Approximation and generic properties of McKean-Vlasov stochastic equations with continuous coefficients

TL;DR

This paper investigates approximation methods and stability properties of McKean-Vlasov stochastic differential equations with continuous coefficients, establishing new results on solution stability and the construction of strong solutions without Yamada-Watanabe theorem.

Contribution

It demonstrates the stability of solutions under perturbations and introduces approximation procedures for strong solutions without relying on Yamada-Watanabe theorem.

Findings

Solutions are stable under small perturbations of initial conditions and parameters.

Strong solutions can be constructed via effective approximation methods.

The set of coefficients ensuring unique strong solutions is topologically large (second category).

Abstract

We consider various approximation properties for systems driven by a Mc Kean-Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover, the unique strong solutions may be constructed by effective approximation procedures, without using the famous Yamada-Watanabe theorem. Finally we show that the set of bounded uniformly continuous coefficients for which the corresponding MVSDE have a unique strong solution is a set of second category in the sense of Baire.