Ordinary Differential Equations with Fractalnoisef

The diierential equation dx(t) = a(x(t); t) dZ (t) + b(x(t); t) dt for fractal-type functions Z (t) is determined via fractional calculus. Under appropriate conditions we prove existence and uniqueness of a local solution by means of its representation x(t) = h(y(t) +Z(t); t) for certain C 1-functions h and y. The method is also applied to It^ o stochastic diierential equations and leads to a general pathwise representation. Finally we discuss fractal sample path properties of the solutions. 1. Introduction In this paper we study the diierential equation on the real line dx(t) = a(x(t); t) dZ(t) + b(x(t); t) dt (1.1) x(0) = x 0 under the following conditions: (C1) Z is HH older continuous of order > 1 2 , Z(0) = 0