Prof Angelos Dassios COL 7.14

This course is compulsory on the BSc in Actuarial Science. This course is available on the BSc in Data Science, BSc in Financial Mathematics and Statistics, BSc in Mathematics with Data Science, BSc in Mathematics with Economics and BSc in Mathematics, Statistics and Business. This course is available as an outside option to students on other programmes where regulations permit and to General Course students.

Students must have completed either Probability, Distribution Theory and Inference (ST202) or Probability and Distribution Theory (ST206).

A second course in stochastic processes and applications to insurance. Markov chains (discrete and continuous time), processes with jumps; Brownian motion and diffusions; Martingales; stochastic calculus; applications in insurance and finance. Content: Stochastic processes in discrete and continuous time; Markov chains: Markov property, Chapman-Kolmogorov equation, classification of states, stationary distribution, examples of infinite state space; filtrations and conditional expectation; discrete time martingales: martingale property, basic examples, exponential martingales, stopping theorem, applications to random walks; Poisson processes: counting processes, definition as counting process with independent and stationary increments, compensated Poisson process as martingale, distribution of number of events in a given time interval as well as inter-event times, compound Poisson process, application to ruin problem for the classical risk process via Gerber's martingale approach; Markov processes: Kolmogorov equations, solution of those in simple cases, stochastic semigroups, birth and death chains, health/sickness models, stationary distribution; Brownian motion: definition and basic properties, martingales related to Brownian motion, reflection principle, Ito-integral, Ito's formula with simple applications, linear stochastic differential equations for geometric Brownian motion and the Ornstein-Uhlenbeck process, first approach to change of measure techniques, application to Black-Scholes model. The items in the course content that also appear in the content of ST227 are covered here at greater depth. However, ST227 is not a pre-requisite for this course.

This course will be delivered through a combination of classes, lectures, and Q&A sessions totalling a minimum of 29 hours across Autumn Term.

Compulsory written answers to two sets of problems.

Lecture notes will be provided. Relevant books include R Durrett, Essentials of Stochastic Processes; T Mikosch, Elementary Stochastic Calculus with Finance in View; Institute of Actuaries core reading notes.

Exam (100%, duration: 3 hours) in the January exam period.