A New Approach to Fractional Brownian Motion of Order n Via Random Walk in the Complex Plane

✍ Scribed by Guy Jumarie

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 370 KB
• Volume: 10
• Category: Article
• ISSN: 0960-0779
• DOI: 10.1016/s0960-0779(98)00103-9

✦ Synopsis

By using a very simple model of random walk de_ned on the roots of the unity in the complex plane\ one can obtain the model of fractional brownian motion of order n which has been previously introduced in the form of rotating Gaussian white noise[ This de_nition of fractional Brownian motion of order n as the limit of complex random walk\ provides new insights in its genuine practical meaning\ and in the derivation of most of the related theoretical results[ Ito ¼ |s stochastic calculus can be extended in a straightforward manner to the path integral so generated in the complex plane[ The corresponding prob! ability distribution is stable in Levy|s sense\ a Lindeberg|s like central limit theorem is stated\ together with a FeymanÐKac|s formula and a Dinkin|s formula[ Then one exhibits the relation between the Hausdor}|s dimension and the pattern entropy of the process[ The probabilistic approach here is di}erent from Hochberg|s and Mandelbrot|s[ Like Sainty|s\ it uses the complex roots of the unity\ but it is much more straightforward and simple\ and it is the only one which provides results which are fully consistent with the so!called KramersÐMoyal expansion[ Þ 0888 Elsevier Science Ltd[ All rights reserved[