The calculus of differentials for the weak Stratonovich integral

TL;DR

This paper derives an explicit expression and an Itô-type formula for the weak Stratonovich integral involving fractional Brownian motion with Hurst parameter 1/6, revealing new correction terms related to signed cubic variation.

Contribution

It introduces a novel explicit formula for the weak Stratonovich integral and an associated Itô-type formula with correction terms involving signed cubic variation.

Findings

Explicit expression for the weak Stratonovich integral with fractional Brownian motion

An Itô-type formula including a correction term related to signed cubic variation

A new formula for differentials involving stochastic correction terms

Abstract

The weak Stratonovich integral is defined as the limit, in law, of Stratonovich-type symmetric Riemann sums. We derive an explicit expression for the weak Stratonovich integral of $f(B)$ with respect to $g(B)$, where $B$ is a fractional Brownian motion with Hurst parameter 1/6, and $f$ and $g$ are smooth functions. We use this expression to derive an It\^o-type formula for this integral. As in the case where $g$ is the identity, the It\^o-type formula has a correction term which is a classical It\^o integral, and which is related to the so-called signed cubic variation of $g(B)$. Finally, we derive a surprising formula for calculating with differentials. We show that if $dM = X dN$, then $Z dM$ can be written as $ZX dN$ minus a stochastic correction term which is again related to the signed cubic variation.