Itô’s formula with respect to fractional Brownian motion and its application. (English) Zbl 0867.60029

Dai, W.; Heyde, C. C.

J. Appl. Math. Stochastic Anal. 9, No. 4, 439-448 (1996).

Summary: 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 \(1/2<H<1\). 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.

MSC:

60H05	Stochastic integrals
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

fractional Brownian motion; Itô’s formula; long range dependence; stochastic differential equations; Black-Scholes model