MATH3251/4091 Stochastic Processes III/IV
A stochastic process is a mathematical model for a system evolving randomly in time. For example, the size of a biological population or the price of a share may vary with time in an unpredictable manner. These and many other systems in the physical sciences, biology, economics, engineering and computer sciences may best be modelled in a non-deterministic manner.
More technically, a stochastic process is a collection of random quantities indexed by a time parameter. Typically these quantities are not independent, but have their dependency structure specified via the time parameter. Specific models to be covered include martingales, branching processes, Markov chains in continuous time, and Poisson processes.
Outline of Course
Aim: This module continues on from the treatment of probability in Probability II. It is designed to introduce mathematics students to the wide variety of models of systems in which sequences of events are governed by probabilistic laws. Students completing this course should be equipped to read for themselves much of the vast literature on applications to problems in mathematical finance, physics, engineering, chemistry, biology, medicine, psychology and many other fields.
- Probability revision: conditional expectation, sigma fields, generating functions.
- Branching processes and their applications; survival and extinction; multi-type branching processes.
- Coupling of stochastic processes and applications; mixing of Markov chains.
- Martingales and applications: discrete-time martingales and examples; stopping times, optional stopping theorem; martingale convergence theorem; applications.
- Poisson processes: Poisson process on the line, relation to exponential distribution, thinning and superposition; compound Poisson processes, Cramér's ruin problem, spatial Poisson processes.
- Continuous-time Markov chains: infinitesimal rates, exponential holding times, Kolmogorov equations; stationary distributions, resolvents and hitting times; birth and death processes, simple queueing models.
- Brownian motion as a Gaussian process, a Markov process, a diffusion, and a martingale; scaling limit of random walk; reflection principle; the Dirichlet problem.
- Topics chosen from: renewal theory, interacting particle systems, stochastic epidemics; further applications of martingale theory.