Adapted integral representations of random variables

TL;DR

This paper explores integral representations of random variables with respect to H"older continuous processes, including fractional and mixed fractional Brownian motion, demonstrating conditions for proper and improper integral representations and their distributional properties.

Contribution

It introduces new integral representation techniques for random variables relative to H"older processes, including fractional Brownian motion, and characterizes when proper or improper integrals can be used.

Findings

Any random variable can be represented as an improper integral with respect to these processes.

Proper integral representations are possible for adapted H"older continuous processes.

In the case of mixed fractional Brownian motion, all adapted random variables admit proper integral representations.

Abstract

We study integral representations of random variables with respect to general H\"older continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that arbitrary random variable can be represented as an improper integral, and that the stochastic integral can have any distribution. If in addition the random variable is a final value of an adapted H\"older continuous process, then it can be represented as a proper integral. It is also shown that in the particular case of mixed fractional Brownian motion, any adapted random variable can be represented as a proper integral.