Date of Award


Degree Type


Degree Name

Doctor of Philosophy


The concept of Gaussian white noise has been very valuable in the application of stochastic differential equations to problems ranging from physics to finance. White noise has links with the diffusion equation, the solution of which gives the density of random walkers on a Euclidean space.;The use of white noise is not always appropriate in the study of random processes. In this thesis a new family of noise processes is defined. These "fractal walk" noise processes have connections with a "diffusion" equation in which the order {dollar}\gamma{dollar} of the time derivative takes on a continuous range of values between 0 and 2. The solution of this equation, in the case {dollar}0<\gamma<1,{dollar} describes the density of random walkers on some fractal space. This provides a spatial limit on randomness. When {dollar}\gamma{dollar} = 1 the white noise process is recovered. In the case {dollar}1<\gamma<2{dollar} the solution may be interpreted as a description of the density of a group of walkers whose behaviour is, to an extent, deterministic. This provides a deterministic limit on randomness.;This new family of fractal walk noise processes is then applied to the problem of determining the value of a financial instrument called a European call option. The current technique for valuing this instrument is based upon white noise and is called the Black-Scholes formula. It is shown that the extension of this formula to account for fractal walk noise processes produces results that are, in certain cases. qualitatively different from those produced by the Black-Scholes model.



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