Singular control of SPDEs with space-mean dynamics

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.


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 ∈ [0, T ] and at the point x ∈ D is modelled by a stochastic reaction-diffusion equation with neighbouring interactions.By this we mean a stochastic partial differential equation of the form        du(t, x) = 1 2 ∆u(t, x) + αu(t, x) dt + βu(t, x)dB(t) − λ 0 ξ(dt, x); (t, x) ∈ (0, T ) × D u(0, x) = u 0 (x) > 0; x ∈ D, u(t, x) = u 1 (t, x) ≥ 0; (t, x) ∈ (0, T ) × ∂D, where u(t, x) is the space-averaging operator where V (•) denotes Lebesgue volume and is the ball of radius r > 0 in R d centerd at 0,where D is a bounded Lipschitz domain in R d and u 0 (x), u 1 (t, x) are given deterministic functions.
In the above B(t) = B(t, ω); (t, ω) ∈ [0, ∞) × Ω, is an m-dimensional Brownian motion on a filtered probability space (Ω, F = {F t } t∈[0,∞) , P).Moreover, α, β and λ 0 > 0 are given constants and is the Laplacian differential operator on R d .We may regard ξ(dt, x) as the harvesting effort rate, and λ 0 > 0 as the harvesting efficiency coefficient.The performance functional is assumed to be of the form (h 0 (t, x)u(t, x) − c(t, x))ξ(dt, x)dx + D h 0 (T, x)u(T, x)dx , (1.2) {eq248} {eq248} 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(ξ) represents the expected total net income from the harvesting.The problem is to maximise J(ξ) over all (admissible) harvesting strategies ξ(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 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: be a given measurable mapping and L(t, x) : [0, T ] × D → R a given continuous function.
Consider the problem to find an F-adapted random fields Y (t, x) ∈ R, Z(t, x) ∈ R m , ξ(t, x) ∈ R + left-continuous and increasing with respect to t, such that 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.

The optimization problem
We now give a general formulation of the problem discussed in the Introduction: Let T > 0 and let D ⊂ R n be an open set with C 1 boundary ∂D.Specifically, we assume that the state u(t, x) at time t ∈ [0, T ] and at the point (2.1) {eq2.1}{eq2.1} Here B = {B(t)} t∈[0,T ] is a d-dimensional Brownian motion, defined in a complete filtered probability space (Ω, F , F, P).The filtration F = {F t } t≥0 is assumed to be the P-augmented filtration generated by B. We denote by A the second order partial differential operator acting on x given by , where dm(x) = dx is the Lebesgue measure on 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 where P A t = e tA 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 * of an operator A on C ∞ 0 (R) is defined by the identity where In our case we have We interpret u as a weak (variational) solution to (2.1), in the sense that for φ ∈ C ∞ 0 (D), where •, • represents the duality product between W 1,2 (D) and W 1,2 (D) * , with W 1,2 (D) the Sobolev space of order 1.In the above equation, we have not written all the arguments of b, σ, γ, for simplicity.We want to maximize the performance functional J(ξ), given by ) {ju} {ju} over all ξ ∈ A, where A is the set of all adapted processes ξ(t, x) that are nondecreasing and left continuous with respect to t for all x, with ξ(0, x) = 0, ξ(T, x) < ∞ and such that J(ξ) < ∞.We call A the set of admissible singular controls.Thus we want to find ξ ∈ A, such that J( ξ) = sup ξ∈A J(ξ).
(2.6) {Ham} {Ham} where and We assume that H, f, b, σ, γ and g admit Fréchet derivatives with respect to u and ϕ.

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 ξ ∈ A, with corresponding u(t, x), p(t, x), q(t, x).Suppose the functions (u, ϕ) → g(x, u, ϕ) and (u, ϕ, ξ) → H(t, x, u, ϕ, p(t, x), q(t, x))(dt, ξ(dt, dx)) are concave for each (t, x) ∈ (0, T ) × D.Moreover, suppose that Then ξ is an optimal singular control. where and By concavity on g together with the identity (2.9)-(2.10),we get Applying the Itô formula to p(t, x) u(t, x), we have By the first Green formula (see e.g.Wloka [19], page 258) there exist first order boundary differential operators where the last integral is the surface integral over ∂D.We have that for all (t, x) ∈ (0, T ) × ∂D.Substituting (2.15) in (2.16), yields Using the definition of the Hamiltonian H, we get Summing the above we end up with By the maximum condition of H (2.12), we have

