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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2023

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

Résumé

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.
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Dates et versions

hal-03868434 , version 1 (07-12-2022)
hal-03868434 , version 2 (16-01-2023)

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  • HAL Id : hal-03868434 , version 2

Citer

Yifeng Qin. Approximation schemes for McKean-Vlasov and Boltzmann type equations (error analysis in total variation distance). 2023. ⟨hal-03868434v2⟩
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