Indefinite linear-quadratic optimal control of mean-field stochastic differential equation with jump diffusion: an equivalent cost functional method

Guangchen Wang; Wencan Wang

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arXiv · 2111.08854

Indefinite linear-quadratic optimal control of mean-field stochastic differential equation with jump diffusion: an equivalent cost functional method

Guangchen Wang·Wencan Wang

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Abstract

In this paper, we consider a linear-quadratic optimal control problem of mean-field stochastic differential equation with jump diffusion, which is also called as an MF-LQJ problem. Here, cost functional is allowed to be indefinite. We use an equivalent cost functional method to deal with the MF-LQJ problem with indefinite weighting matrices. Some equivalent cost functionals enable us to establish a bridge between indefinite and positive-definite MF-LQJ problems. With such a bridge, solvabilities of stochastic Hamiltonian system and Riccati equations are further characterized. Optimal control of the indefinite MF-LQJ problem is represented as a state feedback via solutions of Riccati equations. As a by-product, the method provides a new way to prove the existence and uniqueness of solution to mean field forward-backward stochastic differential equation with jump diffusion (MF-FBSDEJ, for short), where existing methods in literature do not work. Some examples are provided to illustrate our results.

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BibTeX RIS

Guangchen Wang, Wencan Wang. 2021-11-17. Indefinite linear-quadratic optimal control of mean-field stochastic differential equation with jump diffusion: an equivalent cost functional method. https://arxiv.org/abs/2111.08854

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