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arXiv · 1802.03974

McKean-Vlasov SDEs under Measure Dependent Lyapunov Conditions

William Hammersley·David Šiška·Lukasz Szpruch
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Abstract

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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William Hammersley, David Šiška, Lukasz Szpruch. 2020-09-30. McKean-Vlasov SDEs under Measure Dependent Lyapunov Conditions. https://arxiv.org/abs/1802.03974

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