^{\infty}(\Omega,H_{\alpha})}$, $\theta$, $\alpha$, $C_{1}$. This implies that $Y_{T-\delta_{1}}$ belongs to $L^{\infty}(\Omega;H_{\alpha})$ and it can be taken as final value for the problem

[TABLE]

on an interval $[T-\delta_{1}-\delta_{2},T-\delta_{1}]$, for some $\delta_{2}>0$. As in the proof of Theorem 4, we fix a positive number $R_{2}$ such that

[TABLE]

By Theorem 4 there exists a pair of progressively measurable processes $(Y^{2},Z^{2})$ in $L^{2}(\Omega;C([T-\delta_{1}-\delta_{2},T-\delta_{1}];H_{\alpha}))\times L^{2}(\Omega\times[T-\delta_{1}-\delta_{2},T-\delta_{1}];{\mathcal{L}}^{2}(K,H))$ which solves (19) on the interval $[T-\delta_{1}-\delta_{2},T-\delta_{1}]$ where $\delta_{2}$ depends on the operator $A$, $\alpha$, $R_{2}$. We note that the continuity in $T-\delta_{1}$ of $Y^{2}$ follows from the fact that $Y_{T-\delta_{1}}$ takes values in $D_{A}(\alpha,1)$ (see Remark 3), so that $Y_{T-\delta_{1}}$ takes values in $\overline{D(A)}^{H_{\alpha}}$. Now, the process $Y_{t}$ defined by $Y_{t}^{1}$ on the interval $[T-\delta_{1},T]$ and by $Y_{t}^{2}$ on $[T-\delta_{1}-\delta_{2},T-\delta_{1}]$ belongs to $L^{2}(\Omega;C([T-\delta_{1}-\delta_{2},T];H_{\alpha}))$ and it easy to see that it satisfies (6) in the whole interval $[T-\delta_{1}-\delta_{2},T]$. Consequently, by Proposition 2, $\mathbb{P}$-a.s., $|Y_{T-\delta_{1}-\delta_{2}}|_{H}\leq C_{1}$ with $C_{1}$ the constant in (17), and by (18)

[TABLE]

where $C_{2}$ is the same constant as in (18). Again, $Y_{T-\delta_{1}-\delta_{2}}$ can be taken as initial value for problem

[TABLE]

on the interval $[T-\delta_{1}-\delta_{2}-\delta_{3},T-\delta_{1}-\delta_{2}]$, where $\delta_{3}$ will be fixed later. In this case, by (20), we can choose

[TABLE]

and prove that there exists a unique mild solution $(Y^{3},Z^{3})$ of (21) on the interval $[T-\delta_{1}-\delta_{2}-\delta_{3},T-\delta_{1}-\delta_{2}]$, with $\delta_{3}=\delta_{2}$ . So we extend the solution to $[T-\delta_{1}-2\delta_{2},T]$. Proceeding this way we prove the global existence to (6) on $[0,T]$. ∎

4 The general case

We can now study the equation:

[TABLE]

We require that the function $f_{1}$ satisfy the following assumptions:

Hypothesis 6**.**

1.

there exists $K\geq 0$ such that $\mathbb{P}$-a.s.

$\quad\quad|f_{1}(t,y,z)-f_{1}(t,y^{{}^{\prime}},z^{{}^{\prime}})|_{H}\leq K|y-y^{{}^{\prime}}|_{H}+K||z-z^{{}^{\prime}}||_{{\mathcal{L}}^{2}(K;H)},$

for every $t\in[0,T],y,y^{{}^{\prime}}\in H,z,z^{{}^{\prime}}\in{\mathcal{L}}^{2}(K;H),$

2.

there exists $C\geq 0$ such that $\mathbb{P}$-a.s. $|f_{1}(t,y,z)|_{H}\leq C,$ for every $t\in[0,T],y\in H,z\in{\mathcal{L}}^{2}(K;H)$.

Theorem 7**.**

If Hypotheses 2 and 6 hold, then equation (22) has a unique solution in $L^{2}(\Omega;C([0,T];H_{\alpha}))\times L^{2}(\Omega\times[0,T];{\mathcal{L}}^{2}(K;H))$.

Proof.

