Change-of-variable formula for the bi-dimensional fractional Brownian motion in Brownian time

TL;DR

This paper derives an Ito's type change-of-variable formula for a two-dimensional fractional Brownian motion in Brownian time, covering different Hurst parameter regimes, including a critical case involving additional stochastic elements.

Contribution

It provides the first Ito's formula for 2D fractional Brownian motion in Brownian time, especially addressing the critical case H=1/6 with a novel law-involving formula.

Findings

For H > 1/6, the formula resembles classical calculus.

At H=1/6, the formula involves third derivatives and an independent Brownian motion.

The case H<1/6 is also discussed, extending the theoretical framework.

Abstract

Let X^{1}, X^{2} be two independent (two-sided) fractional Brownian motions having the same Hurst parameter H in (0,1), and let Y be a standard (one-sided) Brownian motion independent of (X^{1},X^{2}). In dimension 2, fractional Brownian motion in Brownian motion time (of index H) is, by definition, the process Z_t:= (Z^1_t, Z^2_t)= (X^{1}_{Y_t},X^{2}_{Y_t}). The main result of the present paper is an Ito's type formula for f(Z_t), when f:\R^2\to\R is smooth and H in [ 1/6,1). When H>1/6, the change-of-variable formula we obtain is similar to that of the classical calculus. In the critical case H=1/6, our change-of-variable formula is in law and involves the third partial derivatives of f as well as an extra Brownian motion independent of (X^1,X^2,Y). We also discuss the case H<1/6.