Fractional Wiener Chaos

TL;DR

This paper introduces a fractional Wiener chaos expansion using a novel power normalised parabolic cylinder function, extending Hermite polynomials into the fractional calculus domain with preserved martingale properties.

Contribution

It generalizes Wiener chaos expansion by developing a fractional version based on a new class of functions that extend Hermite polynomials, maintaining key properties.

Findings

The new functions act as fractional Hermite polynomials.

The fractional Wiener chaos retains martingale properties.

The approach bridges fractional calculus and stochastic analysis.

Abstract

The area of fractional calculus has made its way into various pure and applied scientific fields, as evidenced by its integration into numerous disciplines. An increasing number of researchers are exploring various approaches to incorporating fractional calculus into stochastic analysis. In this paper, we generalise the Wiener chaos expansion by constructing a fractional Wiener chaos expansion based on the parabolic cylinder function with an exponential factor. In the process, we demonstrate that this parabolic cylinder function with an exponential factor, which we call "a power normalised parabolic cylinder function" acts as an extension of a Hermite polynomial and retains the same martingale properties inherent in the Hermite polynomial, as well as other basic properties of Hermite polynomials. Hence, it is accurate to state that power normalised cylindrical functions can be viewed as fractional versions of Hermite polynomials.