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Stochastic Processes (MATH M6006)

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Administrative Information

  1. Unit number and title: MATH M6006 Stochastic Processes
  2. Level: M/7
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 2007
  6. Unit Organiser: Balint Toth
  7. Lecturer: Balint Toth
  8. Teaching block: 1
  9. Prerequisites: MATH21400 Applied Probability 2 essential. Background in PDE's helpful but not essential

Unit aims

The aim of the unit is to introduce theory of Brownian motions, in particular, how to construct it from random walks, various properties, and finally stochastic integration leading to a brief survey of diffusion processes.

General Description of the Unit

This unit aims for an intuitive understanding of Brownian motion and stochastic calculus, although rigorous proofs will be presented for a few of the most beautiful results. Students should be comfortable with reading and understanding rigorous proofs. Understanding Brownian motion, commonly regarded as the canonical example of a martingale and a Markov process with continuous paths, is essential for any future study of stochastic processes and its applications. 

COURSE WEB PAGE: http://www.maths.bris.ac.uk/~mabat/SP_M6006/

Relation to Other Units

This unit is a first course in continuous time stochastic processes and introductory stochastic analysis.

Teaching Methods

Lectures supported by problem sheets and solution sheets.

Learning Objectives

At the end of the unit students should:

  • be able to recall all definitions and main results,
  • be able to understand on an intuitive level the reasoning behind proofs of major results,
  • be able to apply the theory in standard situations,
  • be able to use the ideas of the unit in unseen situations

Assessment Methods

The assessment mark for Stochastic Processes:

  • 100% by means of a standard closed book one and half hour examination in April consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators are NOT permitted in this examination.

Award of Credit Points

Credit points will be awarded if a student passes the unit, i.e. attains a final mark of 50 or more.

If a student obtains a mark between 30 and 49, they will still be awarded the credit points if they have turned in acceptable solutions to at least two thirds of the problems assigned as home work .

Transferable Skills

Understanding the behaviour of diffusion processes so as to be able to use them (e.g.  perform calculations and write simulations) in problems arising in physics, engineering, financial calculus or statistics.

Texts

Geoffrey R. Grimmett and David R. Stirzaker, Probability and Random Processes, Oxford University Press, 3rd edition can be used as the main reference on Brownian motion.

B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 6th edition can be used as a reference for stochastic integration.

Hand written lecture notes available online at the course web page.

Syllabus

Existence and Explicit Construction of Brownian motion

Elementary properties (Markov property, reflection principle, hitting times)

Sample path properties (zero set, nowhere differentiability, quadratic variation)

Stochastic integral and Ito's formula

Stochastic differential equations and Brownian bridge