Singular control of SPDEs with space-mean dynamics

TL;DR

This paper develops maximum principles for optimal singular control of SPDEs with space-mean dependence, modeling population growth in random environments, and introduces a reflected BSPDE framework with existence and uniqueness results.

Contribution

It introduces a novel control framework for SPDEs with space-mean dependence, including maximum principles and a new class of reflected BSPDEs with proven well-posedness.

Findings

Derived necessary and sufficient maximum principles for control.

Established existence and uniqueness of the reflected BSPDEs.

Applied the theory to optimal population harvesting models.

Abstract

We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for such control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.

Equations

$$\left\{\begin{array}[c]{l}du(t,x)=\left[\dfrac{1}{2}\Delta u(t,x)+\alpha\overline{u}(t,x)\right]dt+\beta\overline{u}(t,x)dB(t)-\lambda_{0}\xi(dt,x);\quad(t,x)\in(0,T)\times D\\ \text{ }u(0,x)=u_{0}(x)>0;\quad x\in D,\\ \text{ }u(t,x)=u_{1}(t,x)\geq 0;\quad(t,x)\in(0,T)\times\partial D,\end{array}\right.$$

$$\overline{u}(t,x)=\frac{1}{V(K_{\theta })}{\int }_{K_{\theta }}u(x+y)dy,$$

$$K_{\theta }=\{y\in {\mathbb{R}}^n;|y|<\theta \}$$

$$\Delta =\sum _{i=1}^d\frac{{\partial }^2}{\partial x_i^2}$$

$$J(\xi )=\mathbb{E}\left[{\int }_D{\int }_0^T(h_0(t,x)u(t,x)-c(t,x))\xi (dt,x)dx+{\int }_D{h}_0(T,x)u(T,x)dx\right],$$

$$g:[0,T]\times D\times \mathbb{R}\times {\mathbb{R}}^m\to \mathbb{R}$$

$$\left\{\begin{array}[c]{l}dY(t,x)=-AY(t,x)dt-F(t,x,Y(t,x),Y(t,\cdot),Z(t,x))dt+Z(t,x)dB(t)\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\xi(dt,x),t\in(0,T),\\ Y(t,x)\geq L(t,x),\\ \int_{0}^{T}\int_{D}(Y(t,x)-L(t,x))\xi(dt,x)dx=0,\\ Y(T,x)=\phi(x)\quad\text{a.s.,}\end{array}\right.$$

$$\{\begin{array}{ll}du(t,x)& =Au(t,x)dt+b(t,x,u(t,x),u(t,\cdot ))dt+\sigma (t,x,u(t,x),u(t,\cdot ))dB(t)\\ & +f(t,x,u)\xi (dt,x);(t,x)\in (0,T)\times D,\\ u(0^{-},x)& =u_0(x);x\in \overline{D},\\ u(t,x)& =u_1(t,x);(t,x)\in (0,T)\times \partial D.\end{array}$$

$$A\varphi (x)=\sum _{i,j=1}^n{\alpha }_{ij}(x)\frac{{\partial }^2\varphi }{\partial x_i\partial x_j}+\sum _{i=1}^n{\beta }_i(x)\frac{\partial \varphi }{\partial x_i};\varphi \in C^2({\mathbb{R}}^n),$$

$$\phi$$

$$u$$

$$u(0,x)$$

$$+{\int }_0^t{P}_s^{A}f(s,\cdot ,u(s,x))(x)\xi (ds,x),$$

$$(A\varphi ,\psi )=(\varphi ,A^{\ast }\psi ),\text{ for all }\varphi ,\psi \in C_0^{\infty }(\mathbb{R}),$$

$$A_x^{\ast }\varphi (x)=\sum _{i,j=1}^n\frac{{\partial }^2}{\partial x_i\partial x_j}({\alpha }_{ij}(x)\varphi (x))-\sum _{i=1}^n\frac{\partial }{\partial x_i}({\beta }_i(x)\varphi (x));\varphi \in C^2({\mathbb{R}}^n).$$

