COMP0045 Probability Theory and Stochastic Processes

This database contains the 2018-19 versions of syllabuses. These are still being finalised and changes may occur before the start of the session.

Syllabuses from the 2017-18 session are available here.

Academic session



Probability Theory and Stochastic Processes



Module delivery

1819/A7P/T1/COMP0045 Postgraduate

Related deliveries


Prior deliveries




FHEQ Level


FHEQ credits



Term 1

Module leader

Germano, Guido


Germano, Guido

Module administrator

Nolan, Martin


A systematic introduction to probability theory and stochastic processes as well as some of their applications, with worked-out exercises and without stressing too much the measure-theoretical aspects and other mathematical formalisms. The main target are students with an undergraduate degree in physics, engineering, computer science and the like, who have a good basis in calculus and have already come into contact with aspects of probability and statistics for ad hoc applications like laboratory data treatment, transport equations, and quantum mechanics, but have not attended yet a dedicated course on this subject. The course material unfolds with references to its historical development and early applications in gambling, physics and engineering, ending with current-day applications in finance.

Learning outcomes

Familiarity with probability theory, stochastic processes in discrete and continuous time, stochastic calculus, and basic applications in physics, engineering and finance.

Availability and prerequisites

This module delivery is available for selection on the below-listed programmes. The relevant programme structure will specify whether the module is core, optional, or elective.

In order to be eligible to select this module as optional or elective, where available, students must meet all prerequisite conditions to the satisfaction of the module leader. Places for students taking the module as optional or elective are limited and will be allocated according to the department’s module selection policy.

Programmes on which available:

  • MRes Web Science and Big Data Analytics
  • MSc Computational Finance
  • MSc Financial Risk Management
  • MSc Web Science and Big Data Analytics
  • MSc Scientific Computing


In order to be eligible to select this module, students must have an understanding of Calculus and Linear Algebra; please see the maths part of Introduction to Mathematics and Programming for Finance.


  1. Elementary probability
    • Probability space, Kolmogorov’s axioms
    • Joint and conditional probability, independent events
    • Total probability theorem, Bayes’ theorem
  2. Random variables
    • Random variables; probability distribution and density functions
    • Multivariate, marginal and conditional distribution and density functions
    • Transformation of random variables, sum, product
    • Mean, variance, covariance, correlation, moments, median, entropy
    • Chebyshev inequality, law of large numbers
    • Characteristic function, moment- and cumulant-generating functions
    • Central limit theorem, Lévy stable distributions
    • Gaussian, Poisson and exponential distributions
  3. Random functions or stochastic processes
    • Definitions, auto- and cross-covariance/correlation, stationarity, Wiener-Khinchin theorem
    • Classification with respect to memory; ergodicity, martingales and semimartingales
    • Markov and semi-Markov processes, Chapman-Kolmogorov equation
    • Forward ad backward time-evolution equations
    • Jump processes: the master equation
    • Diffusion processes: the Fokker-Planck equation
    • Deterministic processes: the Liouville equation
    • Random telegraph signal, random walk, hidden Markov model
    • Poisson, compound Poisson and renewal processes, continuous-time random walk
    • Wiener and Ornstein-Uhlenbeck processes
    • Stochastic integral: Ito and Stratonovich, Ito’s formula
    • Langevin equation, stochastic differential equation, Feynman-Kac theorem
    • Geometric Brownian motion, Black-Scholes equation
    • Cox-Ingersoll-Ross, Rayleigh and Bessel processes

An indicative reading list is available via


The module is delivered through a combination of lectures (3 hours per week for 10 weeks), homework assignments, and self-directed learning.


This module delivery is assessed as below:



Weight (%)



Written examination (2hrs 30mins)




In-class test (2hrs 30mins)



In order to pass this module delivery, students must achieve an overall weighted module mark of 50%.