Mean-field stochastic differential equations and associated PDEs

Abstract

In this paper we consider a mean-field stochastic differential equation, also called Mc Kean-Vlasov equation, with initial data (t, x) ∈ [0, T ] × R d , which coefficients depend on both the solution X t,x s but also its law. By considering square integrable random variables ξ as initial condition for this equation, we can easily show the flow property of the solution X t,ξ s of this new equation. Associating it with a process X t,x,P ξ s which coincides with X t,ξ s , when one substitutes ξ for x, but which has the advantage to depend only on the law P ξ of ξ, we characterise the function)] under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a non local PDE of mean-field type, involving the first and second order derivatives of V with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first and second order derivatives of the solution of the mean-field stochastic differential equation with respect to the probability law and a corresponding Itô formula. In our approach we use the notion of derivative with respect to a square integrable probability measure introduced in [6] and we extend it in a direct way to second order derivatives.

Smoothing properties of McKean–Vlasov SDEs

“…A similar result has been proved in [8,12] under different conditions than ours and for an initial condition g that is sufficiently smooth. For the stochastic flow (X x t ) t≥0 solving a classical SDE with initial condition x ∈ R N , the standard strategy to show that the function u(t, x) := E g(X x t ) is a classical solution of a linear PDE is to show, using the flow property of X x t , that for h > 0, u(t + h, x) = E [u(t, X x h )] and then show that u is regular enough to apply Itô's formula to u(t, X x h ).…”

“…As mentioned previously, the PDE (1.2) is also studied in [8] and [12]. Let us explain the relationship between the results in those works and the results in this paper.…”

“…We also prove this as part of Theorem 3.2, although we extend this to derivatives of any order (assuming sufficient smoothness of the coefficients). In [8], the hypotheses on the continuity and differentiability of the coefficients are the same as ours The authors then consider initial conditions g : R N × P 2 (R N ) → R N for which the derivatives up to second order exist and are bounded, which they use to prove regularity of U . Since g is sufficiently smooth, they do not need to impose any non-degeneracy condition on the coefficients.…”

“…Considering functions on R N × P 2 (R N ) raises a question about whether the order in which we take derivatives matters. A result from [8] says that derivatives commute when the mixed derivatives are Lipschitz continuous.…”

Dynamic Programming for Optimal Control of Stochastic McKean--Vlasov Dynamics

“…Lions in his course at Collège de France [35]. We provide a brief introduction to this concept and refer to the lecture notes [11] (see also [10], [19]) for the details. This notion is based on the lifting of functions u :…”