Abstract
In this paper, we first establish well-posedness of McKean–Vlasov stochastic differential equations (McKean–Vlasov SDEs) with common noise, possibly with coefficients of super-linear growth in the state variable. Second, we present stable time-stepping schemes for this class of McKean–Vlasov SDEs. Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for an interacting particle system associated with the McKean–Vlasov equation. We prove stability and strong convergence of order and 1, respectively. To obtain our main results, we employ techniques from calculus on the Wasserstein space. The proof for the strong convergence of the tamed Milstein scheme only requires the coefficients to be once continuously differentiable in the state and measure component. To demonstrate our theoretical findings, we present several numerical examples, including mean-field versions of the stochastic volatility model and the stochastic double well dynamics with multiplicative noise.
Funding Statement
The first author gratefully acknowledges financial support provided by the Science and Engineering Research Board (SERB), India, under its MATRICS program (Grant Number SER-1329-MTD). The last author gratefully acknowledges support by a special Upper Austrian Government grant.
Acknowledgements
We would like to thank David Šiška and Łukasz Szpruch, both from University of Edinburgh, for pointing us to Lemma A.5 in [38] (Lemma 1.2 above), used in the proof of our Theorem 2.1, before the preprint became publicly available. We are also thankful to the anonymous referees for their insightful comments and suggestions.
Citation
Chaman Kumar. Neelima. Christoph Reisinger. Wolfgang Stockinger. "Well-posedness and tamed schemes for McKean–Vlasov equations with common noise." Ann. Appl. Probab. 32 (5) 3283 - 3330, October 2022. https://doi.org/10.1214/21-AAP1760
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