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

Thomas Cavallazzi

Stochastic Processes and their Applications, 2026, vol. 192, issue C

Abstract: 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 Pβ(Rd), the space of probability measures on Rd having a finite moment of order β∈(0,2]. As an application, we exhibit the backward Kolmogorov partial differential equation stated on [0,T]×Pβ(Rd) associated with a McKean–Vlasov stochastic differential equation driven by a Poisson random measure. It describes the dynamics of the semigroup acting on functions defined on Pβ(Rd) 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. α-stable processes with α∈(1,2).

Keywords: Itô’s formula; Flow of probability measures; Poisson random measure; Stable processes; McKean–Vlasov stochastic differential equations; Propagation of chaos; Backward Kolmogorov partial differential equation (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1016/j.spa.2025.104788

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