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 $X_t^{\mathcal{P}}$ of this equation converges to $X_t$ in Wasserstein distance. In this paper, under more restricted assumptions, we show that the Euler scheme $X_t^{\mathcal{P}}$ 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 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.

翻译：我们研究McKean-Vlasov与Boltzmann型跳跃方程。这意味着随机方程的系数依赖于解的概率分布，且方程由强度测度同样依赖于解分布的泊松点测度驱动。Alfonsi与Bally在文献[3]中证明了，在适当条件下，此类方程的解$X_t$存在且唯一。他们还证明了$X_t$是某解析弱方程的概率解释。此外，该方程的欧拉格式$X_t^{\mathcal{P}}$在Wasserstein距离下收敛于$X_t$。本文在更强假设下，证明欧拉格式$X_t^{\mathcal{P}}$在全变差距离下收敛于$X_t$，且$X_t$具有光滑密度（该密度是该解析弱方程的函数解）。另一方面，为便于模拟，我们采用截断欧拉格式$X^{\mathcal{P},M}_t$，其在任意紧区间内仅包含有限次跳跃。我们证明$X^{\mathcal{P},M}_{t}$亦在全变差距离下收敛于$X_t$。最后，我们给出基于与$X^{\mathcal{P},M}_t$关联的粒子系统的算法，以近似$X_t$概率分布的密度函数，并得到了完整的误差估计。