$$⟨u(t),\varphi {⟩}_{L^2(D)}$$

$$+{\int }_0^t⟨\sigma (s,u(s)),\varphi {⟩}_{L^2(D)}dB(s)+{\int }_0^t⟨f(s.u(s)),\varphi {⟩}_{L^2(D)}\xi (ds,x),$$

$$\begin{array}[c]{c}J(\xi)=\mathbb{E}\Big{[}{\displaystyle\int_{0}^{T}}{\displaystyle\int_{D}}h_{0}(t,x,u(t,x),u(t,\cdot))dxdt+{\displaystyle\int_{0}^{T}}{\displaystyle\int_{D}}h_{1}(t,x,u(t,x),u(t,\cdot))\xi(dt,dx)\\ +{\displaystyle\int_{D}}g(x,u(T,x),u(T,\cdot))dx\Big{]},\end{array}$$

$$J(\hat{\xi })=\underset{\xi \in \mathcal{A}}{\sup }J(\xi ).$$

$$H(t,x,u,\phi ,p,q)(dt,\xi (dt,x))=H_0(t,x,u,\phi ,p,q)dt+H_1(t,x,u,\phi ,p)\xi (dt,x).$$

$$H_0(t,x,u,\phi ,p,q)=$$

$$H_1(t,x,u,\phi ,p)=$$

$$⟨{\nabla }_{\phi }h,\psi ⟩(x)={\int }_D{\nabla }_{\phi }^{\ast }h(x,y)\psi (y)dy;\text{ for all }\psi \in L^2(D).$$

$${\overline{\nabla }}_{\phi }^{\ast }h(x)={\int }_D{\nabla }_{\phi }^{\ast }h(y,x)dy.$$

$$\left\{\begin{array}[c]{ll}dp(t,x)&=-A_{x}^{\ast}p(t,x)dt-\left\{\frac{\partial H_{0}}{\partial u}(t,x)+\overline{\nabla}_{\varphi}^{\ast}H_{0}(t,x)\right\}dt\\ &-\left\{\frac{\partial H_{1}}{\partial u}(t,x)+\overline{\nabla}_{\varphi}^{\ast}H_{1}(t,x)\right\}\xi(dt,x)\\ &\text{ \ }+q(t,x)dB(t);\quad(t,x)\in(0,T)\times D,\\ p(T,x)&=\frac{\partial g}{\partial u}(T,x)+\overline{\nabla}_{\varphi}^{\ast}g(T,x);\quad x\in D,\\ p(t,x)&=0;\quad(t,x)\in(0,T)\times\partial D,\end{array}\right.$$

$$H_i(t,x)=H_i(t,x,u,\phi ,p,q){|}_{u=u(t,x),\phi =u(t,\cdot ),p=p(t,x),q=q(t,x)},i=0,1$$

$$\hat{\xi }(dt,x)\in \arg \underset{\hat{\xi }\in \mathcal{A}}{\max }H(t,x,\hat{u}(t,x),\hat{u}(t,\cdot ),\hat{p}(t,x),\hat{q}(t,x))(dt,\xi (dt,x));$$

$$\left\{H(t,x,\hat{u}(t,x),\hat{u}(t,\cdot ),\hat{p}(t,x),\hat{q}(t,x)\right\}\xi (dt,x)$$

$$\le \left\{H(t,x,\hat{u}(t,x),\hat{u}(t,\cdot ),\hat{p}(t,x),\hat{q}(t,x)\right\}\hat{\xi }(dt,x);\text{ for all }\xi \in \mathcal{A}.$$

$$J(\xi )-J(\hat{\xi })=I_1+I_2+I_3,$$

$$I_1=\mathbb{E}\left[{\int }_0^T{\int }_D\{h_0(t,x,u(t,x),u(t,\cdot ))-h_0(t,x,\hat{u}(t,x),\hat{u}(t,\cdot ))\}dxdt\right],$$

$$I_2=\mathbb{E}\left[{\int }_0^T{\int }_D\{h_1(t,x,u(t,x),u(t,\cdot ))\xi (dt,x)-{\int }_0^T{\int }_D{h}_1(t,x,\hat{u}(t,x),\hat{u}(t,\cdot ))\hat{\xi }(dt,x)\right],$$

Full text

MSC(2010):

60H05, 60H15, 93E20, 91G80,91B70.

