Well-posedness and tamed schemes for McKean-Vlasov Equations with Common Noise

TL;DR

This paper proves well-posedness for McKean-Vlasov SDEs with common noise and introduces explicit tamed Euler and Milstein schemes that are stable and converge strongly, supported by numerical examples.

Contribution

It establishes well-posedness for McKean-Vlasov equations with super-linear coefficients and develops stable, convergent tamed numerical schemes for these equations.

Findings

Proved well-posedness of McKean-Vlasov SDEs with common noise.

Developed explicit tamed Euler and Milstein schemes with proven stability and convergence.

Numerical examples demonstrate the effectiveness of the proposed methods.

Abstract

In this paper, we first establish well-posedness of McKean-Vlasov stochastic differential equations (McKean-Vlasov SDEs) with common noise, possibly with coefficients having 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 $1/2$ 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 $3/2$ volatility model and the stochastic double well dynamics with multiplicative noise.