Let $\mathbb{M}$ be the space of progressive processes $(Y,Z)$ in the space $L^{2}(\Omega\times[0,T];H)\times L^{2}(\Omega\times[0,T];{\mathcal{L}}^{2}(K;H))$ endowed with the norm

[TABLE]

where $\beta$ will be fixed later. We define $\Phi:\mathbb{M}\rightarrow\mathbb{M}$ as follows: given $(U,V)\in\mathbb{M}$, $(Y,Z)=\Phi(U,V)$ is the unique solution on the interval $[0,T]$ of the equation

[TABLE]

By Theorem 5 the above equation has a unique mild solution $(Y,Z)$ which belongs to $L^{2}(\Omega;C([0,T];H_{\alpha}))\times L^{2}(\Omega\times[0,T];{\mathcal{L}}^{2}(K;H))$. Therefore $\Phi(\mathbb{M})\subset\mathbb{M}$. We will show that $\Phi$ is a contraction for a suitable choice of $\beta$; clearly, its unique fixed point is the required solution of (22). We take another pair $(U^{{}^{\prime}},V^{{}^{\prime}})\in\mathbb{M}$ and apply Proposition 3.1 in [3] to the difference of two equations. We obtain

[TABLE]

where we have used 4.iv) of Hypothesis 2 and 1. of Hypothesis 6. Choosing $\beta=4K^{2}+1$, we obtain the required contraction property. ∎

5 Applications

In this section we present some backward stochastic partial differential problems which can be solved with our techniques.

5.1 The reaction-diffusion equation

Let $D$ be an open and bounded subset of $\mathbb{R}^{n}$ with a smooth boundary $\partial D$. We choose $K=L^{2}(D)$. This choice implies that $dW_{t}/dt$ is the so-called ”space-time white noise”. Moreover, since Hilbert-Schmidt operators on $L^{2}(D)$ are represented by square integrable kernels, the space $\mathcal{L}^{2}(L^{2}(D),L^{2}(D))$ can be identified with $L^{2}(D\times D)$. We are given a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a filtration $(\mathcal{F}_{t})_{t\in[0,T]}$ generated by $W$ and augmented in the usual way. Let us consider a non symmetric bilinear, coercive continuous form $a:H_{0}^{1}(D)\times H_{0}^{1}(D)\rightarrow\mathbb{R}$ defined by $a(u,v):=-\int_{D}\sum_{i,j}^{n}a_{ij}(x)D_{i}u(x)D_{j}v(x)dx,$ where the coefficients $a_{ij}$ are Lipschitz continuous and there exists $\alpha>0$ such that $\sum_{i,j=1}^{n}a_{ij}(x)\xi_{i}\xi_{j}\geq\alpha|\xi|^{2}$ for every $x\in\overline{D}$, $\xi\in{\mathbb{R}}^{n}$. Let $A$ be the operator associated with the bilinear form $a$ such that $<Au,v>_{L^{2}(D)}=a(u,v)$, $v\in H_{0}^{1}(D)$ and $u\in D(A)$. It is known that, in this case, $D(A)=H^{2}(D)\cap H_{0}^{1}(D)$, where $H^{2}(D)$ and $H_{0}^{1}(D)$ are the usual Sobolev spaces.

We consider for $t\in[0,T]$ and $x\in D$ the backward stochastic problem written formally

[TABLE]

We suppose the following.

Hypothesis 8**.**

1.

$r:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous, increasing and locally Lipschitz function;

2.

$r$ satisfies the following growth condition: $|r(x)|\leq S(1+|x|^{\gamma})\quad\forall x\in\mathbb{R}$ for some $\gamma>1$;

3.

$g$ is a measurable real function defined on $[0,T]\times\mathbb{R}\times L^{2}(D\times D)\times D$ and there exists a constant $K>0$ such that

[TABLE]

for all $t\in[0,T]$, $y_{1},y_{2}\in\mathbb{R}$, $z_{1},z_{2}\in L^{2}(D)$, $x\in D$;

4.

there exists a real function $h$ in $L^{2}(D\times D)$ such that $\mathbb{P}$-a.s. $|g(t,y,z,x)|\leq K_{1}h(x)$ for all $t\in[0,T]$, $y\in\mathbb{R}$, $z\in L^{2}(D)$, $x\in D$;

5.

$\xi$ belongs to $L^{\infty}(\Omega;H^{2}(D)\cap H_{0}^{1}(D))$.

