Course syllabus for Basic stochastic processes

Perform basic calculations of continuous time and discrete time Fourier transforms and inverse transforms as well as basic understanding of their usage in probability theory (that is, characteristic functions) as well as in the context of stationary random processes and linear time invariant systems (that is, spectral analysis).

Describe how continuous time and discrete time Markov chains (including queueing systems) work from a principal theoretical point of view and by means of implementation of corresponding computer code (or so called pseudo code) as well as performing computational examples on these objects. 

Describe the importance of dependence and independence between different stochastic process values (random variables) from a principal theoretical point of view as well as performing corresponding computational examples. 

Describe the basic defining properties of stationary processes, wide (weak) sense stationary processes, Gaussian processes and martingales from a principal theoretical point of view as well as performing corresponding computational examples.

Introduction to Fourier transforms (characteristic functions), convolutions and Dirac’s delta-function together with a review of some important concepts from multivariate probability theory. Definition of discrete time and continuous time stochastic processes together with their finite dimensional distributions, mean functions, correlation functions and covariance functions. Processes with independent stationary increments (Levy processes) and Gaussian processes (normal processes). A quite substantial treatment of discrete time and continuous time Markov chains including applications to queueing systems (that adds up to upwards half of the total course material). Discrete time and continuous time martingales. Continuity and differentiability of stochastic processes together with integration and summation of them. Spectral densities together with white noise in continuous and discrete time. Stochastic processes as inputs to and outputs from linear time invariant systems. Computer implementation of the majority of the above mentioned classes of stochastic processes.

Lectures and tutorials.

Hwei Hsu: Probability, Random Variables, and Random Processes, 2nd Edition. Schaum's Outlines, McGraw-Hill 2010.

Geoffrey Grimmett och David Stirzaker: Probability and Random Processes, 3rd Edition. Oxford 2001.

Written exam.