Ordinary differential equations with fractal noise

Source abstract

The differential equation $$d x ( t ) = a ( x ( t ) , t ) d Z ( t ) + b ( x ( t ) , t ) d t dx(t) \, = \, a(x(t),t) \,dZ(t) \:+\: b(x(t),t) \,dt$$ for fractal-type functions Z ( t ) 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 ) x(t)\, =\, h(y(t)+Z(t),t) for certain C 1 C^1 -functions h h and y y . The method is also applied to Itô stochastic differential equations and leads to a general pathwise representation. Finally we discuss fractal sample path properties of the solutions.