Itô's formula with respect to fractional Brownian motion and its application. (1996)

• Record Type: Journal Article
• Authors: Dai, W.; Heyde, C. C.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 9:Number 4(1996)
• Page Start: 439
• Page End: 448
• Publication Date: 1996
• Subjects: fractional Brownian motion -- Itô's formula -- long range dependence -- stochastic differential equations -- Black-Scholes model; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S104895339600038X
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : 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.