# An Optimal Gauss-Markov Approximation for a Process with Stochastic Drift and Applications

## TL;DR

This paper develops an optimal approximation method for stochastic differential equations with stochastic drift using calculus of variations, with applications to neural activity modeling.

## Contribution

It introduces a novel approach to approximate SDEs with stochastic drift via Ornstein-Uhlenbeck processes, ensuring existence, uniqueness, and bounds for the approximation.

## Key findings

- Existence and uniqueness of the approximation under general power cost functionals
- Bounds on the approximation quality
- Application to neural activity modeling

## Abstract

We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein-Uhlenbeck type process, by using direct methods of calculus of variations. We show that general power cost functionals satisfy the conditions for existence and uniqueness of the approximation. We provide some examples of general interest and we give bounds on the goodness of the corresponding approximations. Finally, we focus on a model of a neuron embedded in a simple network and we study the approximation of its activity, by exploiting the aforementioned results.