推荐优秀的大学数学课程

今天在B站看 R-S积分发现这个老师讲的不错：Riemann-Stieltjes Integrals_哔哩哔哩_bilibili

可以用优秀来说，板书也不错！授课老师：吴庆堂老师（国立交通大学，目前台湾阳明大学和台湾交通大学合并而成的台湾“国立”阳明交通大学） 参考书目为shreve的金融随机分析此课程可以作为金融数学的基础课程

后来检索到这个老师的开放课：

财务数学导论（金融数学基础）Part 1_哔哩哔哩_bilibili
财务数学导论（金融数学基础）Part 2_哔哩哔哩_bilibili

课程目录：

课程大纲
Objective
This course is to make students understand and familiar with mathematical methods while studying Finance.
Outline
Chapter Overview
Introduction
1 Probability Theory

• 1.1 Probability space
• 1.2 Random variables
• 1.3 Expectation
2 Discrete-Time Martingales
• 2.1 Conditional probability and conditional expectation
• 2.2 Discrete time Martingales
• 2.3 Martingale transform and Doob decomposition
3 One-Period Model
• Introduction
• 3.1 Portfolios
• 3.2 Derivative securities
• 3.3 Absence of arbitrage
• 3.4 No arbitrage and price system
• 3.5 Martingale measures
• 3.6 Pricing
• 3.7 Complete market model
4 Multi-Period Model
• Introduction
• 4.1 The market model
• 4.2 Arbitrage opportunities
• 4.3 Martingale measures
• 4.4 Arbitrage-free prices for European contingent claim
5 American Contingent Claim
• 5.1 Stopping time
• 5.2 American claims
• 5.3 Arbitrage-free prices
6 Measures of Risk
• Introduction
• 6.1 Monetary measure of risk
• 6.2 Coherent and convex risk measures
• 6.3 Acceptance sets
• 6.4 Robust representation of coherent risk measure
• 6.5 Robust representation of convex risk measures

•

Unit Overview
7 Continuous-Time Martingales
• 7.1 Stochastic processes
• 7.2 Uniform integrability
• 7.3 Martingale theory in continuous-time
• 7.4 Local martingales
• 7.5 Doob-Meyer decomposition
• 7.6 Semimartingales
8 Brownian Motions
• 8.1 Scaled random walk
• 8.2 Brownian motions
• 8.3 The Brownian sample paths
• 8.4 Exponential martingales
• 8.5 d-dimensional Brownian motions

9 Stochastic Integrals
• 9.1 Construction of stochastic integrals with respect to martingales
• 9.2 Stochastic integrals with respect to semimartingales
• 9.3 Itô formula
• 9.4 Integration by parts
• 9.5 Martingale representation theorem
• 9.6 Girsanov theorem
• 9.7 Local times
10 Stochastic Differential Equations
• 10.1 Examples and some solution methods
• 10.2 An existence and uniqueness result
• 10.3 Weak and strong solutions
• 10.4 Feynman-Kac theorem
11 Continuous-Time Models
• 11.1 Market portfolios and arbitrage
• 11.2 Equivalent local martingale measures
• 11.3 Completeness
• 11.4 Pricing for attainable contingent claim
• 11.5 Black-Scholes-Merton formula
• 11.6 Parity relations
• 11.7 The greeks
12 Hedging
• 12.1 Hedging strategy for the simple contingent claim
• 12.2 Delta and gamma hedging
• 12.3 Superhedging
• 12.4 Quantile hedging
6 Volatility
• 13.1 Historical volatility
• 13.2 Implied volatility
Appendix

• A. Limits of Sequences of Numbers
• B.  Convergence of Sequences of Functions and Stochastic Processes I
• C. Distribution Functions
• D. Convergence of Sequences of Functions and Stochastic Processes II
• E. Riemann-Stieltjes Integrals
• F . Convex Analysis

Textbook
• S. E. Shreve: Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004.

References
• T. M. Apostol: Mathematical Analysis, Second Edition
• M. Baxter and A. Rennie: Financial Calculus.
• T. Björk: Arbitrage Theory in Continuous Time.
• K. L. Chung: A Course in Probability Theory, Second Edition.
• F. Delbaen and W. Schachermayer: The Mathematics of Arbitrage.
• J. Elstrodt: Maβ- und Integrationstheorie, Third Edition.
• H. Föllmer and A. Schied: Stochastic Finance. An Introduction in Discrete Time.
• J. Jacod and Ph. Protter: Probability Essentials.
• J. C. Hull: Options, Futures, & Other Derivatives, Sixth Edition.
• I. Karatzas: Lectures on the Mathematics of Finance.
• I. Karatzas and S. E. Shreve: Brownian Motion and Stochastic Calculus, Second  Edition.
• I. Karatzas and S. E. Shreve: Method of Mathematical Finance.
• D. Lamberton and B. Lapeyre: Introduction to Stochastic Calculus Applied to Finance.
• B. Øksendal: Stochastic Differential Equations, An Introduction with Applications,S ixth Edition.
• R. T. Rockafellar: Convex Analysis.
• H. L. Royden: Real Analysis, Third Edition.
• A.N. Shiryaev: Probability Theory, Second Edition.
• S. E. Shreve: Stochastic Calculus for Finance I: The Binomial Asset Pricing Model.
• R. L. Wheeden and A. Zygmund: Measure and integral.