Journal of Applied Mathematics and Stochastic Analysis
Volume 9 (1996), Issue 4, Pages 439-448

Itô's formula with respect to fractional Brownian motion and its application

W. Dai1 and C. C. Heyde1,2

1Australian National University, School of Mathematical Sciences, ACT, Canberra 0200, Australia
2Columbia University, 2990 Broadway, Mail Code 4403, New York 10027, NY, USA

Received 1 July 1996; Revised 1 October 1996

Copyright © 1996 W. Dai and C. C. Heyde. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Fractional Brownian motion (FBM) with Hurst index 1/2<H<1 is not a semimartingale. Consequently, the standard Itô calculus is not available for stochastic integrals with respect to FBM as an integrator if 1/2<H<1. In this paper we derive a version of Itô's formula for fractional Brownian motion. Then, as an application, we propose and study a fractional Brownian Scholes stochastic model which includes the standard Black-Scholes model as a special case and is able to account for long range dependence in modeling the price of a risky asset. This article is dedicated to the memory of Roland L. Dobrushin.