Optimal control of mean-field jump-diffusion systems with noisy memory

ABSTRACT

This paper is concerned with an optimal control problem under mean-field jump-diffusion systems with delay and noisy memory. First, we derive necessary and sufficient maximum principles using Malliavin calculus technique. Meanwhile, we introduce a new mean-field backward stochastic differential equation as the adjoint equation which involves not only partial derivatives of the Hamiltonian function but also their Malliavin derivatives. Then, applying a reduction of the noisy memory dynamics to a two-dimensional discrete delay optimal control problem, we establish the second set of necessary and sufficient maximum principles under partial information. Moreover, a natural link between the above two approaches is established via the adjoint equations. Finally, we apply our theoretical results to study a mean-field linear-quadratic optimal control problem.

KEYWORDS:

• Maximum principle
• mean-field
• jump diffusion
• noisy memory
• Malliavin calculus

Acknowledgments

We would like to express our gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which led to the improvement of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by NNSF of China [grant number 11571126].