Computation of sensitivities for the invariant measure of a parameter dependent diffusion

TL;DR

This paper develops a numerical method to efficiently compute sensitivities of invariant measures for parameter-dependent stochastic differential equations, enabling accurate derivative estimation via long-term simulations.

Contribution

It introduces conditions ensuring uniform integrability of the derivative process, facilitating reliable Monte Carlo estimation of sensitivities.

Findings

Provided sufficient conditions for uniform-in-time square integrability.

Demonstrated the effectiveness of the method for computing derivatives of invariant measure averages.

Enabled efficient sensitivity analysis in stochastic systems.

Abstract

We consider the solution to a stochastic differential equation with a drift function which depends smoothly on some real parameter $\lambda$, and admitting a unique invariant measure for any value of $\lambda$ around $\lambda$ = 0. Our aim is to compute the derivative with respect to $\lambda$ of averages with respect to the invariant measure, at $\lambda$ = 0. We analyze a numerical method which consists in simulating the process at $\lambda$ = 0 together with its derivative with respect to $\lambda$ on long time horizon. We give sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to $\lambda$ of the mean of an observable through Monte Carlo simulations.