## Mean-square fractional calculus and some applications.

##### Abstract

The fractional calculus of deterministic functions is well known and
widely used. Mean-square calculus is a calculus that is suitable for use
when dealing with second-order stochastic processes. In this dissertation
we explore the idea of extending the fractional calculus of deterministic
functions to a mean-square setting. This exploration includes the development
of some of the theoretical aspects of mean-square fractional calculus
– such as definitions and properties – and the consideration of the application
of mean square fractional calculus to fractional random differential
and integral equations. The development of mean-square calculus follows
closely that of the calculus of deterministic functions making mean square
calculus more accessible to a large audience. Wherever possible, our development
of mean-square fractional calculus is done in a similar manner to
that of ordinary fractional calculus so as to make mean-square fractional
calculus more accessible to people with some exposure to ordinary fractional
calculus.