Exact corrections for finite-time drift and diffusion coefficients

TL;DR

This paper derives exact finite-time corrections for drift and diffusion coefficients in stochastic models, enabling more accurate parameter estimation from discretely sampled data.

Contribution

It provides explicit formulas for correcting finite-time estimates of drift and diffusion, including higher-order moments, based on Itô-Taylor expansions.

Findings

Exact correction formulas for finite-time drift and diffusion coefficients.

Higher-order finite-time expressions for third and fourth moments.

Numerical validation with artificial time-series data.

Abstract

Real data are constrained to finite sampling rates, which calls for a suitable mathematical description of the corrections to the finite-time estimations of the dynamic equations. Often in the literature, lower order discrete time approximations of the modeling diffusion processes are considered. On the other hand, there is a lack of simple estimating procedures based on higher order approximations. For standard diffusion models, that include additive and multiplicative noise components, we obtain the exact corrections to the empirical finite-time drift and diffusion coefficients, based on It\^o-Taylor expansions. These results allow to reconstruct the real hidden coefficients from the empirical estimates. We also derive higher-order finite-time expressions for the third and fourth conditional moments, that furnish extra theoretical checks for that class of diffusive models. The theoretical predictions are compared with the numerical outcomes of some representative artificial time-series.