¹û¶³Ó°Ôº

Aims:

A systematic introduction to probability theory and stochastic processes as well as some of their applications, with many worked-out examples and exercises and only the indispensable mathematical formalism. The module starts with basic knowledge in set and measure theory and unfolds with references to historical developments and early motivations from gambling and natural sciences, ending with current-day applications in finance, information theory and machine learning. The main target is students with an undergraduate degree in physics, engineering, computer science, economics or similar who have a good basis in calculus and linear algebra and may have already come into contact with aspects of probability and statistics for ad hoc applications like data analysis, transport equations, and quantum mechanics, but have not attended yet a dedicated course on this subject.

Intended learning outcomes:

On successful completion of the module, a student will be able to:

1. Demonstrate familiarity with probability theory, stochastic processes in discrete or continuous time, and stochastic calculus.
2. Apply these tools to solve problems in physics, engineering, finance and machine learning.

Indicative content:

The following are indicative of the topics the module will typically cover:

1. Elementary probability

• Set theory, sample space, events, sigma-algebra, measure.
• Probability measure, Kolmogorov’s axioms, probability space.
• Joint and conditional probability, independence.
• Bayes’, total probability, and extended Bayes’ theorems.
• Bernoullian trials, factorial, Stirling’s approximation.

2. Random variables

• Random variables, probability distribution function, CDF, PDF, median.
• Joint, marginal and conditional CDF and PDF, independence.
• Function of a random variable in one and many dimensions.
• Expectation, variance, covariance, correlation, moments.
• Bienaymé-Chebyshev inequality, law of large numbers.
• Fourier transform, characteristic function, moment- and cumulant-generating functions.
• Central limit theorem, Lévy stable distributions, standardised moments, mode, skewness, kurtosis, heavy-tailed distributions
• Discrete distributions: Bernoulli, binomial, Poisson.
• Entropy, information theory and statistical mechanics, maximum entropy principle, differential entropy.
• Continuous distributions: Gaussian, exponential, uniform, beta, gamma, chi squared, chi, Student t.

3. Stochastic processes

• Definitions, auto- and cross-covariance/correlation, stationarity, ergodicity.
• Classification with respect to memory; martingales and semimartingales.
• Markov and semi-Markov processes, Chapman-Kolmogorov equation.
• Time-evolution equation for processes with discrete states: master equation.
• Random telegraph signal, hidden Markov model, random walk.
• Poisson, compound Poisson and renewal processes, continuous-time random walk.
• Time-evolution equation for processes with continuous states: Kramers-Moyal expansion, Fokker-Planck equation, Kolmogorov backward equation.
• Wiener and Ornstein-Uhlenbeck processes, method of characteristics.
• Laplace transform, solution of the diffusion equation in Fourier-Laplace space.
• Langevin equation, stochastic differential equation.
• Stochastic integral: Ito and Stratonovich; Ito’s formula, Feynman-Kac theorem.
• Geometric Brownian motion, Black-Scholes-Merton equation.
• Bessel, squared Bessel, Feller square-root and Rayleigh processes, and their generalisation.

Requisites:

To be eligible to select this module as optional or elective, a student must: (1) be registered on a programme and year of study for which it is a formally available; and (2) have knowledge of calculus and linear algebra,Ìýwhich can be refreshed taking the non-examined support course ‘Introduction to Mathematics and Programming for Finance’.