Abstract
In this article, we develop integration by parts formulae on Wiener space for solutions of SDEs with general McKean–Vlasov interaction and uniformly elliptic coefficients. These integration by parts formulae hold both for derivatives with respect to a real variable and derivatives with respect to a measure understood in the sense of Lions. They allows us to prove the existence of a classical solution to a related PDE with irregular terminal condition. We also develop bounds for the derivatives of the density of the solutions of McKean–Vlasov SDEs.
Highlights
We explore to what extent the same is true for the PDE (1.2) under a uniform ellipticity assumption
We introduce the uniform ellipticity assumption (UE) in Assumption 3.3, used throughout the rest of the paper
We prove this as part of Theorem 3.2, we extend this to derivatives of any order
Summary
The main object of study in this paper is the McKean–Vlasov stochastic differential equation (MVSDE). MVSDEs are equations whose coefficients depend on the law of the solution They are referred to as mean-field SDEs and their solutions are often called nonlinear diffusions. These MVSDEs provide a probabilistic representation to the solutions of a class of nonlinear PDEs. A particular example of such nonlinear PDEs was first studied by McKean [29]. We consider the question of whether the PDE (1.2) has classical solutions when the initial condition g is not differentiable For this we exploit a probabilistic representation for the classical solution of the PDE (1.2) given in terms of a functional of θ t and of the solution of the following de-coupled equation: Xtx,[θ] = x +. We give further details of our results
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