Optimal stopping of conditional McKean-Vlasov jump diffusions

TL;DR

This paper develops a framework for solving optimal stopping problems involving conditional McKean-Vlasov jump diffusions, combining variational inequalities with stochastic Fokker-Planck equations, and applies it to financial and project management scenarios.

Contribution

It introduces a novel approach to optimal stopping for conditional McKean-Vlasov jump diffusions using a Markovian system and variational inequalities, with explicit solutions for practical problems.

Findings

Derived sufficient variational inequalities for the value function.

Solved explicitly two optimal stopping problems involving jumps and common noise.

Provided a new method to handle conditional McKean-Vlasov equations in optimal stopping.

Abstract

We study the problem of optimal stopping of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. To achieve this, we combine the state equation for the conditional McKean-Vlasov equation with the associated stochastic Fokker-Planck equation for the conditional law of the solution of the state. This gives us a Markovian system which can be handled by using a version of the Dynkin formula. We illustrate our result by solving explicitly two optimal stopping problems for conditional McKean-Vlasov jump diffusions. More specifically, we first find the optimal time to sell in a market with common noise and jumps, and, next, we find the stopping time to quit a project whose state is modelled by a jump diffusion, when the performance functional involves the conditional mean of the state.