Intervention in Ornstein-Uhlenbeck SDEs

TL;DR

This paper introduces a causal intervention framework for stochastic differential equations, specifically demonstrating that interventions in Ornstein-Uhlenbeck SDEs preserve their form and analyzing their stationary distributions.

Contribution

It defines a notion of intervention for SDEs, shows that interventions in Ornstein-Uhlenbeck SDEs result in similar SDEs, and discusses stationary distribution criteria.

Findings

Interventions in Ornstein-Uhlenbeck SDEs produce similar SDEs.

Stationary distribution existence criteria are discussed.

Mean and variance of the stationary distribution are computed in a simple case.

Abstract

We introduce a notion of intervention for stochastic differential equations and a corresponding causal interpretation. For the case of the Ornstein-Uhlenbeck SDE, we show that the SDE resulting from a simple type of intervention again is an Ornstein-Uhlenbeck SDE. We discuss criteria for the existence of a stationary distribution for the solution to the intervened SDE. We illustrate the effect of interventions by calculating the mean and variance in the stationary distribution of an intervened process in a particularly simple case.