The Donsker delta function and local time for McKean–Vlasov processes and applications

Abstract

The purpose of this paper is to establish a stochastic differential equation for the Donsker delta measure of the solution of a McKean–Vlasov (mean-field) stochastic differential equation. If the Donsker delta measure is absolutely continuous with respect to Lebesgue measure, then its Radon–Nikodym derivative is called the Donsker delta function. In that case it can be proved that the local time of such a process is simply the integral with respect to time of the Donsker delta function. Therefore we also get an equation for the local time of such a process. For some particular McKean–Vlasov processes, we find explicit expressions for their Donsker delta functions and hence for their local times.