Existence and Uniqueness of Solutions to Vlasov-McKean Equations
A talk in the Bielefeld Stochastic Afternoon series by
Sima Mehri from TU Berlin
| Abstract: | Existence and uniqueness of weak solutions to Vlasov-McKean Equations and more general law-dependent stochastic differential equations under linear growth conditions and non-degeneracy is well-known in the existing literature.
We present a Lyapunov type approach to the problem of existence and uniqueness of general law-dependent stochastic differential equations that allows us to relax the assumption concerning the growth conditions.
In the existing literature most results concerning existence and uniqueness are obtained under regularity assumptions of the coeffcients w.r.t the Wasserstein distance. There are also existence and uniqueness results with irregular coeffcients by taking total variation distance in the existing literature (see [1]). Controlling the law of the solution in some weighted total variational distance allows us now to derive a rather general uniqueness result, merely assuming measurability and certain integrability on the drift coeffcient and non-degeneracy on the dispersion coeffcient. We also present an abstract existence result for law-
dependent stochastic differential equations with merely measurable coeffcients, based on an approximation with law-dependent stochastic differential equations with regular coeffcients under Lyapunov-type assumptions.
The talk is based on joint work with Prof. Dr. Wilhelm Stannat (TU Berlin).
[1] Mishura, Y. S., and Veretennikov, A. Y. Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations. arXiv preprint:1603.02212v7 (2018). |