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.

arXiv comment33

References in corpus (13)

• Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
• Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients
• A note on tamed Euler approximations
• Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients
• Global solvability of a networked integrate-and-fire model of McKean-Vlasov type
• Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes
• Nonlinear SDEs driven by Lévy processes and related PDEs
• Propagation of Chaos for Stochastic Spatially Structured Neuronal Networks with Delay driven by Jump Diffusions
• First order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems
• On Explicit Milstein-type Scheme for Mckean-Vlasov Stochastic Differential Equations with Super-linear Drift Coefficient
• Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients
• Simulation of McKean Vlasov SDEs with super linear growth
• Milstein-type Schemes of SDE Driven by Lévy Noise with Super-linear Diffusion Coefficients

Cited by in corpus (5)

• On Explicit Milstein-type Scheme for Mckean-Vlasov Stochastic Differential Equations with Super-linear Drift Coefficient
• A flexible split-step scheme for solving McKean-Vlasov Stochastic Differential Equations
• Full history recursive multilevel Picard approximations for ordinary differential equations with expectations
• Well-posedness and propagation of chaos for L{é}vy-driven McKean-Vlasov SDEs under Lipschitz assumptions
• Explicit Numerical Approximations for McKean-Vlasov Neutral Stochastic Differential Delay Equations