Approximation Schemes for McKean–Vlasov and Boltzmann-Type Equations (Error Analysis in Total Variation Distance)

Abstract

We deal with McKean–Vlasov and Boltzmann-type jump equations. This means that the coefficients of the stochastic equation depend on the law of the solution, and the equation is driven by a Poisson point measure with intensity measure which depends on the law of the solution as well. Alfonsi and Bally (Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach), 2021, arXiv:2105.12677) have proved that under some suitable conditions, the solution \(X_t\) of such equation exists and is unique. One also proves that \(X_t\) is the probabilistic interpretation of an analytical weak equation. Moreover, the Euler scheme \(X_t^{{\mathcal {P}}}\) of this equation converges to \(X_t\) in Wasserstein distance. In this paper, under more restrictive assumptions, we show that the Euler scheme \(X_t^{{\mathcal {P}}}\) converges to \(X_t\) in total variation distance and \(X_t\) has a smooth density (which is a function solution of the analytical weak equation). On the other hand, in view of simulation, we use a truncated Euler scheme \(X^{{\mathcal {P}},M}_t\) which has a finite numbers of jumps in any compact interval. We prove that \(X^{{\mathcal {P}},M}_{t}\) also converges to \(X_t\) in total variation distance. Finally, we give an algorithm based on a particle system associated with \(X^{{\mathcal {P}},M}_t\) in order to approximate the density of the law of \(X_t\). Complete estimates of the error are obtained.