C

Stochastic Differential Equations

I. Course Discription

Stochastic differential equations have been used extensively in many areas of application, including finance and social science as well as in physics, chemistry. This course develops the theory of Itô's calculus and stochastic differential equations.The course will introduce stochastic integrals with respect to general semi- martingales, stochastic differential equations based on these integrals, integration with respect to Poisson random measures, stochastic differential delayed equations, stability of stochastic differential equations, and applications to mathematical biology.Through this course, students will master the basic concepts of stochastic differential equations, basic principles and methods of stochastic stability, and to lay a foundation for future professional development.

II. Professionals

Mathematics and Applied Mathematics, Probability

III. Prerequisite

This is a graduate level course that requires only upper division probability and differential equations, since we will approach the analysis of questions about SDE through the associated differential equations and the inference through the normal (gaussian) distribution.

IV. Course Outline

1 Introduction and preliminaries

1.1 Probability and measure theory

1.2 Expectation and conditional expectation

1.3 Brownian motion

2 Stochastic integrals

2.1 The Ito integral

2.2 Martingales

2.3 Ito’s formula

2.4 Vector valued stochastic differentials

2.5 The Stratonovich integral

3Stochastic differential equations

3.1 Analytical solutions

3.2 Existence and uniqueness of solutions

3.3 Strong and weak solutions

4.Stochastic stability, attraction and boundedness

4.1Stability

4.2 Attraction

4.3 Boundedness

5.Stochastic differential delay equations

5.1 Existence-and-uniqueness theorems

5.2 Stability of stochastic differential delay equations

6.Applications of stochastic differential equations

6.1. Stochastic population systems

6.2 Stochastic epidemic model

6.3Stochastic synchronization of complex networks

V. Class hours assignment table

VI. Texts and reading materials

1. B. Oksendal (2013). Stochastic Differential Equations: An Introduction with Applications (sixth edition, sixth corrected printing. Springer.) Chapters II, III, IV, V, part of VI, Chapter VIII (F)

2.Mao, X., Stochastic Differential Equations and Applications, 2nd Edition, Horwood, 2008.

3.M. Yor and D. Revaz, Continuous Martingales and Brownian Motion (Springer).

4.R. Durrett, Stochastic Calculus (CRC Press).

5.Futher Online information:

http://math.stanford.edu/~papanico/Math236/CourseInfo.html

https://courses.maths.ox.ac.uk/node/154

http://personal.strath.ac.uk/x.mao/talks/

VII. Grading

30% homework and discussion, 70% final