Keywords:

Stochastic partial differential equations; space-mean dependence; maximum principle; backward stochastic partial differential equations; space-mean reaction diffusion equation; optimal harvesting.

11footnotetext: Department of Mathematics, Linnaeus University (LNU), Sweden.

Emails: nacira.agram@lnu.se, astrid.hilbert@lnu.se.22footnotetext: Department of Mathematics, University of Oslo, Norway.

Email: oksendal@math.uio.no. This research was carried out with support of the Norwegian Research Council, within the research project Challenges in Stochastic Control, Information and Applications (STOCONINF), project number 250768/F20.

1 Introduction

We start by a motivation for the problem that will be studied in this paper:

Consider a problem of optimal harvesting from a fish population in a lake $D$. We assume that the density $u(t,x)$ of the population at time $t\in[0,T]$ and at the point $x\in D$ is modelled by a stochastic reaction-diffusion equation with neighbouring interactions. By this we mean a stochastic partial differential equation of the form

[TABLE]

where $\overline{u}(t,x)$ is the space-averaging operator

[TABLE]

where $V(\cdot)$ denotes Lebesgue volume and

[TABLE]

is the ball of radius $r>0$ in $\mathbb{R}^{d}$ centered at [math],where $D$ is a bounded Lipschitz domain in $\mathbb{R}^{d}$ and $u_{0}(x),u_{1}(t,x)$ are given deterministic functions.

In the above $B(t)=B(t,\omega);(t,\omega)\in[0,\infty)\times\Omega,$ is an $m$-dimensional Brownian motion on a filtered probability space $(\Omega,\mathbb{F}=\{\mathcal{F}_{t}\}_{t\in[0,\infty)},\mathbb{P})$. Moreover, $\alpha$, $\beta$ and $\lambda_{0}>0$ are given constants and

[TABLE]

is the Laplacian differential operator on $\mathbb{R}^{d}$.

We may regard $\xi(dt,x)$ as the harvesting effort rate, and $\lambda_{0}>0$ as the harvesting efficiency coefficient. The performance functional is assumed to be of the form

[TABLE]

where $h_{0}(t,x)>0$ is the unit price of the fish and $c(t,x)$ is the unit cost of energy used in the harvesting and $T>0$ is a fixed terminal time. Thus $J(\xi)$ represents the expected total net income from the harvesting. The problem is to maximise $J(\xi)$ over all (admissible) harvesting strategies $\xi(t,x)$.

** **Remark 1.1

This population growth model, which was first introduced in Agram et al [1], is a generalisation of the classical stochastic reaction-diffusion model, in that we have added the term $\overline{u}(t,x)$ which represents an average of the neighbouring densities. Thus our model allows for the growth at a point to depend on interactions from the whole vicinity. This space-mean interaction is different from the pointwise interaction represented by the Laplacian.

The problem above turns out to be related to a problem of the following form:

Let $\phi(x)=\phi(x,\omega)$ be an $\mathcal{F}_{T}$-measurable $H=L^{2}(D)$-valued random variable. Let

[TABLE]

be a given measurable mapping and $L(t,x):[0,T]\times D\rightarrow\mathbb{R}$ a given continuous function. Consider the problem to find an $\mathbb{F}$-adapted random fields $Y(t,x)\in\mathbb{R},Z(t,x)\in\mathbb{R}^{m},\xi(t,x)\in\mathbb{R}^{+}$ left-continuous and increasing with respect to $t$, such that

[TABLE]

where $A$ is a second order linear partial differential operator. We call the equation (1.3) a reflected stochastic partial differential equation (SPDE) with space-mean dynamics. We will come back to this equation in the last section.

