Stochastic calculus methods in continuous-time information theory

Abstract

Continuous time information theory has been studied extensively after the introduction of white Gaussian channel in Shannon [35]. This channel is important due to its realistic and mathematically neat modelling nature. In this thesis, we follow the formulation of a continuous-time Gaussian channel as in [17], in which the channel is modeled by a stochastic functional differential equation (SFDE) and study it under a stochastic analytic prospective. A detailed discussion of the model and the underlying theory is presented in Chapter 2. Computing the mutual information and channel capacity is one of the major research problems in information theory. Related works based on continuous-time Gaussian channels can be found in [10, 14, 18, 24, 29]. Although an explicit formula of mutual information between the message and channel output of a continuous-time Gaussian channel is well-known, it is not feasible in actual calculations. As an alternative, an approximation scheme using the Picard's iteration of the underlying SFDE of the channel is proposed in Chapter 4. Several possible future directions are discussed in Chapter 5. Preliminaries on It^o's stochastic calculus and information theory are given in Chapter 1. In Chapter 3, the Girsanov's theorem and several variants which are useful in the sequel are discussed.