Stochastic differential equations with fractal noise

Abstract Stochastic differential equations in ℝ n with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W β 2 for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed which combines the stochastic Itô calculus with fractional calculus via norm estimates of associated integral operators in W α 2 for 0 < α < 1. Linear equations are considered as a special case. This approach leads to fast computer algorithms basing on Picard's iteration method. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)