Summary: | 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 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
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