## Doctoral Dissertations

#### Title

Functions of finite {\it p\/}-variation and stochastic integration

January 1993

Mathematics

Ph.D.

#### Abstract

Functions of finite p-variation in one variable were first introduced by Wiener in 1924. Wiener showed that the Fourier series of such functions converges almost everywhere. In 1938 L.C Young developed an integration theory with respect functions of finite p-variation and showed that the Fourier series of such functions converges everywhere. We study the functions of finite p-variation in one and higher dimensions (several variables). We show that Fourier series of such functions converges everywhere. Furthermore if $p < 2$ then Fourier series of such functions converges absolutely provided they are Lipschits. This generalises results of Bernstein and Zygmund. We show that $\int\sb{\lbrack 0,1\rbrack\sp2}fdg$ exists if f is a function of finite p-variation and g is a function of finite q-variation provided ${1\over p}+{1\over q}>1.$^ We use the integration theory of functions of finite p- variation to show that if X and Y are two stochastic processes which are independent and have finite expectations then the stochastic integral $\int YdX$ is a well defined integrable random variable. ^

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