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Pré-Publication, Document De Travail Année : 2022

Itô's formula for the flow of measures of Poisson stochastic integrals and applications

Résumé

We prove Itô's formula for the flow of measures associated with a jump process defined by a drift, an integral with respect to a Poisson random measure and with respect to the associated compensated Poisson random measure. We work in $\mathcal{P}_{\beta}(\mathbb{R}^d)$, the space of probability measures on $\mathbb{R}^d$ having a finite moment of order $\beta \in (0, 2]$. As an application, we exhibit the backward Kolmogorov partial differential equation stated on $[0,T] \times \mathcal{P}_{\beta}(\mathbb{R}^d)$ associated with a McKean-Vlasov stochastic differential equation driven by a Poisson random measure. It describes the dynamics of the semigroup associated with the McKean-Vlasov stochastic differential equation, under regularity assumptions on it. Finally, we use the semigroup and the backward Kolmogorov equation to prove new quantitative weak propagation of chaos results for a mean-field system of interacting Ornstein-Uhlenbeck processes driven by i.i.d. $\alpha$-stable processes with $\alpha \in (1,2)$.
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Dates et versions

hal-03873590 , version 1 (27-11-2022)

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  • HAL Id : hal-03873590 , version 1

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Thomas Cavallazzi. Itô's formula for the flow of measures of Poisson stochastic integrals and applications. 2022. ⟨hal-03873590⟩
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