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arXiv · 2308.00173

Optimal control of SPDEs driven by time-space Brownian motion

Nacira Agram·Bernt Øksendal·Frank Proske·Olena Tymoshenko
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Abstract

In this paper we study a Pontryagin type stochastic maximum principle for the optimal control of a system, where the state dynamics satisfy a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion (also called Brownian sheet). We first discuss some properties of a Brownian sheet driven linear SPDE which models the growth of an ecosystem. Further, applying time-space white noise calculus we derive sufficient conditions and necessary conditions of optimality of the control. Finally, we illustrate our results by solving a linear quadratic control problem and an optimal harvesting problem in the plane. We also study possible applications to machine learning.

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Nacira Agram, Bernt Øksendal, Frank Proske, Olena Tymoshenko. 2023-07-31. Optimal control of SPDEs driven by time-space Brownian motion. https://arxiv.org/abs/2308.00173

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