We define the operator $A$ by $(Ay)(x)=Ay(x)$ with domain $D(A)=H^{2}(D)\cap H_{0}^{1}(D)$. We set $f_{0}(t,y)(x)=-r(y(t,x))$ for $t\in[0,T]$, $x\in D$ and $y$ in a suitable subspace of $H$ which will be determined below. For $t\in[0,T]$, $x\in D$, $y\in L^{2}(D)$, $z\in L^{2}(D\times D)$ we define $f_{1}$ as the operator $f_{1}(t,y,z)(x)=-g(t,y(t,x),z(t,x),x)$. Then problem (23) can be written in abstract way as

$\quad\quad dY_{t}=-AY_{t}dt-f_{0}(t,Y_{t})dt-f_{1}(t,Y_{t},Z_{t})dt+Z_{t}dW_{t},\quad Y_{T}=\xi.$

Under the conditions in Hypothesis 8, the assumptions in Hypotheses 2, 6 are satisfied. The operator $A$ is a closed operator in $L^{2}(D)$ and it is the infinitesimal generator of an analytic semigroup in $L^{2}(D)$ satisfying $\|e^{tA}\|_{\mathcal{L}(H)}\leq 1$ (see [17], Chapter 3). In particular, by Lumer-Philips theorem, $A$ is dissipative. The non linear function $f_{0}(t,\cdot):L^{2\gamma}(D)\rightarrow L^{2}(D)$, $y\mapsto-r(y)$ is locally Lipschitz. We look for a space of class $J_{\alpha}$ between $H$ and $D(A)$ where $f_{0}$ is well defined and locally Lipschitz. It is well known (see [18]) that the fractional order Sobolev space $W^{\beta,2}(D)$ is of class $J_{\beta/2}$ between $L^{2}(D)$ and $H^{2}(D)$ for every $\beta\in(0,2)$. Hence the space $H_{\alpha}$ defined by $H_{\alpha}=W^{\beta,2}(D)$ if $\beta<1$, by $W^{\beta,2}(D)\cap H_{0}^{1}(D)$ if $\beta\geq 1$ is of class $J_{\beta/2}$ between H and $D(A)$. Moreover the restriction of $A$ on $H_{\alpha}$ is a sectorial operator ([18]). By the Sobolev embedding theorem, $W^{\beta,2}$ is contained in $L^{q}(D)$ for all $q$ if $\beta\geq\frac{n}{2}$, and in $L^{2n/(n-2\beta)}(D)$ if $\beta<\frac{n}{2}$. If we choose $\beta\in(0,2)$ we have $W^{\beta,2}(D)\subset L^{2\gamma}(D)$ for $n<4\frac{\gamma}{\gamma-1}$. It is clear that $f_{0}$ is locally Lipschitz with respect to $y$ from $H_{\alpha}$ into $H$. It is easy to verify that $f_{0}$ satisfies 4.ii) of Hypothesis 2 with $\gamma=2n+1$ and that it is dissipative with constant $\mu=0$. The function $f_{1}$ is Lipschitz uniformly with respect to $y$ and $z$ and it is bounded. The final condition $\xi$ takes values in $\overline{D(A)}^{H_{\alpha}}$ and belongs to $L^{\infty}(\Omega;H_{\alpha})$. Hence we can apply the global existence theorem and state that the above problem has a unique mild solution $(Y,Z)\in L^{2}(\Omega;C([0,T];H_{\alpha}))\times L^{2}(\Omega\times[0,T];{\mathcal{L}}^{2}(K,H))$.

5.2 A spin system

Let $\mathbb{Z}$ be the one-dimensional lattice of integers. Its elements will be interpreted as atoms. A configuration is a real function $y$ defined on $\mathbb{Z}$. The value $y(n)$ of the configuration $y$ at the point $n$ can be viewed as the state of the atom $n$.

We consider an infinite system of equations

[TABLE]

$\quad Y_{n}(T)=\xi_{n}\quad n\in\mathbb{Z},$

where $Y^{n}$ and $Z^{n}$ are real processes, and $V:\mathbb{R}\rightarrow\mathbb{R}$.

Let $l^{2}(\mathbb{Z})$ be the usual Hilbert space of square summable sequences. To study system (24) we apply results of previous sections. To fit our assumption in Hypotheses 2 and 6, we suppose the following

Hypothesis 9**.**

1.

$W^{n}$, $n\in\mathbb{Z}$ are independent standard real Wiener processes;

2.

$a=\{a_{n}\}_{n\in\mathbb{Z}}$ is a sequence of nonnegative real numbers;

3.

