[Paper Review] A general maximum principle for mean-field stochastic differential equations with jump processes

Mokhtar Hafayed, Syed Abbas|arXiv (Cornell University)|Jan 30, 2013

Stochastic processes and financial applications18 references10 citations

TL;DR

This paper establishes a general stochastic maximum principle for optimal control of mean-field stochastic differential equations with jump processes, where the control affects both diffusion and jump coefficients. Using the spike variation method, it derives necessary optimality conditions involving a linear mean-field backward SDE with jumps for the first-order adjoint process, extending classical results to include expectation-dependent coefficients and jump dynamics.

ABSTRACT

In this paper, we investigate the optimal control problems for stochastic differential equations (SDEs in short) of mean-field type with jump processes. The control variable is allowed to enter into both diffusion and jump terms. This stochastic maximum principle differs from the classical one in the sense that here the first-order adjoint equation turns out to be a linear mean-field backward SDE with jumps, while the second-order adjoint equation remains the same as in Tang and Li's stochastic maximum principle [32]. Finally, for the reader's convenience we give some analysis results used in this paper in the Appendix.

Motivation & Objective

• To develop necessary conditions of optimality for stochastic control problems governed by mean-field SDEs with jump processes.
• To extend the classical stochastic maximum principle to systems where coefficients depend on the expected value of the state process.
• To incorporate control-dependent diffusion and jump coefficients in the mean-field framework.
• To address optimal control problems with cost functionals that are also of mean-field type.
• To establish a general maximum principle valid for non-convex control domains and non-convex coefficients.

Proposed method

• Applies the spike variation method to perturb the optimal control and derive first-order optimality conditions.
• Derives the first-order adjoint equation as a linear mean-field backward SDE with jumps, distinguishing it from classical SDEs.
• Uses the martingale representation theorem for jump processes to characterize the adjoint processes.
• Employs an integration by parts formula for jump diffusions to handle the variation in the cost functional.
• Relies on Itô's formula for jump diffusions and moment estimates for stochastic integrals with respect to Poisson random measures.
• Establishes the optimality condition through a variational inequality involving the Hamiltonian and second-order terms.

Experimental results

Research questions

• RQ1 How can the stochastic maximum principle be extended to mean-field SDEs with jump processes?
• RQ2 What are the necessary conditions for optimality when the control enters both diffusion and jump coefficients in a mean-field context?
• RQ3 How does the adjoint equation differ in structure when the coefficients depend on the expected value of the state?
• RQ4 Can the spike variation method be adapted to derive optimality conditions in the presence of jumps and mean-field dependence?
• RQ5 What conditions ensure that the derived optimality condition is both necessary and general for non-convex control domains?

Key findings

• The first-order adjoint equation is a linear mean-field backward SDE with jumps, which generalizes the classical adjoint equation.
• The second-order adjoint equation remains unchanged from Tang and Li's framework, preserving the structure of the classical maximum principle.
• The necessary optimality condition is expressed via a variational inequality involving the Hamiltonian and quadratic terms in the control perturbation.
• When the coefficients and cost functional do not depend explicitly on the expected value, the result reduces to the classical stochastic maximum principle of Tang and Li (2003).
• The method yields a complete characterization of optimal controls through a Hamiltonian system involving the adjoint processes and jump contributions.
• The analysis includes rigorous moment estimates and integrability conditions ensuring the validity of the derived optimality conditions.