Fractional Brownian motion and asymptotic Bayesian estimation

TL;DR

This paper investigates the asymptotic behavior of Bayesian estimators for the Hurst parameter in fractional Brownian motion, demonstrating that the posterior distribution uniquely identifies the parameter in the limit.

Contribution

It provides a rigorous proof of the limiting behavior of the posterior distribution for the Hurst parameter without relying on common simplifying assumptions.

Findings

Posterior distribution converges to a point mass at the true parameter

Strong laws of large numbers are established for the problem

The approach avoids additional technical assumptions

Abstract

In this paper, we study the recovery of the Hurst parameter from a given discrete sample of fractional Brownian motion with statistical inverse theory. In particular, we show that in the limit the posteriori distribution of the parameter given the sample determines the parameter uniquely. In order to obtain this result, we first prove various strong laws of large numbers related to the problem at hand and then employ these limit theorems to verify directly the limiting behaviour of posteriori distributions without making additional technical or simplifying assumptions that are commonly used.