2013/12/03 by Ivan Nourdin, Nourdin, Ivan, Raghid Zeineddine +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1312.0818
openalex publication_date 2013/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \cite13, 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(Zt), 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.