Smoothing properties of McKean–Vlasov SDEs

By D. Crisan, Eamon McMurray. 2017

106 citations recorded by the source corpus.

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.

Subfields: Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/dd20614219639e8056e8463c2a5bec3a987e1ac3

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Citations in the corpus

Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “That is, the functions (t, x,m) 7→ E[F (XT , μT ) |Xt = x, μt = m] should satisfy a certain linear PDE on [0, T ] × R × P(Rd), provided F is sufficiently nice [25, 22], and regularity estimates on the solution are useful for quantitative propagation of chaos arguments [34, 21].”
Weak quantitative propagation of chaos via differential calculus on the space of measures (2019, methodology): “The method of Malliavin calculus is adopted in [10].”
Distribution dependent SDEs driven by fractional Brownian motions (2021, background): “mention, for examples, the works in [10, 11, 18] for large scale social interactions within the memory of mean-field games, [8, 13, 23] for value functions and related PDEs, [19, 20, 32, 36] for (shift) Harnack type inequalities and gradient estimates, and the references therein.”
Bismut formula for Lions derivative of distribution dependent SDEs and applications (2018, methodology): “Moreover, formulas of the L-derivative and integration by parts have been presented in [8] for the following de-coupled SDE: dX t = b(t, X x,μ t , P ∗ t μ)dt+ σ(t, X x,μ t , P ∗ t μ)dWt, X x,μ t = x, 4”
Freidlin–Wentzell LDP in path space for McKean–Vlasov equations and the functional iterated logarithm law (2017, background): “Mean-field stochastic differential equations and associated PDEs.”
Bismut formula for Lions derivative of distribution-path dependent SDEs (2020, methodology): “See also [3, 11, 28] for the study of derivative in the initial points for McKean-Vlasov SDEs, and Lions derivative for solutions to the de-coupled SDEs (which do not depend on the distribution of its own solution) associated with McKean-Vlasov SDEs.”