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.


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


[1] Decreusefond, L.; Ustunel, A. S., Stochastic analysis of the fractional Brownian motion, Potential Anal., 10, 177-214 (1999) · Zbl 0924.60034
[2] Duncan, T. E.; Hu, Y.; Pasik-Duncan, B., Stochastic calculus for fractional Brownian motion, I. Theory, SIAM J. Control Optim., 38, 582-612 (2000) · Zbl 0947.60061
[3] Hu, Y.; Øksendal, B., Fractional white noise calculus and applications to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6, 1-32 (2003) · Zbl 1045.60072
[4] Jumarie, G., Stochastic differential equations with fractional Brownian motion input, Int. J. Syst. Sci., 6, 1113-1132 (1993) · Zbl 0771.60043
[5] Jumarie, G., Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker-Planck equations, Chaos Solitons Fractals, 4, 907-925 (2004) · Zbl 1068.60053
[7] Kober, H., On fractional integrals and derivatives, Quart. J. Math. Oxford, 11, 193-215 (1940) · Zbl 0025.18502
[8] Letnivov, A. V., Theory of differentiation of fractional order, Math. Sb., 3, 1-7 (1868)
[9] Liouville, J., Sur le calcul des differentielles à indices quelconques, J. Ecole Polytechnique, 13, 71 (1832), (in French)
[10] Mandelbrot, B. B.; van Ness, J. W., Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422-437 (1968) · Zbl 0179.47801
[11] Mandelbrot, B. B.; Cioczek-Georges, R., A class of micropulses and antipersistent fractional Brownian motions, Stochastic Proces. Appl., 60, 1-18 (1995) · Zbl 0846.60055
[12] Mandelbrot, B. B.; Cioczek-Georges, R., Alternative micropulses and fractional Brownian motion, Stochastic Process. Appl., 64, 143-152 (1996) · Zbl 0879.60076
[13] Osler, T. J., Taylor’s series generalized for fractional derivatives and applications, SIAM. J. Math. Anal., 2, 1, 37-47 (1971) · Zbl 0215.12101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.