Fractional Hida Malliavin Derivatives and Series Representations of Fractional Conditional Expectations

TL;DR

This paper develops series representations for fractional conditional expectations of functionals of fractional Brownian motion, enabling practical computation through backward Taylor and exponential operator-based series expansions.

Contribution

It introduces novel series representations for fractional conditional expectations, including backward Taylor and exponential operator methods, for both discrete and continuous fractional filtrations.

Findings

Series representations converge in L^2 space.

Backward Taylor series for discrete fractional Brownian motion trajectories.

Exponential operator series for continuous fractional filtrations.

Abstract

We represent fractional conditional expectations of a functional of fractional Brownian motion as a convergent series in L^2 space. When the target random variable is some function of a discrete trajectory of fractional Brownian motion, we obtain a backward Taylor series representation; when the target functional is generated by a continuous fractional filtration, the series representation is obtained by applying a "frozen path" operator and an exponential operator to the functional. Three examples are provided to show that our representation gives useful series expansions of ordinary expectations of target random variables.