A forward-backward stochastic analysis of diffusion flows

Pierre del Moral (1) , S.S. Singh (2)

1 CQFD - Quality control and dynamic reliability
2 Department of Engineering [Cambridge]

We present forward-backward stochastic semigroup formulae to analyse the difference of diffusion flows driven by different drift and diffusion functions. These formulae are expressed in terms of tangent and Hessian processes and can be interpreted as an extension of the Aleeksev-Gröbner lemma to diffusion flows. We present some natural spectral conditions that allows to derive in a direct way a series of uniform estimates with respect to the time horizon. We illustrate the impact of these results in the context of diffusion perturbation theory, interacting diffusions and discrete time approximations.

Keywords: Skorohod stochastic integral; Aleeksev-Gröbner lemma; Perturbation semigroups; Tangent and Hessian processes; Bismut-Elworthy-Li formulae; Malliavin differential; Stochastic flows; Variational equations

Domains: Probability [math.PR]; Dynamical Systems [math.DS]

Pierre del Moral, S.S. Singh. A forward-backward stochastic analysis of diffusion flows. [Research Report] INRIA. 2019. ⟨hal-02161914v1⟩