Course syllabus - Stochastic Processes

Objectives

Stochastic processes play a key role in analytical finance and insurance, and in financial engineering. The course presents the basic models of stochastic processes such as random walks, Markov chains, Poisson process, Brownian motion and diffusion processes, elements of stochastic calculus as well as simulation of stochastic processes. This basic part of the course can also be interesting for students from other specialties than analytical finance and financial engineering. The final part of the course will present applications of stochastic processes in finance and insurance.

Learning outcomes

At the end of the course the student is expected to be able to
- use formulas for evaluation of various distributional parameters, prices for financial contracts, and other characteristics connected with stochastic processes.
- know different models stochastic processes (random walks, Markov chains with discrete and continuous time, Poisson process, Brownian motion, diffusion processes and related models).
- know and to be able to use exchange of measure algorithms.
- use replacing portfolio and martingale based methods for evaluation of values for contingent claims.
- perform basic calculations from stochastic calculus based on Ito formula.
- solve linear stochastic differential equations.
- build algorithms for Monte Carlo simulation of various models of stochastic processes.

Course content

Random walks (transition probabilities, reflection principle, change of measure). Markov chains (Markov property, transition probabilities and Kolmogorov equations, ergodic properties, absorbing Markov chains, autoregressive models). Poisson process and Brownian motion (approximation by random walks, distributions of functionals). Basic stochastic processes in continuous time (diffusion processes, martingales, elements of stochastic calculus). Simulation of stochastic processes (Monte Carlo method, generation of random numbers, variance reduction, generation of realisation of stochastic processes,). Discrete and continuous time models for pricing processes, risk processes.

Tuition

Lectures combined with exercises. Continuous examination of problems/projects combined with written tests. Examination of seminars through oral presentation of written reports.

Specific requirements

At least totally 120 credits in the engineering, natural sciences, business administration or economics areas including Probability 7.5 credits, of which 4.5 credits must be completed at the beginning of the course and Basic Calculus Continuation Course, 7.5 credits, of which 1.5 credits must be completed at the beginning of the course, or the equivalent. In addition Swedish course B/Swedish course 3 and English course A/English course 6 are required. For courses given entirely in English exemption is made from the requirement in Swedish course B/Swedish course 3.

Examination

Continuous examination/projects (PRO1), 4.5 credits, marks Pass (G) or Pass with distinction (VG)
Seminars (SEM1), 3 credits, marks Pass (G) or Pass with distinction (VG)

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Three-grade scale