Synopsis for C10.1a: Stochastic Differential Equations

Level: M-level Method of Assessment: Written examination. Weight: Half-unit (OSS paper code 2A83) Recommended Prerequisites Part A integration and B10a Martingales Through Measure Theory, is expected.

Overview

Stochastic differential equations have been used extensively in many areas of application, including finance and social science as well as chemistry. This course develops the basic theory of Itô's calculus and stochastic differential equations.

Learning Outcomes

The student will have developed an appreciation of stochastic calculus as a tool that can be used for defining and understanding diffusive systems.

Synopsis

Brownian motion: basic properties, reflection principle, quadratic variation. Itô's calculus: stochastic integrals with respect to martingales, Itô's lemma, Levy's theorem on characteristic of Brownian motion, exponential martingales, exponential inequality, Girsanov's theorem, The Martingale Representation Theorem. Stochastic differential equations: strong solutions, questions of existence and uniqueness, diffusion processes, Cameron–Martin formula, weak solution.

Reading — Main Texts

1. Dr Qian's online notes: www.maths.ox.ac.uk/courses/course/15721
2. B. Oksendal, Stochastic Differential Equations: An introduction with applications (Universitext, Springer, 6th edition). Chapters II, III, IV, V, part of VI, Chapter VIII (F).
3. F. C. Klebaner, Introduction to Stochastic Calculus with Applications (Imperial College Press, 1998, second edition 2005). Sections 3.1 – 3.5, 3.9, 3.12. Chapters 4, 5, 11.

Alternative Reading

1. H. P. McKean, Stochastic Integrals (Academic Press, New York and London, 1969).
   

Further Reading

1. N. Ikeda & S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North–Holland Publishing Company, 1989).
2. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics 113 (Springer-Verlag, 1988).
3. L. C. G. Rogers & D. Williams, Diffusions, Markov Processes and Martingales Vol 1 (Foundations) and Vol 2 (Ito Calculus) (Cambridge University Press, 1987 and 1994).