A Poisson-Alekseev-Gröbner formula through Malliavin calculus for Poisson random integrals

Abstract: In this paper, we establish an Alekseev--Gr\"obner formula for stochastic differential equations (SDEs) driven by a Poisson random measure, which express the global error between a functional of two processes solution of SDEs started at the same initial condition, in terms of the infinitesimal error (i.e, the difference between the SDEs coefficients). In particular, we consider the situation where the flow process is only assumed to be jointly stochastically continuous with respect to space and time. Our proof relies on a new approach for the definition of the Skorohod--Poisson integral to treat the anticipating term appearing in the formula, based on a definition of the Malliavin derivative on a class of smooth random variables, instead of the more standard polynomial chaos approach.