Stochastic Processes module (MA51005)

Probability theory deals with events for which we can recover robust probabilities, but not certainties, of occurrence. You will learn how measure theory can be used to perform a calculus of events, thus giving rise to the mathematical notion of probability measure. You will study the laws of large numbers and the central limit theorem, the principal ways we extract certainties in a probabilistic context. Moving on to the evolution of probabilistic systems in time, you will study the most straightforward such systems, namely Markov chains. Due to their versatility and simplicity, Markov chains have many applications – from search algorithms to textual analysis. You will engage with some of these applications. You will learn about continuous functions that evolve randomly in time – mainly the 2nd-order theory of Gaussian processes and specifically about Brownian motion.  

Topics include 

• Probability space and random variables - representations of probability measures, moments, the law of large numbers, and the central limit theorem; operations and transformations of random variables. 
• Markov chains - the Markov property, ergodic theorem and invariant measures, first passage time, discrete random walks. 
• Brownian motion - Brownian motion as the continuous limit of a random walk, Gaussian processes, mean-square calculus of random processes, introduction to stochastic differential equations.