2022/12/14 by Yifeng Qin, Qin, Yifeng
Economics, Econometrics and Finance · Engineering · Mathematics · #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Numerical Analysis (math.NA) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2212.07411
openalex publication_date 2022/12/14 · openalex created_date 2022/12/27 · openalex updated_date 2026/07/28
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 XtP of this equation converges to Xt in Wasserstein distance. In this paper, under more restricted assumptions, we show that the Euler scheme XtP 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 XP,Mt which has a finite numbers of jumps in any compact interval. We prove that XP,Mt also converges to Xt in total variation distance. Finally, we give an algorithm based on a particle system associated to XP,Mt in order to approximate the density of the law of Xt. Complete estimates of the error are obtained.