2 The optimization problem

We now give a general formulation of the problem discussed in the Introduction:

Let $T>0$ and let $D\subset\mathbb{R}^{n}$ be an open set with $C^{1}$ boundary $\partial D.$ Specifically, we assume that the state $u(t,x)$ at time $t\in[0,T]$ and at the point $x\in\overline{D}:=D\cup\partial D$ satisfies

[TABLE]

Here $B=\{B(t)\}_{t\in[0,T]}$ is a $d$-dimensional Brownian motion, defined in a complete filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P}).$ The filtration $\mathbb{F=}\left\{\mathcal{F}_{t}\right\}_{t\geq 0}$ is assumed to be the $\mathbb{P}$-augmented filtration generated by $B$.

We denote by $A$ the second order partial differential operator acting on $x$ given by

[TABLE]

where $(\alpha_{ij}(x))_{1\leq i,j\leq n}$ is a given nonnegative definite $n\times n$ matrix with entries $\alpha_{ij}(x)\in C^{2}(D)\cap C(\overline{D})$ for all $i,j=1,2,...,n$ and $\beta_{i}(x)\in C^{2}(D)\cap C(\overline{D})$ for all $i=1,2,...,n.$

Let $L(\mathbb{R}^{n})$ denote the set of real measurable functions on $\mathbb{R}^{n}$. For each $t,x,u,\zeta$ the functions

[TABLE]

are $C^{1}$ functionals on $L^{2}(D)=L^{2}(D,m)$, where $dm(x)=dx$ is the Lebesgue measure on $\mathbb{R}^{n}$. Here $Au(t,x)$ is interpreted in the sense of distribution. Thus $u$ is understood as a weak (mild) solution to (2.1), in the sense that

[TABLE]

where $P_{t}^{A}$ is the semigroup associated to the operator $A$. Thus we see that we can in the usual way apply the Itô formula to such SPDEs.

Moreover, the adjoint operator $A^{\ast}$ of an operator $A$ on $C_{0}^{\infty}(\mathbb{R})$ is defined by the identity

[TABLE]

where $\langle\phi_{1},\phi_{2}\rangle_{L^{2}(\mathbb{R})}:=(\phi_{1},\phi_{2})=\int_{\mathbb{R}}\phi_{1}(x)\phi_{2}(x)dx$ is the inner product in $L^{2}(\mathbb{R}).$ In our case we have

[TABLE]

We interpret $u$ as a weak (variational) solution to (2.1), in the sense that for $\phi\in C_{0}^{\infty}(D),$

[TABLE]

where $\langle\cdot,\cdot\rangle$ represents the duality product between $W^{1,2}(D)$ and $W^{1,2}(D)^{\ast}$, with $W^{1,2}(D)$ the Sobolev space of order $1$. In the above equation, we have not written all the arguments of $b,\sigma,\gamma$, for simplicity.

We want to maximize the performance functional $J(\xi),$ given by

[TABLE]

over all $\xi\in\mathcal{A}$, where $\mathcal{A}$ is the set of all adapted processes $\xi(t,x)$ that are nondecreasing and left continuous with respect to $t$ for all $x$, with $\xi(0,x)=0,$ $\xi(T,x)<\infty$ and such that $J(\xi)<\infty.$ We call $\mathcal{A}$ the set of admissible singular controls. Thus we want to find $\widehat{\xi}\in\mathcal{A},$ such that

[TABLE]

For each $t,x,u$ we assume that the functions $\varphi\mapsto h_{0}(t,x,u,\varphi):[0,T]\times D\times\mathbb{R}\times L(\mathbb{R}^{n})\rightarrow\mathbb{R},$ and $\varphi\mapsto g(x,u,\varphi):D\times\mathbb{R}\times L(\mathbb{R}^{n})\rightarrow\mathbb{R},$ are $C^{1}$ functionals on $L^{2}(D)$.

