Mean-square fractional calculus and some applications.
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.