Stochastic Evolution Equations of Jump Type with Random Coefficients: Existence, Uniqueness and Optimal Control

TL;DR

This paper investigates stochastic evolution equations with jumps and random coefficients, establishing existence, uniqueness, and optimal control solutions, including a practical example involving controlled stochastic PDEs with jumps.

Contribution

It introduces new methods for proving solution existence and uniqueness, and develops a stochastic maximum principle for jump-type equations with random coefficients.

Findings

Proved existence and uniqueness of solutions using continuous dependence and parameter extension.

Established stochastic maximum principle and verification theorem for optimal control.

Provided an example of controlled stochastic PDE with jumps demonstrating the theoretical results.

Abstract

We study a class of stochastic evolution equations of jump type with random coefficients and its optimal control problem. There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual technique. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation with jumps which our theoretical results can solve.