Existence and smoothness of density function of solution to Mckean--Vlasov Equation with general coefficients

TL;DR

This paper tackles the existence and smoothness of density functions for solutions to Mckean-Vlasov equations with general coefficients using Malliavin calculus. The authors develop a strategy based on approximating SDEs to handle cases where Lions derivatives may not exist, establish the invertibility of the Malliavin covariance $Q(t)$, and prove the density exists for all $t>0$ under Lipschitz and uniform ellipticity conditions. When coefficients possess higher-order differentiability up to order $N+2$, they obtain that $X(t)\in \mathbb{D}^{N+2,\infty}$ and the density $p(t,\cdot)$ lies in $C_b^{N}(\mathbb{R}^d)$, providing finite-order regularity. Complementary numerical experiments solve the associated Fokker-Planck equations to approximate the density independently, demonstrating practical applicability of the theoretical results.

Abstract

In this paper, we study the existence and smoothness of a density function to the solution of a Mckean-Vlasov equation with the aid of Malliavin calculus. We first show the existence of the density function under assumptions that the coefficients of equation are only Lipschitz continuity and satisfy a uniform elliptic condition. Furthermore, we derive a precise regularity order and bounded a priori estimate for the density function under optimal smoothness assumptions for the coefficients. Finally, we present several numerical experiments to illustrate the approximation of the density function independently determined by solving a Fokker-Planck equation.

Figures (2)

• Figure 1: An approximate density function.
• Figure 2: Development of approximate joint density functions.

Theorems & Definitions (24)

• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Remark 1
• Lemma 7
• proof