On MAP estimates and source conditions for drift identification in SDEs

TL;DR

This paper investigates the inverse problem of estimating the drift in stochastic differential equations using MAP estimates, analyzing differentiability, source conditions, and convergence rates with numerical validation.

Contribution

It derives properties of the MAP estimate for drift identification in SDEs, including differentiability and tangential cone conditions, and reviews related theoretical results.

Findings

Numerical simulations in 1D support the convergence rates of MAP estimates.

Differentiability and source conditions are established for the forward operator.

Theoretical review suggests convergence rates under stronger tangential cone conditions.

Abstract

We consider the inverse problem of identifying the drift in an SDE from $n$ observations of its solution at $M+1$ distinct time points. We derive a corresponding MAP estimate, we prove differentiability properties as well as a so-called tangential cone condition for the forward operator, and we review the existing theory for related problems, which under a slightly stronger tangential cone condition would additionally yield convergence rates for the MAP estimate as $n\to\infty$. Numerical simulations in 1D indicate that such convergence rates indeed hold true.