Approximation schemes for McKean-Vlasov and Boltzmann type equations (error analyses in total variation distance)
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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 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^𝒫 of this equation converges to X_t in Wasserstein distance. In this paper, under more restricted assumptions, we show that the Euler scheme X_t^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^𝒫,M_t which has a finite numbers of jumps in any compact interval. We prove that X^𝒫,M_t also converges to X_t in total variation distance. Finally, we give an algorithm based on a particle system associated to X^𝒫,M_t in order to approximate the density of the law of X_t. Complete estimates of the error are obtained.
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