Well-posedness and tamed Euler schemes for McKean-Vlasov equations driven by Lévy noise

Abstract

We prove the well-posedness of solutions to McKean-Vlasov stochastic differential equations driven by Lévy noise under mild assumptions where, in particular, the Lévy measure is not required to be finite. The drift, diffusion and jump coefficients are allowed to be random, can grow super-linearly in the state variable, and all may depend on the marginal law of the solution process. We provide a propagation of chaos result under more relaxed conditions than those existing in the literature, and consistent with our well-posedness result. We propose a tamed Euler scheme for the associated interacting particle system and prove that the rate of its strong convergence is arbitrarily close to $1/2$. As a by-product, we also obtain the corresponding results on well-posedness, propagation of chaos and strong convergence of the tamed Euler scheme for McKean-Vlasov stochastic delay differential equations (SDDE) and McKean-Vlasov stochastic differential equations with Markovian switching (SDEwMS), both driven by Lévy noise. Furthermore, our results on tamed Euler schemes are new even for ordinary SDEs driven by Lévy noise and with super-linearly growing coefficients.