Prediction Law of fractional Brownian Motion

TL;DR

This paper derives the conditional future law of fractional Brownian motion given its past, demonstrating its continuity and analyzing its path properties and covariance behavior.

Contribution

It provides a detailed analysis of the conditional law of fractional Brownian motion, including continuity and asymptotic covariance, which was previously not well-understood.

Findings

Conditional law is continuous with respect to the past path.

Characterization of the path properties of the conditional process.

Analysis of the asymptotic behavior of the conditional covariance.

Abstract

We calculate the regular conditional future law of the fractional Brownian motion with index $H\in(0,1)$ conditioned on its past. We show that the conditional law is continuous with respect to the conditioning path. We investigate the path properties of the conditional process and the asymptotic behavior of the conditional covariance.