Mean field game equations with underlying jump-diffusion process

TL;DR

This paper analyzes mean field game equations linked to jump-diffusion processes, providing explicit solutions for certain controls and applying the results to optimal portfolio selection in finance.

Contribution

It extends existing results for pure diffusion processes to jump-diffusion processes, offering explicit solutions for the MFG system under specific controls.

Findings

Explicit expectation formulas for jump-diffusion processes

Extension of MFG results from diffusion to jump-diffusion

Application to optimal investment portfolio evaluation

Abstract

We consider a couple of integrodifferential PDEs arising from a stochastic Markovian control problem subjected to initial-terminal conditions. These equations correspond to the MFG system for a controlled jump-diffusion process. We prove that for a specific choice of the control function the expectation of the jump-diffusion process can be found explicitly. The study is an extension of similar results known for the pure diffusion process. As an example, we show how this can be applied to the problem of investors evaluating the trend of an asset when choosing an optimal portfolio.