Course syllabus - Introduction to Stochastic Processes

Objectives

The course aims to give the student the opportunity to acquire basic knowledge in the area of stochastic processes.

Learning outcomes

At the end of the course the student is expected to be able to:

1. compute probabilities, expectations, and variances by conditioning.
2. classify states of discrete time Markov chains, calculate their long-run probabilities and mean time spent in transient states, apply Markov chain Monte Carlo methods for statistical simulation.
3. explain properties of homogeneous, non-homogeneous and compound Poisson processes and apply them to real-life problems.
4. derive the forward and backward differential equations for continuous time Markov chains, determine their long-run probabilities.
5. explain properties of Brownian motions, such as stationary and independent increments, nowhere differentiability, hitting times.
6. perform basic calculations for Gaussian and stationary processes, such as computing the autocorrelation function and linear filtering.
7. apply theoretical knowledge to engineering areas such as option pricing, renewal, queueing, and reliability theories.

Course content

- Conditional probabilities and conditional expectations. Computing probabilities, expectations, and variances by conditioning.
- Discrete time Markov chains. Chapman-Kolmogorov equations. Classification of states. Long-run probabilities.
- Counting processes. The Poisson process and its generalizations. Interarrival and waiting time distributions.
- Continuous time Markov chains. Birth and death processes. The forward and backward differential equations. Long-run probabilities.
- Brownian motions. Application to pricing stock options. Gaussian and stationary processes.Applications to renewal, queueing, and reliability theories.

Specific requirements

At least totally 60 credits in the technical, 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.

Examination

TEN1, Written examination, 4.5 credits, individual written examination concerning learning outcomes 1-6, grades Fail (U), Pass (G) or Pass with distinction (VG)
SEM1, Seminar, 3 credits, active participation concerning learning outcome 7, grades Fail (U) or Passed (G)

The course grade is the same as the grade for TEN1.

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

Pass with distinction, Pass, Fail