Functional It\^o formula for fractional Brownian motion

TL;DR

This paper develops a functional Itô calculus for fractional Brownian motion with Hurst parameter greater than 1/2, extending classical formulas to non-semimartingale cases and applying it to fractional BSDEs and path-dependent PDEs.

Contribution

It introduces a new functional Itô formula for fractional Brownian motion, extending existing calculus to non-semimartingale processes and linking fractional BSDEs with path-dependent PDEs.

Findings

Established Stratonovich and Wick-Itô integrals for fractional Brownian motion.

Extended functional Itô formulas to non-semimartingale cases.

Connected fractional BSDEs with path-dependent PDEs.

Abstract

We develop the functional It\^o/path-dependent calculus with respect to fractional Brownian motion with Hurst parameter $H> \frac{1}{2}$. Firstly, two types of integrals are studied. The first type is Stratonovich integral, and the second type is Wick-It\^o integral. Then we establish the functional It\^o formulas for fractional Brownian motion, which extend the functional It\^o formulas in Dupire (2009) and Cont-Fourni\'e (2013) to the case of non-semimartingale. Finally, as an application, we deal with a class of fractional backward stochastic differential equations (BSDEs). A relation between fractional BSDEs and path-dependent partial differential equations (PDEs) is established.