The Hamiltonian $H$ is defined by

[TABLE]

where

[TABLE]

and

[TABLE]

We assume that $H,f,b,\sigma,\gamma$ and $g$ admit Fréchet derivatives with respect to $u$ and $\varphi.$

In general, if $h:L^{2}(D)\mapsto L^{2}(D)$ is Fréchet differentiable, we denote its Fréchet derivative (gradient) at $\varphi\in L^{2}(D)$ by $\nabla_{\varphi}h$, and we denote the action of $\nabla_{\varphi}h$ on a function $\psi\in L^{2}(D)$ by $\left\langle\nabla_{\varphi}h,\psi\right\rangle$.

Definition 2.1

We say that the Fréchet derivative $\nabla_{\varphi}h$ of a map $h:L^{2}(D)\mapsto L^{2}(D)$ has a dual function $\nabla_{\varphi}^{\ast}h\in L^{2}(D\times D)$ if

[TABLE]

By Fubini’s theorem, we get

[TABLE]

We associate to the Hamiltonian the following reflected BSPDE

[TABLE]

where we have used the simplified notation

[TABLE]

and similarly with $g$.

2.1 A sufficient maximum principle

We now formulate a sufficient version ( a verification theorem) of the maximum principle for the optimal control of the problem (2.1)-(2.5).

Theorem 2.2** (Sufficient Maximum Principle)**

*Suppose $\widehat{\xi}\in\mathcal{A}$, with corresponding

$\widehat{u}(t,x),\widehat{p}(t,x),\widehat{q}(t,x).$ Suppose the functions $(u,\varphi)\mapsto g(x,u,\varphi)$ and

$(u,\varphi,\xi)\mapsto H(t,x,u,\varphi,\widehat{p}(t,x),\widehat{q}(t,x))(dt,\xi(dt,dx))$ are concave for each $(t,x)\in(0,T)\times D$. Moreover, suppose that*

[TABLE]

i.e.,

[TABLE]

Then $\widehat{\xi}$ is an optimal singular control.

Proof.  Consider

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

By concavity on $g$ together with the identity (2.9)-(2.10), we get

[TABLE]

where $\tilde{u}(t,x)=u(t,x)-\hat{u}(t,x);t\in[0,T]$.

Applying the Itô formula to $\widehat{p}(t,x)\widetilde{u}(t,x)$, we have

[TABLE]

By the first Green formula (see e.g. Wloka [19], page 258) there exist first order boundary differential operators $A_{1},A_{2},$ such that

[TABLE]

where the last integral is the surface integral over $\partial D$. We have that

[TABLE]

for all $(t,x)\in(0,T)\times\partial D.$

Substituting $\left(\ref{**}\right)$ in $\left(\ref{1}\right)$, yields

[TABLE]

Using the definition of the Hamiltonian $H$, we get

[TABLE]

Summing the above we end up with

[TABLE]

By the maximum condition of $H$ (2.12), we have

[TABLE]

$\square$

2.2 A necessary maximum principle

The concavity conditions in the sufficient maximum principle imposed on the involved coefficients are not always satisfied. Hence, we will derive now a necessary optimality conditions which do not require such an assumptions. We shall first need the following Lemmas:

For $\xi\in\mathcal{A}$, we let** $\mathcal{V}(\xi)$** denote the set of adapted processes $\zeta(dt,x)$ of finite variation with respect to $t$, such that there exists $\delta=\delta(\xi)>0$, such that $\xi+y\zeta\in\mathcal{A}$ for all $y\in[0,\delta].$

Lemma 2.3

Let $\xi(dt,x)\in\mathcal{A}$ and choose $\zeta(dt,x)\in\mathcal{V}(\xi)$. Define the derivative process

