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：

1. 财务数学导论（金融数学基础）Part 1_哔哩哔哩_bilibili
2. 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 It? 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. Bj?rk: 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. F?llmer 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. ?ksendal: 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.
?

其他资源：

台大开方式课程：最新上線 - 臺大開放式課程 (NTU OpenCourseWare)

?MIT开方式课程：Free Online Courses from MIT OCW | Open Learning

国立清华大学开方式课程：國立清華大學開放式課程OpenCourseWare(NTHU, OCW) - 課程列表?

国立阳明交通大学开方式课程：https://www.youtube.com/@NYCUOCW/videos?