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 ξ ∈ A, we let V(ξ) denote the set of adapted processes ζ(dt, x) of finite variation with respect to t, such that there exists δ = δ(ξ) > 0, such that ξ + yζ ∈ A for all y ∈ [0, δ].
Lemma 2.3 Let ξ(dt, x) ∈ A and choose ζ(dt, x) ∈ V(ξ).Define the derivative process (2.17) {z} {z} Then Z satisfies the following singular linear SPDE Proof.By (2.4) and (2.18), we have lim Using the definition (2.6) of the Hamiltonian, yields where we have used the simplified notation etc.
Applying the Itô formula to p(T, x)Z(T, x), we get Therefore, substituting (2.21) and (2.20) into (2.19),we get lim We can now state our necessary maximum principle:
( Then ξ is a directional sub-stationary point for J(•), in the sense that Proof.
The proof is just a consequence of Lemma 2.4 and Theorem 3 in Øksendal et al [13].

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 ∈ [0, T ] and at the point x ∈ D is given by the stochastic reaction-diffusion equation ) {spde} {spde} where λ 0 > 0 is a constant and, as in (1.1), The performance criterion is assumed to be where h 10 > 0 and g 0 > 0 are given deterministic functions.We can interpret ξ(dt, x) as the harvesting effort at x. Problem 3.1 We want to find ξ ∈ A such that sup ξ∈A J(ξ) = J( ξ).
In this case the Hamiltonian is Recall that for the map L : L 2 (D) → L 2 (D) given by L(u) = ū we know that See Example 3.1 in Agram et al [1].Therefore the adjoint equation is + βq(t, x) dt The variational inequalities for an optimal control ξ(dt, x) and the associated p are: We claim that u(t, x) > 0 for all (t, x) ∈ [0, T ] × D.
Suppose this claim is proved.Then, choosing first ξ = 2 ξ and then ξ = 1 2 ξ in the above we obtain that In addition we get that p(t, x) − 1 λ 0 h 10 (t, x) ξ(dt, x) ≤ 0; which implies that p(t, x) − 1 λ 0 h 10 (t, x) ≤ 0 always.Summarising, we have proved the following: Theorem 3.2 Suppose that u > 0 and (p, ξ) satisfies the following variational inequality Then ξ is an optimal singular control for the space-mean SPDE singular control problem 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 ξ(dt, x) ∈ A is an optimal singular control for the harvesting problem where u(t, x) is given by the SPDE (3.1).Then ξ(dt, x) solves the reflected BSPDE (3.2), (3.4).
Heuristically we can interpret the optimal harvesting strategy as follows: • As long as p(t, x) < 1 λ 0 h 1 (t, x), we do nothing.
x), we harvest immediately from u(t, x) at a rate ξ(dt, x) which is exactly enough to prevent p(t, x) from dropping below we harvest immediately what is necessary to bring p(t, x) up to the level of 1 λ 0 h 1 (t, x).
Remark 3.4 Note that if p(t, x) = 1 λ 0 h 10 (t, x) and λ 0 > h 10 (t, x), then an immediate harvesting of an amount ∆ξ > 0 from u(t, x) produces an immediate decrease in the process p(t, x) and hence pushes p(t, x) below 1 λ 0 h 10 (t, x).This follows from the comparison theorem for reflected BSPDEs of the type (3.2).

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 where we have denoted by W * the topological dual of V .Let A be a bounded linear operator from W to W * satisfying the following Gårding inequality (coercivity hypothesis): There exist constants α > 0 and λ ≥ 0 so that

) {COE} {COE}
where Au, u = Au(u) denotes the action of Au ∈ W * on u ∈ W and || • || H (respectively • W ) the norm associated to the Hilbert space H (respectively W ). We will also use the following spaces: 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 η is a -valued continuous process, nonnegative, nondecreasing in t and η(0, x) = 0. We now consider the adjoint equation (2.11) as a reflected backward stochastic evolution equation where Y (t, x) stands for the W -valued continuous process Y (t, x) and the solution of equation (4.2) is understood as an equation in the dual space W * 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 The following result is essential due to Proof.For the proof of the theorem, we introduce the penalized backward SPDEs: 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≥1 forms a Cauchy sequence, i.e., lim n,m→∞ Now we estimate each of the terms on the right side: It follows from (4.5) and (4.6) that By Donalti-Martin and Pardoux [5] (see also Øksendal et al [13]), under the conditions of Theorem ///.There exists a constant C, such that