McKean–Vlasov SDEs under measure dependent Lyapunov conditions

By W. Hammersley, D. vSivska, Lukasz Szpruch. Annales De L Institut Henri Poincare-probabilites Et Statistiques, 2018

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We prove the existence of weak solutions to McKean-Vlasov SDEs defined on a domain $D \subseteq \mathbb{R}^d$ with continuous and unbounded coefficients that satisfy Lyapunov type conditions, where the Lyapunov function may depend on measure. We propose a new type of {\em integrated} Lyapunov condition, where the inequality is only required to hold when integrated against the measure on which the Lyapunov function depends , and we show that this is sufficient for the existence of weak solutions to McKean-Vlasov SDEs defined on $D$. The main tool used in the proofs is the concept of a measure derivative due to Lions. We prove results on uniqueness under weaker assumptions than that of global Lipschitz continuity of the coefficients.

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Citations in the corpus

Weak quantitative propagation of chaos via differential calculus on the space of measures (2019, background): “[37] or, for more up-to-date panorama on research on existence and uniqueness, see [34, 22, 2, 13]).”
Large and Moderate Deviation Principles for McKean-Vlasov SDEs with Jumps (2020, background): “rge number of researchers under various settings due to their wide range of applications in several ﬁelds, including physics, chemistry, biology, economics, ﬁnancial mathematics etc. One can refer to [13, 25, 30, 48, 49] and the references therein for the existence and uniqueness of solutions to MVS”