A new approach to fractional Brownian motion of order \(n\) via random walk in the complex plane
The fractional Brownian motion \(b(.,n)\) of order \(n\) is defined as the limit of a random walk \(z_j: z_{j+1}= z_j+ R_jW_j\), where the complex random variable \(R_j\) takes its values in the set of the \(n\)th roots of the unit, \(\{\omega_k, k= 0,\dots, n-1\}\), and given \(R_j= \omega_k\), \(W_j\) is a Bernoulli variable in the direction \(\omega_k\), with values \(\pm|\Delta z|\), \(z\in\mathbb{C}\). An Itô stochastic integral is defined with respect to this process. A weak central limit theorem is proved for independent identically distributed random variables with \(n-1\) moments equal to \(0\) and \(n\)th moment denoted as \(\sigma^n\): their \(k\)-sum divided by \(\sigma^n\sqrt k\) tends to distribution of \(b(1,n)\). Finally, a Feynman-Kac formula and a Dynkin formula are done; entropy and Hausdorff dimension of the fractional Brownian motion \(b(.,n)\) are studied.
scientific article; zbMATH DE number 1371654