Contraction criteria for Brownian filtrations samplings

TL;DR

This paper provides a general criterion to determine if a stochastic process generates a sampled Brownian filtration, using contraction criteria and properties of sampled Wiener measures, with applications to stochastic differential equations.

Contribution

It introduces a novel sufficient condition based on contraction criteria and quasi-invariance properties for sampled Brownian filtrations, extending analysis to non-Markovian SDEs.

Findings

Established a sufficient condition for sampled Brownian filtrations.

Identified the Cameron-Martin space for sampled Wiener measures.

Applied results to existence of solutions for non-Markovian SDEs.

Abstract

This paper investigates the problem to determine whether a given stochastic process generates a sampled Brownian filtration. A fairly general sufficient condition is obtained by applying the Frank H. Clarke contraction criteria to a functional whose construction relies on the quasi-invariance properties of a sampled Wiener measure. The latter are investigated from the precise analytic structure underlying this sampled Brownian motion. In particular, the Cameron-Martin space is identified from usual Lebesgue integrals over a Guseinov measure. As an application we obtain sufficient conditions for the existence of a solution to a class of not necessarily Markovian stochastic differential equations driven by a sampled Brownian motion. By taking a sampling times set which coincides with the unit interval, the results apply to continuous time models.