[TABLE]

Then $\mathcal{Z}$ satisfies the following singular linear SPDE

[TABLE]

Lemma 2.4

Let $\xi(dt,x)\in\mathcal{A}$ and $\zeta(dt,x)\in\mathcal{V}(\xi)$. Put $\eta=\xi+\epsilon\zeta;\epsilon\in[0,\delta(\xi)]$. Then

[TABLE]

Proof.  By (2.4) and (2.18), we have

[TABLE]

Using the definition (2.6) of the Hamiltonian, yields

[TABLE]

where we have used the simplified notation

[TABLE]

etc.

Applying the Itô formula to $p(T,x)\mathcal{Z}(T,x)$, we get

[TABLE]

Since $p(t,x)=\mathcal{Z}(t,x)=0$ for $x\in\partial D$, we deduce that

[TABLE]

Therefore, substituting (2.21) and (2.20) into (2.19), we get

[TABLE]

$\square$

We can now state our necessary maximum principle:

Theorem 2.5** (Necessary Maximum Principle)**

(i) Suppose $\xi^{\ast}\in\mathcal{A}$ is optimal, i.e.

[TABLE]

Let $u^{\ast},(p^{\ast},q^{\ast})$ be the corresponding solution of (2.1) and (2.11), respectively, and assume that (2.17) holds with $\xi=\xi^{\ast}$. Then

[TABLE]

and

[TABLE]

(ii) Conversely, suppose that there exists $\hat{\xi}\in\mathcal{A},$ such that the corresponding solutions $\widehat{u}(t,x),(\widehat{p}(t,x),\widehat{q}(t,x))$ of (2.1) and (2.11), respectively, satisfy

[TABLE]

and

[TABLE]

Then $\widehat{\xi}$ is a directional sub-stationary point for $J(\cdot)$, in the sense that

[TABLE]

Proof.  The proof is just a consequence of Lemma 2.4 and Theorem 3 in Øksendal et al [13]. $\square$

3 Application to Optimal Harvesting

We now return to the problem of optimal harvesting from a fish population in a lake $D$ stated in the Introduction. Thus we suppose the density $u(t,x)$ of the population at time $t\in[0,T]$ and at the point $x\in D$ is given by the stochastic reaction-diffusion equation

[TABLE]

where $\lambda_{0}>0$ is a constant and, as in (1.1),

[TABLE]

The performance criterion is assumed to be

[TABLE]

where $h_{10}>0$ and $g_{0}>0$ are given deterministic functions. We can interpret $\xi(dt,x)$ as the harvesting effort at $x$.

Problem 3.1

We want to find $\hat{\xi}\in\mathcal{A}$ such that $\sup_{\xi\in\mathcal{A}}J(\xi)=J(\hat{\xi}).$

In this case the Hamiltonian is

[TABLE]

Recall that for the map $L:L^{2}(D)\mapsto L^{2}(D)$ given by $L(u)=\bar{u}$ we know that

[TABLE]

See Example 3.1 in Agram et al [1]. Therefore the adjoint equation is

[TABLE]

The variational inequalities for an optimal control $\hat{\xi}(dt,x)$ and the associated $\hat{p}$ are:

[TABLE]

We claim that

[TABLE]

Suppose this claim is proved. Then, choosing first $\xi=2\hat{\xi}$ and then $\xi=\frac{1}{2}\hat{\xi}$ in the above we obtain that

[TABLE]

In addition we get that

[TABLE]

which implies that $\hat{p}(t,x)-\frac{1}{\lambda_{0}}h_{10}(t,x)\leq 0$ always.

Summarising, we have proved the following:

Theorem 3.2

Suppose that $\widehat{u}>0$ and $(\hat{p},\hat{\xi})$ satisfies the following variational inequality

[TABLE]

Then $\hat{\xi}$ is an optimal singular control for the space-mean SPDE singular control problem (3.1)

We see that this, together with (3.2) constitute a reflected BSPDE, albeit of a slightly different type than the one that will be discussed in the next section.

