# Stochastic Differential Equations

Yaozhong Hu (University of Alberta)Jan 1, 2021 — Jun 1, 2021

## About the course

This is a one semester three credit hour course and meet twice a week, tentatively Tuesdays and Thursdays from 11:00-12:20. It is about the theory and applications of stochastic differential equations driven by Brownian motion. The stochastic differential equations have found applications in finance, signal processing, population dynamics and many other fields. It is the basis of some other applied probability areas such as filtering theory, stochastic control and stochastic differential games. To balance the theoretical and applied aspects and to include as much audience as possible, we shall focus on the stochastic differential equations driven only by Brownian motion (white noise). We will focus on the theory and not get into specific applied area (such as finance, signal processing, filtering, control and so on). We shall first briefly introduce some basic concepts and results on stochastic processes, in particular the Brownian motions. Then we will discuss stochastic integrals, Ito formula, the existence and uniqueness of stochastic differential equations, some fundamental properties of the solution. We will concern with the Markov property, Kolmogorov backward and forward equations, Feynman-Kac formula, Girsanov formula. We will also concern with the ergodic theory and other stability problems. We may also mention some results on numerical simulations, Malliavin calculus and so on.

# Registration

This course is available for registration under the Western Dean's Agreement. To register, you must obtain the approval of the course instructor and you must complete the Western Dean's agreement form , using the details below. The completed form should be signed by your home institution department and school of graduate studies, then returned to the host institution of the course.

## Enrollment Details

- Course Name
- Stochastic Differential Equations
- Date
- Jan 1, 2021 — Jun 1, 2021
- Course Number
- Math 663
- Section Number
- Topics in Applied Mathematics I: Stochastic differential equations
- Section Code
- N/A

## Instructor(s)

For help with completing the Western Dean’s agreement form, please contact the graduate student program coordinator at your institution. For more information about the agreement, please see the Western Dean's Agreement website

# Other Course Details

#### Reference Texts

- The main reference book for this course is
- Øksendal, B. Stochastic differential equations. An introduction with applications. Sixth edition. Universitext. Springer-Verlag, Berlin, 2003. xxiv+360 pp. ISBN: 3-540-04758-1
- Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
- Klebaner, Fima C. Introduction to stochastic calculus with applications. Third edition. Imperial College Press, London, 2012. xiv+438 pp. ISBN: 978-1-84816-832-9; 1-84816-832-2

- Other references
- Ikeda, N.; Watanabe, S. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3 * Protter, P. E. Stochastic integration and differential equations. Second edition. Version 2.1. Corrected third printing. Stochastic Modelling and Applied Probability, 21. Springer-Verlag, Berlin, 2005. xiv+419 pp. ISBN: 3-540-00313-4
- Revuz, D.; Yor, M. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp.
- Durrett, R. Stochastic calculus. A practical introduction. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. x+341 pp. ISBN: 0-8493-8071-5
- Jeanblanc, M.; Yor, M.; Chesney, M. Mathematical methods for financial markets. Springer Finance. Springer-Verlag London, Ltd., London, 2009. xxvi+732 pp. ISBN: 978-1-85233-376-8
- Hasminskii, R. Z. Stochastic stability of differential equations. Translated from the Russian by D. Louvish. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7. Sijthoff & Noordhoff, Alphen aan den RijnGermantown, Md., 1980. xvi+344 pp. ISBN: 90-286-0100-7
- Hu, Y. Analysis on Gaussian spaces. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. xi+470 pp. ISBN: 978-981-3142-17-6
- Kloeden, P. E.; Platen, E. Numerical solution of stochastic differential equations. Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. xxxvi+632 pp. ISBN: 3-540-54062-8