Type of Credit: Elective
Credit(s)
Number of Students
The main purpose of this course will be the introduction of Stochastic differential equations. Before the formal introduction, we will first study :
1. Continuous semimartigales. This includes local martingales and its quadratic variation process. This is the main tool of stochastic calculus.
2. Stochastic Integration. This will include Burkholder-Davis-Gundy inequality, Girsanov's theorem and Ito's formula.
3. Stochastic differential equation. Many interesting examples of SDE will be introduced and their application to other field will be discussed.
能力項目說明
1. To understand Brownian Motion and Diffusion processes, which are the most used in stochastic analysis.
2. To be able to use Martingale theory in probability.
3. To be able to construct mathematical models in different fields, especially in financial engineering and mathematical physics.
| 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 |
|---|---|---|
1. Finite variation process
2.Continuous local martingales
3 and 4. Quadratic variation of local matingales and the bracket.
5 and 6, Stochastic integrals for local martingales and semi-martingales.
7. and 8. Ito's formula and Girsanov's theorem.
9 and 10. Applications of Girsanov's theorem.
11. and 12. Burkholder-Davis-Gundy's theorem.
13 and 14, Representation theorem and time change Brownian motion.
15 and 16. Solution of SDE as Markov processes.
17. Final
學生學習投入時間:
每週課堂教學 3小時
每週預習/複習 6小時
Home work 30%
Final 40%
Medterm 20%
Class participation 10%
Graduate Texts in Mathematics, 274, Springer, J. Le Gall, 2013
Probability Essentials, J. Jacod and P.Protter. 2000