Stochastic Processes

Aim

In broad terms, the course treats statistical models and methods for randomly varying quantities which are also functions of time. These are fundamental for the advanced study of telecommunications theory, signal theory, control theory, robotics, and many important phenomena in biology, physics, computer networks, and economy. After a completed course the student is expected to be able to:

• describe the basic concepts and theorems in the theory of stochastic processes, e.g., expectation and autocovariance function and spectral density.
• describe important classes of stochastic processes, e.g., the Wiener process, martingales, wide sense stationary processes, and Markov chains, and their special properties.
• make use of stochastic processes to construct relevant models for randomly varying quantities which are functions of time.
• carry out important computations for stochastic processes, such as linear time invariant filtration, and prediction of the values of a process at unobserved times.
• understand and assess models based on stochastic processes and analyses of such models occurring in other undergraduate courses, or the media.

Content

Multivariate distributions, in particular the multivariate normal distribution. Conditioning and conditional expectation. The moment generating function. Stochastic processes: basic properties and examples. Expectation function, autocovariance function, cross covariance function. The Poisson process and the Wiener process. Martingales in discrete time. Stationary and wide sense stationary processes. Gaussian processes. Mean square convergence and the mean square integral. Linear time invariant filtering. Spectral densities. ARMA processes. Prediction. Markov chains in discrete and continuous time.