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

60H10Stochastic ordinary differential equations
60H05Stochastic integrals
60G18Self-similar processes
Full Text: DOI
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