Particle Method for the McKean-Vlasov equation with common noise

TL;DR

This paper develops a numerical approach combining Euler and particle methods to simulate McKean-Vlasov equations with common noise, providing convergence analysis and practical examples.

Contribution

It introduces a novel numerical scheme for McKean-Vlasov equations with common noise, extending existing methods and providing convergence rates.

Findings

Convergence rates established for the Euler and particle methods.

Simulation examples demonstrate practical applicability.

Extension of results to equations with common noise.

Abstract

This paper studies the numerical simulation of the solution to the McKean-Vlasov equation with common noise. We begin by discretizing the solution in time using the Euler scheme, followed by spatial discretization through the particle method, inspired by the propagation of chaos property. Assuming H{\"o}lder continuity in time, as well as Lipschitz continuity in the state and measure arguments of the coefficient functions, we establish the convergence rate of the Euler scheme and the particle method. These results extend those for the standard McKean-Vlasov equation without common noise. Finally, we present two simulation examples : a modified conditional Ornstein Uhlenbeck process with common noise and an interbank market model.