Forward stochastic integration for adapted processes w.r.t. Riemann-Liouville fractional Brownian motion (Full version)

TL;DR

This paper develops a new $L^2$-martingale representation for forward stochastic integrals driven by Riemann-Liouville fractional Brownian motion with Hurst parameter between 1/2 and 1, including an exact isometry formula.

Contribution

It introduces a novel time-dependent $L^2$-martingale representation and derives the exact $L^2$-isometry for forward stochastic integrals with fractional Brownian motion.

Findings

Established the $L^2$-martingale representation for the integrals.

Derived the exact $L^2$-isometry under new conditions.

Connected the representation with Nelson's stochastic derivative.

Abstract

This paper provides the time-dependent $L^2$-martingale representation of the forward stochastic integral where the driving noise is the Riemann-Liouville fractional Brownian motion with parameter $\frac{1}{2} < H < 1$ and the integrand is a square-integrable adapted process. As a by-product, we obtain the exact $L^2$-isometry of the forward stochastic integrals based on suitable conditions on time-dependent martingale representations of adapted integrands combined with the Nelson's stochastic derivative of the underlying Gaussian driving noise.