On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion. (English) Zbl 1075.60068

Jumarie, Guy

Appl. Math. Lett. 18, No. 7, 817-826 (2005).

The author considers stochastic differential equations of the form \(dx=\sigma x \,db(t,a),\) \(t\geq 0,\) \(x(0)=x_0,\) \(\sigma \in R\), where \(b(t,a) := 1/(\Gamma(a+1/2))\int_0^t (t-\tau)^{a-1/2} w(\tau)\,d\tau\), and \(a\) is between \(0\) and \(1\). Here \(w\) is a normalized Gaussian white noise and \(b(t,a)\) is a normalized fractional Brownian motion of order \(a\). First some basic results on fractional derivatives, integrals and a Taylor expansion of fractional order are given. Then the solutions of some deterministic fractional differential equations are discussed and, finally, an application to geometric fractional Brownian motion is given. The solutions obtained involve the Mittag-Leffler function. Somewhat irritating is a misprint, namely \(\triangleleft\) (or \(\triangleright\)), which from the context may mean \(<\) or \(\leq\) (or \(>\), \(\geq\)) in different places, sometimes \(\leq\) appears, too.

Reviewer: Evelyn Buckwar (Berlin)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05	Stochastic integrals
60G18	Self-similar stochastic processes

Keywords:

fractional derivative; fractional Taylor series; Mittag-Leffler function

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