Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Stochastic Differential Equations in Infinite Dimensional Hilbert Space and its Optimal Control Problem with L´evy Processes

Version 1 : Received: 8 April 2021 / Approved: 12 April 2021 / Online: 12 April 2021 (11:21:33 CEST)

How to cite: Wang, M.; Shi, Q.; Meng, Q.; Tang, M. Stochastic Differential Equations in Infinite Dimensional Hilbert Space and its Optimal Control Problem with L´evy Processes. Preprints 2021, 2021040278 (doi: 10.20944/preprints202104.0278.v1). Wang, M.; Shi, Q.; Meng, Q.; Tang, M. Stochastic Differential Equations in Infinite Dimensional Hilbert Space and its Optimal Control Problem with L´evy Processes. Preprints 2021, 2021040278 (doi: 10.20944/preprints202104.0278.v1).

Abstract

The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugel's martingales which are more general processes. and its optimal control problem. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with L\'{e}vy 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 classicconvex 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.

Subject Areas

Stochastic Evolution Equation; Teugels Martingales; Optimal Control; Stochastic Maximum Principle; Verification Theorem

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