$\xi=\{\xi_{n}\}_{n\in\mathbb{Z}}$ is a random variable belonging to $L^{\infty}(\Omega,l^{2}(\mathbb{Z}))$;

4.

the function $V:\mathbb{R}\rightarrow\mathbb{R}$ is defined by $V(x)=x^{2k+1}\quad k\in\mathbb{N}.$

We will study system (24) regarded as a backward stochastic evolution equation for $t\in[0,T]$

[TABLE]

on a properly chosen Hilbert space $H$ of functions on $\mathbb{Z}$.

To reformulate problem (24) in the abstract form (25), we set $K=H=l^{2}(\mathbb{Z})$. We set $W_{t}=\{W_{t}^{n}\}_{n\in\mathbb{Z}},t\in[0,T]$. By 1. of Hypothesis 9, $W$ is a cylindrical Wiener process in $H$ defined on $(\Omega,\mathcal{F},P)$. We define the operator $A$ by

$\quad\quad A(y)=(a_{n}y_{n})_{n},\quad D(A)=\{y\in l^{2}(\mathbb{Z})\mbox{ such that }\sum_{n\in\mathbb{Z}}a_{n}^{2}y_{n}^{2}<\infty\}.$

It is easy to prove that $A$ is a self-adjoint operator in $l^{2}(\mathbb{Z})$, hence the infinitesimal generator of a sectorial semigroup. The coefficient $f_{0}$ is given by $(f_{0}(t,y))_{n}=(V(y_{n+1}-y_{n})+V(y_{n-1}-y_{n})),\quad t\in[0,T],y\in D(f_{0})$ where $D(f_{0})=\{y\in l^{2}(\mathbb{Z})\mbox{ such that }\sum_{n\in\mathbb{Z}}|x_{n+1}-x_{n}|^{2(2k+1)}<+\infty\}.$ Under Hypothesis 9, $A,f_{0},\xi$ satisfy Hypotheses 2 and 6. We observe that in this case the domain of $f_{0}$ is the whole space $H$: if $y\in l^{2}(\mathbb{Z})$ then

[TABLE]

Consequently, we can take $H_{\alpha}$ with $\alpha=0$, i.e. $H_{0}=H$. The function $f_{0}$ is dissipative. Namely

[TABLE]

and the last term is negative. Moreover, $f_{0}$ satisfies 4.ii) of Hypothesis 2 with $\gamma=2k+1$. The map $f_{0}$ is also locally Lipschitz from $H$ in to $H$. Then by Theorem 7, problem (25) has a unique mild solution $(Y,Z)$ which belongs to $L^{2}(\Omega,C([0,T];H))\times L^{2}(\Omega\times[0,T];\mathcal{L}^{2}(K,H))$.

6 Appendix

This section is devoted to the proof of Lemma 1. Assume first that $\beta=1$. Using recursively the inequality $\displaystyle{U_{t}\leq a(T-t)^{-\alpha}+b\int_{t}^{T}\mathbb{E}^{\mathcal{F}_{t}}U_{s}ds}$ we can easily prove that

$U_{t}\leq$

$\quad\,\displaystyle{\leq a(T-t)^{-\alpha}+\int_{t}^{T}a\sum_{k=1}^{n-1}b^{k}\frac{(r-t)^{k-1}}{(k-1)!}\frac{1}{(T-r)^{\alpha}}dr+}\\ \qquad\qquad\qquad\qquad\displaystyle{\qquad\qquad+b\mathbb{E}^{\mathcal{F}_{t}}\int_{t}^{T}\frac{(b(r-t))^{n-1}}{(n-1)!}U_{r}dr.}$

The last term in the above inequality tends to zero as $n$ tends to infinity for each $t$ in the interval $[0,T]$. Thus

[TABLE]

where $M=1+be^{bT}\frac{1}{1-\alpha}{T}$.

In the case $\beta\neq 1$ a similar proof can be given, based on recursive use of the inequality $\quad\displaystyle{U_{t}}\leq\displaystyle{a(T-t)^{-\alpha}+b\int_{t}^{T}(s-t)^{\beta-1}\mathbb{E}^{\mathcal{F}_{t}}U_{s}ds}.$

Acknowledgments:

I wish to thank Giuseppe Da Prato for hospitality at the Scuola Normale Superiore in Pisa, suggestions and helpful discussions. I would like to express my gratitude to Marco Fuhrman: I am indebted to him for his precious help and encouragement. Special thanks are due to Alessandra Lunardi, who gave me valuable advice and support.