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
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 is not a
semimartingale. Consequently, the standard Itô calculus is not available for
stochastic integrals with respect to FBM as an integrator if . 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.