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:我们处理的是Mckean- Vlasov 和 Boltzmann 的跳跃方程式。 这意味着, 蒸汽方程式的系数取决于解决方案的定律, 而方程式的系数则由Poisson点度量值驱动, 强度度量也取决于解决方案的定律。 在 [3] 中, Alfonsi 和 Bally 证明, 在某些合适的条件下, 此方程式的解决方案存在并且是独一无二的。 其中一项还证明, $xt$是分析疲软方程的概率解释。 此外, 这个方程式的 Euler 方案 $X_ tmathcal{P} 的系数由瓦瑟斯坦距离的 $X 点度度量度度度量值趋同 $X_ t math{ P} 的度量度度量值。 在本文中, Euler $_ txx 的精确度量值和 0.X 值的精确度度值也由 美元 美元=xxxx 的精确度计算。