Bismut formula for Lions derivative of distribution dependent SDEs and applications

By Panpan Ren, Feng-Yu Wang. Journal of Differential Equations, 2018

89 citations recorded by the source corpus.

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of $P_tf(\mu):=\mathbb E f(X_t^\mu)$, where $t>0,$ $ f $ is a bounded measurable function, and $X_t^\mu$ solves a distribution dependent SDE with initial distribution $\mu$. As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semigroup property and the invalidity of the formula $P_tf (\mu)= \int P_tf(x)\mu(d x)$, essential difficulties are overcome in the study.

Subfields: Applied Mathematics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/04a2d600f7d7c2c364a40492c667919a84d45bc0

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Citations in the corpus

Bismut formula for Lions derivative of distribution-path dependent SDEs (2020, methodology): “4 below) of this part generalize those derived in [4] for SDEs with memory and in [26] for McKean-Vlasov SDEs without memory.”
Strong convergence order for slow–fast McKean–Vlasov stochastic differential equations (2019, background): “Further properties, such as the Harnack inequality or the Bismut formula for the Lions Derivative have been investigated in [37] and [33] respectively.”
Distribution dependent SDEs driven by fractional Brownian motions (2021, methodology): “, [5, 29]), when taking the special choices of b((1)), b((2)) and σ, the above model will reduce to distribution dependent stochastic Hamiltonian system with fractional noise.”