Construction of a complex-valued fractional Brownian motion of order N

Abstract

In this paper, a Brownian motion of order n (n≳2) is defined by a probabilistic approach different from Hochberg’s and Mandelbrot’s. This process is constructed from sums of independent R+1/n-valued random variables (rv) (where R+1/n={z∈C; zn∈R+}). Many properties of the real standard Brownian motion are generalized at order n, but in the case n≳2, it is interesting to describe the Brownian motion of order n on the σ algebra ⊗[B(R+1/n)] R+ [where B(R+1/n) is the σ algebra generated by sets of type A(0,h)={z∈C; zn∈[0;hn[},(hεR+*)]. This σ algebra is totally different from ⊗[B(R)] R+. Thus this study shows the fractal nature of the Brownian motion of order n, and given invariance scale (self-similarity) properties. Then, a stochastic integral and an Itô–Taylor lemma at order n are given to allow the representation of the solution of the heat equation of order n by a probabilistic average. All these results can be obtained via nonstandard analysis methods (infinitesimal time discretization). Finally, one remarks that this process has a.s (almost surely) continuous sample paths, infinite variance, and independent increments, whereas the fractional Brownian motion of Mandelbrot has a.s continuous sample paths, finite variance, and interdependent increments.