Stochastic Analysis for Finance
This course provides a rigorous yet application-oriented introduction to stochastic analysis with a focus on financial applications. It combines essential probability theory, stochastic calculus, and financial modeling techniques. Students will first build a solid foundation in probability and martingale theory, then progress to stochastic calculus and its use in financial models, and finally explore advanced topics such as stochastic control, filtering theory, and incomplete information markets.
讲师
日期
2025年10月30日 至 2026年01月22日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周四 | 14:20 - 17:50 | A7-201 | Zoom 17 | 442 374 5045 | BIMSA |
修课要求
Ordinary differential equations (ODEs) ; Basic probability theory
课程大纲
Part 1: Probability Theory Foundations (10 hours)
1.Probability spaces and measure-theoretic foundations
2.Conditional expectation: definitions, properties
3.Martingale theory: stopping times, convergence theorems, and financial examples
4.Limit theorems: Law of Large Numbers, Central Limit Theorem
5.Brownian motion
Part 2: Stochastic Integral and Financial Applications (30 hours)
6.Itô integrals: Motivation, construction, and Itô isometry
7.Itô formula: Statement, intuition, and basic examples
8.Stochastic differential equations (SDEs)
9.Martingale Representation Theorem and applications in finance
10.Feymann-Kac formula for PDEs
11.Girsanov’s theorem and its Application on Risk-neutral pricing
12.The Black–Scholes mode and Option pricing formula
13. Extensions: CIR model, stochastic volatility models, HJM model
Module 3: Advanced Models (8 hours)
14. Stochastic control and portfolio optimization
15. Filtering Theory and Markets with Incomplete Information
1.Probability spaces and measure-theoretic foundations
2.Conditional expectation: definitions, properties
3.Martingale theory: stopping times, convergence theorems, and financial examples
4.Limit theorems: Law of Large Numbers, Central Limit Theorem
5.Brownian motion
Part 2: Stochastic Integral and Financial Applications (30 hours)
6.Itô integrals: Motivation, construction, and Itô isometry
7.Itô formula: Statement, intuition, and basic examples
8.Stochastic differential equations (SDEs)
9.Martingale Representation Theorem and applications in finance
10.Feymann-Kac formula for PDEs
11.Girsanov’s theorem and its Application on Risk-neutral pricing
12.The Black–Scholes mode and Option pricing formula
13. Extensions: CIR model, stochastic volatility models, HJM model
Module 3: Advanced Models (8 hours)
14. Stochastic control and portfolio optimization
15. Filtering Theory and Markets with Incomplete Information
参考资料
1. Billingsley, P. Probability and Measure. Wiley, 1995.
2. Øksendal, B. Stochastic Differential Equations: An Introduction with Applications. Springer, 2003.
3. Weinan E, Tiejun Li, Eric, Vanden-eijnden. Applied Stochastic Analysis. AMS, 2019.
4. Karatzas, I., Shreve, S. E. Brownian Motion and Stochastic Calculus. Springer, 1991.
5. Shreve, S. E. Stochastic Calculus for Finance II: Continuous-Time Models. Springer, 2004.
6. Paolo Baldi. Stochastic Calculus: An Introduction Through Theory and Exercises. Springer, 2017.
7. Pham, H. Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer, 2009.
2. Øksendal, B. Stochastic Differential Equations: An Introduction with Applications. Springer, 2003.
3. Weinan E, Tiejun Li, Eric, Vanden-eijnden. Applied Stochastic Analysis. AMS, 2019.
4. Karatzas, I., Shreve, S. E. Brownian Motion and Stochastic Calculus. Springer, 1991.
5. Shreve, S. E. Stochastic Calculus for Finance II: Continuous-Time Models. Springer, 2004.
6. Paolo Baldi. Stochastic Calculus: An Introduction Through Theory and Exercises. Springer, 2017.
7. Pham, H. Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer, 2009.
听众
Undergraduate
, Graduate
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笔记公开
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语言
中文