Stochastic Evolution Equation Driven by Teugels Martingale and Its Optimal Control

TL;DR

This paper studies stochastic evolution equations in Hilbert space driven by Teugels martingales and Brownian motion, establishing existence, uniqueness, and optimal control solutions with practical examples.

Contribution

It introduces a framework for solving stochastic evolution equations with Teugels martingales, including existence, uniqueness, and optimal control principles.

Findings

Proved existence and uniqueness of solutions.

Established stochastic maximum principle and verification theorem.

Provided an example of controlled stochastic PDE driven by Teugels martingales.

Abstract

The paper is concerned with a class of stochastic evolution equations in Hilbert space with random coefficients driven by Teugel's martingales and an independent multi-dimensional Brownian motion and its optimal control problem. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with L\'evy processes (see Nualart and Schoutens). 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 driven by Teugels martingales which our theoretical results can solve.