Parameter estimation of path-dependent McKean-Vlasov stochastic differential equations

TL;DR

This paper studies a class of complex stochastic differential equations with path dependence and unknown parameters, establishing their mathematical properties, proposing parameter estimation methods, and validating through numerical simulations.

Contribution

It introduces the first maximum likelihood estimators for path-dependent McKean-Vlasov equations and proves their strong consistency.

Findings

Existence and uniqueness under non-Lipschitz conditions

Development of a numerical simulation method

Error estimation between solutions and numerical approximations

Abstract

The work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. Third, a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered. Moreover, we estimate the errors between solutions of these equations and that of their numerical equations. Finally, we give an example to explain our result.