On optimal mean-field type control problems of stochastic systems with jump processes under partial information

TL;DR

This paper develops a maximum principle for partially observed mean-field stochastic control systems driven by Brownian motion and jump processes, employing Malliavin calculus to handle randomness and non-Markovian features.

Contribution

It introduces a novel maximum principle for mean-field control problems with jumps under partial information, explicitly expressing the adjoint process and handling non-Markovian dynamics.

Findings

Derived explicit maximum principle using Malliavin calculus.

Extended control framework to systems with jump processes and mean-field interactions.

Provided an example of linear-quadratic control demonstrating the theory.

Abstract

This paper considers the problem of partially observed optimal control for forward stochastic systems which are driven by Brownian motions and an independent Poisson random measure with a feature that the cost functional is of mean-field type. When all the system coefficients and the objective performance functionals are allowed to be random, possibly non-Markovian, Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjointprocess is explicitly expressed. We also investigate the mean-field type optimal control problems for systems driven by mean-field type stochastic differential equations (SDEs in short) with jump processes, in which the coefficients contain not only the state process but also its marginal distribution under partially observed information. The maximum principle is established using convex variational technique with an illustrating example about linear-quadratic optimal control.