Stochastic Processes and Risk

Overview

Logic and Boolean algebra, counting and combinatorics, set algebra, inclustion-exclusion theorem, mutually exclusive events, De Morgan Laws.

Axioms of probability, events and probability spaces, sigma-field, random variables, conditional probability, and expectation, Bayes’ theorem, discrete and continuous random variables, moments and moment generating function. Laws of large numbers and central limit theorem.

Pairs of random variables, marginal probabilities, Cauchy Schwartz Inequality in statistics, correlation and covariance.

Discrete time Markov chains, Chapman Kolmogorov relation, limiting behaviour, transient, recurrent states and periodic states, limiting stationary distribution, hitting times and hitting probabilities.

Continuous time Markov chains, Kolmogorov forward equations, stationary distribution for continuous time Markov chains, Poisson process, MM1 Queue, inhomogeneous Poisson process and compound Poisson process.

Learning Objectives

By the end of this module students will be able to:

- Calculate expectations and variances directly, using the moment generating function and by using the conditional expectation theorem. Students should also be able to explain what predictions can be made given the expectation and/or the variance.

- Recognize which type of random variable is appropriate for modeling a given phenomenon, to identify the assumptions that they have made in constructing this model and to critically assess its validity.


- Explain what it means when we state that a time dependent process has independent and stationary increments and how this differs from a Markov process. By using their understanding of this distinction students should be able to construct probabilistic models for time dependent phenomena, explain the assumptions that they have made in constructing these models and critically assess their validity.

Skills

Write computer programs that generate random variables as well as computer programs for evaluating sample means, histograms and confidence limits.

Discuss the results obtained by running the computer programs described in the previous point.

Assessment

None.