# Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations

## TL;DR

This paper establishes the theoretical identifiability of drift and diffusion coefficients in one-dimensional stochastic differential equations using small-interval observations, employing PDE techniques and the Feynman-Kac theorem.

## Contribution

It provides the first rigorous proof that both coefficients can be uniquely determined from specific observational data in a single SDE.

## Key findings

- Drift coefficient is uniquely determined when diffusion is known.
- Diffusion coefficient is uniquely determined when drift is known.
- Both coefficients are simultaneously identifiable from expectation and variance observations.

## Abstract

In this work, we consider a one-dimensional It{\^o} diffusion process X t with possibly nonlinear drift and diffusion coefficients. We show that, when the diffusion coefficient is known, the drift coefficient is uniquely determined by an observation of the expectation of the process during a small time interval, and starting from values X 0 in a given subset of R. With the same type of observation, and given the drift coefficient, we also show that the diffusion coefficient is uniquely determined. When both coefficients are unknown, we show that they are simultaneously uniquely determined by the observation of the expectation and variance of the process, during a small time interval, and starting again from values X 0 in a given subset of R. To derive these results, we apply the Feynman-Kac theorem which leads to a linear parabolic equation with unknown coefficients in front of the first and second order terms. We then solve the corresponding inverse problem with PDE technics which are mainly based on the strong parabolic maximum principle.