Approximation schemes for McKean-Vlasov and Boltzmann type equations (error analyses in total variation distance)

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. In [3], Alfonsi and Bally have proved that under some suitable conditions, the solution $Xt$ of such equation exists and is unique. One also proves that $Xt$ is the probabilistic interpretation of an analytical weak equation. Moreover, the Euler scheme $Xt^{{\mathcal{P}}$} of this equation converges to $Xt$ in Wasserstein distance. In this paper, under more restricted assumptions, we show that the Euler scheme $Xt^{{\mathcal{P}}$} converges to $Xt$ in total variation distance and $Xt$ 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 $Xt$ in total variation distance. Finally, we give an algorithm based on a particle system associated to $X^{{\mathcal{P},M}}t$ in order to approximate the density of the law of $X_t$. Complete estimates of the error are obtained.