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arXiv · 1809.06068

Bismut Formula for Lions Derivative of Distribution Dependent SDEs and Applications

Panpan Ren·Feng-Yu Wang
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Abstract

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of $P_tf(μ):=\mathbb E f(X_t^μ)$, where $t>0,$ $ f $ is a bounded measurable function, and $X_t^μ$ solves a distribution dependent SDE with initial distribution $μ$. 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 $P_tf (μ)= \int P_tf(x)μ(d x)$, essential difficulties are overcome in the study.

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Panpan Ren, Feng-Yu Wang. 2021-03-11. Bismut Formula for Lions Derivative of Distribution Dependent SDEs and Applications. https://arxiv.org/abs/1809.06068

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