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

arXiv: https://arxiv.org/abs/2111.08854 · HTML: https://mlsanity.com/paper/2111.08854/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2021.

## Built on by (most cited first, 1 of 1)

- [Data-Driven Exploration for a Class of Continuous-Time Indefinite Linear--Quadratic Reinforcement Learning Problems](https://mlsanity.com/paper/2507.00358/) (2025)

## Abstract (arXiv)

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.