On explicit Milstein-type scheme for McKean–Vlasov stochastic differential equations with super-linear drift coefficient

Abstract

We introduce an explicit Milstein-type scheme for McKean–Vlasov stochastic differential equations using the notion of a measure derivative given by P.-L. Lions in his lectures at the Collège de France and presented in [9]. We show that the proposed Milstein-type scheme converges to the true solution in strong sense with a rate equal to 1.0. The drift coefficient is allowed to grow super-linearly in the state variable and both the drift and the diffusion coefficients are assumed to be only once differentiable in variables corresponding to state and measure. Furthermore, derivatives of drift and diffusion coefficients with respect to the measure component are uniformly bounded. The challenges arising due to the dependence of coefficients on measures are tackled and our findings are consistent with the analogous results for stochastic differential equations.