An Itô-type formula for the fractional Brownian motion in Brownian time

Abstract

Let X be a (two-sided) fractional Brownian motion of Hurst parameter H ∈ (0, 1) and let Y be a standard Brownian motion independent of X. Fractional Brownian motion in Brownian motion time (of index H), recently studied in [17], is by definition the process Z = X • Y . It is a continuous, non-Gaussian process with stationary increments, which is selfsimilar of index H/2. The main result of the present paper is an Itô's type formula for f (Z t ), when f : R → R is smooth and H ∈ [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 derivative of f as well as an extra Brownian motion independent of the pair (X, Y ). We also discuss briefly the case H < 1/6.

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

“…The present work may be seen a natural follow-up of [12], in which we proved a changeof-variable for fBmBt in dimension one, that is, for Z 1 . Before stating the results we have obtained, let us start with some historical comments and relationships with the existing literature.…”

“…For simplicity of the exposition and because the computations are rather involved, we will stick on dimension 2, which is representative of the difficulty. In dimension 1, the mathematical definition of fBmBt (together with its terminology) was introduced in our previous paper [12]. Let us give an analogue definition in dimension 2.…”

“…A natural question was then to extend (1.3) for other values of H. We did it in the joint paper [12] with Nourdin, by proving the following theorem.…”