On the existence of optimal controls for backward stochastic partial differential equations

TL;DR

This paper establishes the existence of optimal controls for backward stochastic partial differential equations with random coefficients by analyzing the associated stochastic Hamiltonian system and employing continuation methods.

Contribution

It provides new existence results for optimal controls in backward stochastic PDEs with random coefficients under growth, monotonicity, and Hamiltonian assumptions.

Findings

Existence of solutions to the stochastic Hamiltonian system is proven.

Optimal controls exist under specified conditions.

Linear-quadratic control examples validate the theoretical results.

Abstract

This paper is concerned with the existence of optimal controls for backward stochastic partial differential equations with random coefficients, in which the control systems are represented in an abstract evolution form, i.e. backward stochastic evolution equations. Under some growth and monotonicity conditions on the coefficients and suitable assumptions on the Hamiltonian function, the existence of the optimal control boils down to proving the uniqueness and existence of a solution to the stochastic Hamiltonian system, i.e. a fully coupled forward-backward stochastic evolution equation. Using some a prior estimates, we prove the uniqueness and existence via the method of continuation. Two examples of linear-quadratic control are solved to demonstrate our results.