We summerize the above in the following:

Theorem 3.3

(a)

Suppose $\xi(dt,x)\in\mathcal{A}$ is an optimal singular control for the harvesting problem

[TABLE]

where $u(t,x)$ is given by the SPDE (3.1). Then $\xi(dt,x)$ solves the reflected BSPDE (3.2), (3.4).

(b)

Conversely, suppose $(p,q,\xi)$ is a solution of the reflected BSPDE (3.2), (3.4). Then $\xi(dt,x)$ is an optimal control for the problem to maximize the performance (1.2).

Heuristically we can interpret the optimal harvesting strategy as follows:

• •

As long as $p(t,x)<\frac{1}{\lambda_{0}}h_{1}(t,x)$, we do nothing.

• •

If $p(t,x)=\frac{1}{\lambda_{0}}h_{1}(t,x)$, we harvest immediately from $u(t,x)$ at a rate $\xi(dt,x)$ which is exactly enough to prevent $p(t,x)$ from dropping below $\frac{1}{\lambda_{0}}h_{1}(t,x)$ in the next moment.

• •

If $p(t,x)>\frac{1}{\lambda_{0}}h_{1}(t,x)$, we harvest immediately what is necessary to bring $p(t,x)$ up to the level of $\frac{1}{\lambda_{0}}h_{1}(t,x).$

** **Remark 3.4

Note that if $p(t,x)=\frac{1}{\lambda_{0}}h_{10}(t,x)$ and

[TABLE]

then an immediate harvesting of an amount $\Delta\xi>0$ from $u(t,x)$ produces an immediate decrease in the process $p(t,x)$ and hence pushes $p(t,x)$ below $\frac{1}{\lambda_{0}}h_{10}(t,x).$ This follows from the comparison theorem for reflected BSPDEs of the type (3.2).

4 Existence and uniqueness of solutions of space-mean reflected

backward SPDEs

Let $W,H$ be two separable Hilbert spaces such that $W$ is continuously, densely imbedded in $H$. Identifying $H$ with its dual we have

[TABLE]

where we have denoted by $W^{\ast}$ the topological dual of $V$. Let $A$ be a bounded linear operator from $W$ to $W^{\ast}$ satisfying the following Gårding inequality (coercivity hypothesis): There exist constants $\alpha>0$ and $\lambda\geq 0$ so that

[TABLE]

where $\langle Au,u\rangle=Au(u)$ denotes the action of $Au\in W^{\ast}$ on $u\in W$ and $||\cdot||_{H}$ (respectively $\|\cdot\|_{W}$) the norm associated to the Hilbert space $H$ (respectively $W$). We will also use the following spaces:

• •

$L^{2}(D)$ is the set of all Lebesgue measurable $Y:D\rightarrow\mathbb{R},$ such that

[TABLE]

• •

$L^{2}(H)$ is the set of $\mathcal{F}_{T}$-measurable $H$-valued random variables $\varsigma$ such that $\mathbb{E}[||\varsigma||_{H}^{2}]<\infty$.

We let $W:=W^{1,2}(D)$ and $H=L^{2}(D).$

Denote by $L(t,x)$ the barrier which is a measurable function that is differentiable in time $t$ and twice differentiable in space $x,$ such that

[TABLE]

$\eta$ is a $H$-valued continuous process, nonnegative, nondecreasing in $t$ and $\eta(0,x)=0.$

We now consider the adjoint equation (2.11) as a reflected backward stochastic evolution equation

[TABLE]

where $Y(t,x)$ stands for the $W$-valued continuous process $Y(t,x)$ and the solution of equation $(\ref{r-BSPDE})$ is understood as an equation in the dual space $W^{\ast}$ of $W$.

