Smoothing properties of McKean–Vlasov SDEs

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.

Regularity and Sensitivity for McKean-Vlasov Type SPDEs Generated by Stable-like Processes

“…The well posedness of this nonlinear non-stochastic equation with random coefficients, (7) or (11) follows from the general results on the well-posedness of nonlinear stable-like equations from [16][17][18]20]. Due to the equivalence of this equation with equation (3) we obtain the following well-posedness result for (3).…”

“…yielding (7). An alternative proof can be given by first rewriting equation (7) in the Ito form and then perform the transformation using Ito's formula.…”

“…There is an extensive literature on properties of McKean-Vlasov SPDEs (see e.g. [1][2][3][4][5][6][7][8][9][10][11][12] and references therein), based on the diffusion proccesses. The well-posedness of the McKean-Vlasov SPDE was shown in [1] in the class of L 2 -functions, and for measures in [2], though under an additional monotonicity assumption.…”

Iterative multilevel particle approximation for McKean–Vlasov SDEs

“…Inspired by the work [13], for any k ∈ N, we formally define the higher order derivatives in measures through the following iteration (provided that they actually exist): for any k ≥ 2, (i 1 , . .…”

“…(A.4) Since this notation for higher order derivatives in measure is quite cumbersome, we introduce the following multi-index notation for brevity. This notation was first proposed in [13].…”