Optimal insider control of stochastic partial differential equations

TL;DR

This paper develops a maximum principle for optimal insider control of SPDEs, incorporating inside information and control-dependent operators, with applications to systems with noisy observations.

Contribution

It introduces a novel maximum principle for insider control of SPDEs with control-dependent operators and noisy observation scenarios.

Findings

Derived necessary and sufficient maximum principles for insider SPDE control.

Applied results to control problems with noisy observations using nonlinear filtering.

Provided explicit examples illustrating the theoretical results.

Abstract

We study the problem of optimal inside control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: (i) The controller has access to inside information, i.e. access to information about a future state of the system, (ii) The integro-differential operator of the SPDE might depend on the control. In the first part of the paper, we formulate a sufficient and a necessary maximum principle for this type of control problem, in two cases: (1) When the control is allowed to depend both on time t and on the space variable x. (2) When the control is not allowed to depend on x. In the second part of the paper, we apply the results above to the problem of optimal control of an SDE system when the inside controller has only noisy observations of the state of the system. Using results from nonlinear filtering, we transform this noisy observation SDE inside control problem into a full observation SPDE insider control problem. The results are illustrated by explicit examples.