We mean by $dY(t,x)$ the differential operator with respect to $t$, while $A_{x}$ is the partial differential operator with respect to $x$, and

[TABLE]

The following result is essential due to Agram et al [1]:

Lemma 4.1

For all $\varphi\in H$ we have

[TABLE]

We shall now state and prove our main result of existence and uniqueness of solutions to reflected BSPDE.

Theorem 4.2** (Existence and uniqueness of solutions)**

The space-mean reflected BSPDE $(\ref{r-BSPDE})$ has a unique solution $(Y(t,x),Z(t,x),\eta(t,x))\in W\times L^{2}(D,\mathbb{R}^{m})\times H$-valued progressively measurable process, provided that the following assumptions hold:

(i)

The terminal condition $\phi$ is $\mathcal{F}_{T}$-measurable random variable and satisfies

[TABLE]

(ii)

There exists a constant $C>0$ such that

[TABLE]

for all $t,y_{i},\overline{y}_{i},z_{i},\overline{z}_{i};i=1,2.$

Proof.  For the proof of the theorem, we introduce the penalized backward SPDEs:

[TABLE]

According to Agram et al [1], the solution $(Y^{n},Z^{n})$ of the above equation (4.4) exists and is unique. We are going to show that $(Y^{n},Z^{n})_{n\geq 1}$ forms a Cauchy sequence, i.e.,

[TABLE]

[TABLE]

[TABLE]

Applying Itô’s formula, it follows that

[TABLE]

Now we estimate each of the terms on the right side:

[TABLE]

By the Lipschitz continuity of $b$ and the inequality $ab\leq\varepsilon a^{2}+C_{\varepsilon}b^{2}$, together with inequality (4.3), one has

[TABLE]

It follows from (4.5) and (4.6) that

[TABLE]

Gronwall inequality, yields

[TABLE]

and

[TABLE]

By inequality (4.7) and the Burkholder inequality we get

[TABLE]

Under the conditions of Theorem 4.2 and by Lemma 5 in Øksendal et al [13], there exists a constant $C,$ such that

[TABLE]

Denote by $Y(t,x)$, $Z(t,x)$ the limit of $Y^{n}$ and $Z^{n}$, respectively. Put

[TABLE]

Inequality (4.8) implies that $\overline{\eta}^{n}(t,x)$ admits a non-negative weak limit, denoted by $\overline{\eta}(t,x)$, in the following Hilbert space:

[TABLE]

with inner product

[TABLE]

Set $\eta(t,x)=\int_{0}^{t}\overline{\eta}(s,x)ds$. Then $\eta$ is a continuous $H$-valued process which is increasing in $t$. Letting $n\rightarrow\infty$ in (4.4) we obtain

[TABLE]

Inequality (4.8) and the Fatou Lemma imply that $\mathbb{E}\left[\int_{t}^{T}\int_{D}((Y(s,x)-L(s,x))^{-})^{2}dxds\right]=0$. In view of the continuity of $Y$ in $t$, we conclude $Y(t,x)\geq L(t,x)$ a.e. in $x$, for every $t\geq 0$. Combining the strong convergence of $Y^{n}$ and the weak convergence of $\bar{\eta}^{n}$, we also have

[TABLE]

Hence,

[TABLE]

We have shown that $(Y,Z,\eta)$ is a solution to the reflected backward SPDE $(\ref{r-BSPDE})$.

Uniqueness. Let $(Y_{1},Z_{1},\eta_{1})$, $(Y_{2},Z_{2},\eta_{2})$ be two such solutions to equation $(\ref{r-BSPDE})$. By Itô’s formula, we have

[TABLE]

Similar to the proof of existence, we have

[TABLE]

and

[TABLE]

On the other hand,

[TABLE]

Combining (4.11)-(4.14) we arrive at

[TABLE]

Appealing to the Gronwall inequality, this implies

[TABLE]

which further gives $\eta_{1}=\eta_{2}$ from the equation they satisfy. $\square$