Aims:

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 formalism. 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. The material unfolds with references to its historical development and early applications in gambling and natural sciences, ending with current-day applications in finance and machine learning.

Learning outcomes:

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

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

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.
Random variables:

Random variables, probability distribution function, CDF and PDF.
	Joint, marginal and conditional CDF and PDF, independence.
	Transformation of random variables, sum, product.
	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.
	Discrete distributions: Bernoulli, binomial, Poisson.
	Quantiles, median, mode, skewness, kurtosis, entropy, mutual information.
	Continuous maximum-entropy distributions: Gaussian, exponential, uniform, beta, gamma.
Random functions or stochastic processes:

Definitions, auto- and cross-covariance/correlation, stationarity, Wiener-Khinchin theorem.
	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, 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.
	Stochastic integral: Ito and Stratonovich.
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, which can be refreshed taking the non-examined support course ‘Introduction to Mathematics and Programming for Finance’.