Mean Field Type Control Problems Driven by Jump-diffusions

TL;DR

This work develops a comprehensive probabilistic framework for mean-field type control (MFTC) problems driven by jump-diffusions, allowing state and diffusion coefficients to depend nonlinearly on the state, its law, and the control. By formulating a forward–backward SDE system with jumps through a maximum principle and introducing a joint cone property for adjoint processes, the authors establish global-in-time well-posedness and regularity of the value function $V$, and prove that $V$ is the unique classical solution to the HJB integro-PDE. The approach handles fairly general jump structures beyond Brownian motion and yields explicit derivatives of $V$ with respect to the distribution via Jacobian flows, enabling a rigorous link to the master equation for mean-field games with jumps. This probabilistic method provides a robust alternative to analytic PDE techniques in settings with jump noise and distributional dependence, with potential applications to McKean–Vlasov control and MFGs under Lévy-type perturbations.

Abstract

In this article, we apply a probabilistic approach to study general mean field type control (MFTC) problems with jump-diffusions, and give the first global-in-time solution. We allow the drift coefficient $b$ and the diffusion coefficient $σ$ to nonlinearly depend on the state, distribution and control variables, and both can be unbounded and possibly degenerate; besides, the jump coefficient $γ$ is allowed to be non-constant. To tackle the non-linear and control-dependent diffusion $σ$, we further formulate a joint cone property and estimates for both processes $P$ and $Q$ of the corresponding adjoint process (where $(P,Q,R)$ is the solution triple of the associated adjoint process as a backward stochastic differential equation with jump), in contrast to our previous single cone property of the only process $P$. We study first the system of forward-backward stochastic differential equations (FBSDEs) with jumps arising from the maximum principle, and then the related Jacobian flows, which altogether yield the classical regularity of the value function and thus allow us to show that the value function is the unique classical solution of the HJB integro-partial differential equation. Most importantly, our proposed probabilistic approach can apparently handle the MFTC problem driven by a fairly general process far beyond Brownian motion, in a relatively easier manner than the existing analytic approach.

Theorems & Definitions (24)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem 3.5
• Proposition 3.6: Cone Property
• proof
• Theorem 3.7
• Proposition 3.8
• proof