Type of Credit: Elective
Credit(s)
Number of Students
Stochastic calculus is intended to be a two-semester course. In the first semester we will study stochastic intergral and define what a stochastic differential equation is. In the second semester, we study various properties of SDE and its applications, which , depending on the composition of the class, will include finantial mathematics, partial differential equations and some random phenomena in natural sciences.
The basic tool for stochastic calculus is martingale theory. We shall start from the basic concepts in probability: probability space, sigma algebra, distribution function and some examples in finicial mathematics to motivate the study of stochastic calculus. Then we quickly move to Brownian motion and some martingale theory. Stochastic intergral is our next target and we finish the semester with the definition of stochastic differential equations.
能力項目說明
Students should be able to know why we need stochastic calculus to model some of the random phenomena arising from natural sciences. Also some basic knowledge like martingale theory, Brownian motion and Hilbert spaces will be introduced to students. More precisely students will have the following capability :
1. Understand the nature of randomness
2. How to model natural phenomena with randomness
3. How to use mathematical tools to study these phenomena and make assessments.
| 教學週次Course Week | 彈性補充教學週次Flexible Supplemental Instruction Week | 彈性補充教學類別Flexible Supplemental Instruction Type |
|---|---|---|
In-class Hours: 3; Outside-of-class Hours: 6
week 1-2, Review of the basic concepts : Probability space, sigma algebra, distribution functions, conditional expectation.
week 3-5, Basic martingale theory.
week 6-8, Introduction of Brownian motion.
week 9, Midterm exam.
week 10-11, Stochastic integral
week 12-15, Local martingales and Ito formula
week 16-17, Introduction of stochastic differential equation
week 18, Final exam.
Home work, 40%
Midterm exam, 30%
Final exam. 30%
Financial Calculus, Martin Baxter and Andrew Rennie, Cambridge Press, 1996.
Stochastic Calculus and Financial Applications, Michael Steele, Springer, 2001.
Brownian Motion, Martingales and Stochastic Calculus, Jean-Francois Le Gall, Springer, 2016.