Stochastic Maximum Principle for Optimal Control of SPDEs

Abstract

We prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a stochastic partial differential equation driven by a finite dimensional Wiener process. The equation is formulated in a semi-abstract form that allows direct applications to a large class of controlled stochastic parabolic equations. We allow for a diffusion coefficient dependent on the control parameter, and the space of control actions is general, so that in particular we need to introduce two adjoint processes. The second adjoint process takes values in a suitable space of operators on L ⁴.

Appendix: Stochastic Integrals in L p Spaces

In this appendix we sketch the construction and some basic properties of stochastic integrals with respect to a finite dimensional Wiener process, taking values in an L ^p-space. The few facts collected below are enough for the present paper.

Let $(W^{1}_{t},\ldots,W^{d}_{t})_{t\ge0}$ be a standard, d-dimensional Wiener process defined in some complete probability space $(\varOmega ,\mathcal{F},\mathbb{P})$. We denote by $(\mathcal{F}_{t})_{t\ge0}$ the corresponding natural filtration, augmented in the usual way, and we denote by $\mathcal{P}$ the progressive σ-algebra on Ω×[0,T], where T>0 is a given number. Let $L^{p}:=L^{p}(D,\mathcal{D},m)$ be the usual space, where m is a positive, σ-finite measure and p∈[2,∞). The integrand processes will be functions $H:\varOmega\times [0,T]\times D\to\mathbb{R}^{d}$, which are assumed to be $\mathcal{P}\otimes\mathcal{D}$-measurable. When H is of special type, i.e. it has components of the form

$$H^j(\omega,t,x)=\sum_{i=1}^Nh^j_i( \omega,t)f^j_i(x) $$

for j=1,…,d, $h^{j}_{i}$ bounded $\mathcal{P}$-measurable, $f^{j}_{i}$ bounded $\mathcal{D}$-measurable, then the stochastic integral I _t (x) is defined for fixed x∈D by the formula $I_{t}(x)=\int_{0}^{t}H_{s}^{j}(x) dW^{j}_{s}= f^{j}_{i}(x) \int_{0}^{t}h^{j}_{i}(s) dW^{j}_{s}$. Using the Burkholder–Davis–Gundy inequalities for real-valued stochastic integrals, we have for some constant c _p (depending only on p):

$$\mathbb{E} \bigl|I_t(x)\bigr|^p\le c_p\mathbb{E} \Biggl( \int_0^t \bigl|H_s(x)\bigr|^2 ds \biggr)^{p/2} $$

where $|H_{s}(x)|^{2}=\sum_{j=1}^{d}|H_{s}^{j}(x)|^{2}$. Since p≥2 we have, by en elementary inequality,

$$\mathbb{E} \bigl|I_t(x)\bigr|^p\le c_p \biggl( \int _0^t \bigl(\mathbb{E}\bigl|H_s(x)\bigr|^p \bigr)^{2/p} ds \biggr)^{p/2} = c_p \biggl( \int _0^t \bigl\|H_s(x)\bigr\|^2_{L^p(\varOmega;\mathbb{R}^d)} ds \biggr)^{p/2}. $$

Integrating with respect to m we obtain, again by elementary arguments,

which can be written

$$ \mathbb{E}\|I_t\|_{L^p(D)}^p \le c_p \biggl( \int_0^t \bigl( \mathbb{E}\|H_s\|^p_{L^p(D;\mathbb{R}^d)}\bigr)^{2/p} ds \biggr)^{p/2} $$

(A.1)

or equivalently

$$\|I_t\|_{L^p(\varOmega\times D)} \le c_p^{1/p} \biggl( \int_0^t \|H_s\|^2_{L^p(\varOmega\times D;\mathbb{R}^d)} ds \biggr)^{1/2}. $$

Finally, by standard arguments, the stochastic integral can be extended to the class of $\mathcal{P}\otimes\mathcal{D}$-measurable integrands H for which the right-hand side of (A.1) is finite, and the inequality (A.1) remains true.

We finally note that from (A.1) and the Hölder inequality it follows that

$$ \mathbb{E}\|I_t\|_{L^p(D)}^p \le c_p\int_0^t \mathbb{E} \|H_s\|^p_{L^p(D;\mathbb{R}^d)} ds t^{(p-2)/2}. $$

(A.2)

Now suppose that there exist regular conditional probabilities $\mathbb{P}(\cdot|\mathcal{F}_{t})$ given any $\mathcal{F}_{t}$ (this holds for instance if the Wiener process is canonically realized on the space of $\mathbb{R}^{d}$-valued continuous functions). Then a slight modification of the previous passages shows the validity of the following conditional variant of (A.2): for 0≤r≤t,

$$ \mathbb{E}^{\mathcal{F}_r} \biggl\|\int_r^tH_s^j dW^j_s\biggr\|_{L^p(D)}^p\le c_p\int_r^t \mathbb{E}^{\mathcal{F}_r} \|H_s\|^p_{L^p(D;\mathbb{R}^d)} ds (t-r)^{(p-2)/2}. $$

(A.3)

This is used in the proof of Proposition 5.2.