# Backward It{\^o}-Ventzell and stochastic interpolation formulae

## TL;DR

This paper introduces a new backward Itô-Ventzell formula and extends stochastic interpolation formulas to stochastic flows, providing spectral conditions for uniform estimates and applications in diffusion perturbation and approximation theories.

## Contribution

The paper presents a novel backward Itô-Ventzell formula and extends stochastic interpolation formulas to stochastic flows, with spectral conditions for uniform flow difference estimates.

## Key findings

- New backward Itô-Ventzell formula introduced
- Spectral conditions enable simple proofs of flow estimates
- Applications demonstrated in diffusion perturbation and approximation

## Abstract

We present a novel backward It{\^o}-Ventzell formula and an extension of the Aleeksev-Gr\"obner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between the two stochastic flows when their drift and diffusion functions are not the same, yielding what seems to be the first results of this type for this class of anticipative models. We illustrate the impact of these results in the context of diffusion perturbation theory, interacting diffusions and discrete time approximations