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On the representation of fractional Brownian motion as an integral with respect to $$(dt)^a$$. (English) Zbl 1082.60029
Summary: Maruyama introduced the notation $$db(t)=w(t)(dt)^{1/2}$$ where $$w(t)$$ is a zero-mean Gaussian white noise, in order to represent the Brownian motion $$b(t)$$. Here, we examine in which way this notation can be extended to Brownian motion of fractional order $$a$$ (different from $$1/2)$$ defined as the Riemann-Liouville derivative of the Gaussian white noise. The rationale is mainly based upon the Taylor’s series of fractional order, and two cases have to be considered: processes with short-range dependence, that is to say with $$0 \triangleleft a\leq 1/2$$, and processes with long-range dependence, with $$1/2\triangleleft a\leq 1$$.

MSC:
 60G15 Gaussian processes
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References:
 [1] Decreusefond, L.; Ustunel, A.S., Stochastic analysis of the fractional Brownian motion, Potential anal., 10, 177-214, (1999) · Zbl 0924.60034 [2] Jumarie, G., Stochastic differential equations with fractional Brownian motion input, Int. J. syst. sci., 6, 1113-1132, (1993) · Zbl 0771.60043 [3] 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 [4] Kober, H., On fractional integrals and derivatives, Quart. J. math. Oxford, 11, 193-215, (1940) · Zbl 0025.18502 [5] Letnivov, A.V., Theory of differentiation of fractional order, Math. sb., 3, 1-7, (1868) [6] Liouville, J., J. ecole polytechnique, 13, 71, (1832) [7] Mandelbrot, B.B.; van Ness, J.W., Fractional Brownian motions, fractional noises and applications, SIAM rev., 10, 422-437, (1968) · Zbl 0179.47801 [8] Mandelbrot, B.B.; Cioczek-Georges, R., A class of micropulses and antipersistent fractional Brownian motions, Stochastic process. appl., 60, 1-18, (1995) · Zbl 0846.60055 [9] Mandelbrot, B.B.; Cioczek-Georges, R., Alternative micropulses and fractional Brownian motion, Stochastic process. appl., 64, 143-152, (1996) · Zbl 0879.60076
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