COMP0045 Probability Theory and Stochastic Processes
This database contains the 201819 versions of syllabuses. These are still being finalised and changes may occur before the start of the session.
Syllabuses from the 201718 session are available here.
Academic session 
201819 
Module 
Probability Theory and Stochastic Processes 
Code 
COMP0045 
Module delivery 
1819/A7P/T1/COMP0045 Postgraduate 
Related deliveries 
None 
Prior deliveries  
Level 
Postgraduate 
FHEQ Level 
L7 
FHEQ credits 
15 
Term/s 
Term 1 
Module leader 
Germano, Guido 
Contributors 
Germano, Guido 
Module administrator 
Nolan, Martin 
Aims
A systematic introduction to probability theory and stochastic processes as well as some of their applications, with workedout exercises and without stressing too much the measuretheoretical 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 currentday 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 belowlisted 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: 

Prerequisites: 
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. 
Content
 Elementary probability
 Probability space, Kolmogorov’s axioms
 Joint and conditional probability, independent events
 Total probability theorem, Bayes’ theorem
 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 cumulantgenerating functions
 Central limit theorem, Lévy stable distributions
 Gaussian, Poisson and exponential distributions
 Random functions or stochastic processes
 Definitions, auto and crosscovariance/correlation, stationarity, WienerKhinchin theorem
 Classification with respect to memory; ergodicity, martingales and semimartingales
 Markov and semiMarkov processes, ChapmanKolmogorov equation
 Forward ad backward timeevolution equations
 Jump processes: the master equation
 Diffusion processes: the FokkerPlanck equation
 Deterministic processes: the Liouville equation
 Random telegraph signal, random walk, hidden Markov model
 Poisson, compound Poisson and renewal processes, continuoustime random walk
 Wiener and OrnsteinUhlenbeck processes
 Stochastic integral: Ito and Stratonovich, Ito’s formula
 Langevin equation, stochastic differential equation, FeynmanKac theorem
 Geometric Brownian motion, BlackScholes equation
 CoxIngersollRoss, Rayleigh and Bessel processes
An indicative reading list is available via http://readinglists.ucl.ac.uk/departments/comps_eng.html.
Delivery
The module is delivered through a combination of lectures (3 hours per week for 10 weeks), homework assignments, and selfdirected learning.
Assessment
This module delivery is assessed as below:
# 
Title 
Weight (%) 
Notes 
1 
Written examination (2hrs 30mins) 
60 

2 
Inclass test (2hrs 30mins) 
40 

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