本书是一本英文影印版的随机分析在定量经济学领域中应用方面的著名教材。本书共分2卷。第1卷主要包括随机分析的基础性知识和离散时间模型；第2卷主要包括连续时间模型和该模型经济学中的应用。就其内容而言，第2卷有较为实际的可操作性的定量经济学内容，同时也包含了较为完整的随机微分方程理论。

这是一套随机分析在定量经济学领域中应用方面的著名教材，作者在该领域享有盛誉，全书共分2卷。第1卷主要包括随机分析的基础性知识和离散时间模型；第2卷主要包括连续时间模型和该模型经济学中的应用。就其内容而言，第2卷有较为实际的可操作性的定量经济学内容，同时也包含了较为完整的随机微分方程理论。本书各章有习题，适用于掌握微积积分基础知识的大学高年级本科生和硕士研究生。

1 General Probability Theory

 1.1 Infinite Probability Spaces

 1.2 Random Variables and Distributions

 1.3 Expectations

 1.4 Convergence of Integrals

　1.5 Computation of Expectations

　1.6 Change of Measure

　1.7 Summary

　1.8 Notes

　1.9 Exercises

2　Information and Conditioning

　2.1 Information and a-algebras

　2.2 Independence

　2.3 General Conditional Expectations

　2.4 Summary

　2.5 Notes

　2.6 Exercises

3　Brownian Motion

　3.1 Introduction

　3.2 Scaled Random Walks

　　3.2.1 Symmetric Random Walk

　　3.2.2 Increments of the Symmetric Random Walk

3.2.3 Martingale Property for the Symmetric　Random Walk

3.2.4 Quadratic Variation of the Symmetric Random Walk

3.2.5 Scaled Symmetric Random Walk

　　3.2.6 Limiting Distribution of the Scaled Random Walk

　　3.2.7 Log-Normal Distribution as the Limit of the　Binomial Model

　3.3 Brownian Motion

　　3.3.1 Definition of Brownian Motion

　　3.3.2 Distribution of Brownian Motion

　　3.3.3 Filtration for Brownian Motion

　　3.3.4 Martingale Property for Brownian Motion

　3.4 Quadratic Variation

　　3.4.1 First-Order Variation

　　3.4.2 Quadratic Variation

　　3.4.3 Volatility of Geometric Brownian Motion

　3.5 Markov Property

　3.6 First Passage Time Distribution

　3.7 Reflection Principle

　　3.7.1 Reflection Equality

　　3.7.2 First Passage Time Distribution

　　3.7.3 Distribution of Brownian Motion and Its Maximum

　3.8 Summary

　3.9 Notes

　3.10 Exercises

4  Stochastic Calculus

　4.1 Introduction

　4.2 Ito's Integral for Simple Integrands

　　4.2.1 Construction of the Integral

　　4.2.2 Properties of the Integral

　4.3 Ito's Integral for General Integrands

　4.4 Ito-Doeblin Formula

　　4.4.1 Formula for Brownian Motion

　　4.4.2 Formula for It5 Processes

　　4.4.3 Examples

　4.5 Black-Scholes-Merton Equation

　　4.5.1 Evolution of Portfolio Value

　　4.5.2 Evolution of Option Value

　　4.5.3 Equating the Evolutions

　　4.5.4 Solution to the Black-Scholes-Merton Equation

　　4.5.5 The Greeks

　　4.5.6 Put-Call Parity

　4.6 Multivariable Stochastic Calculus

　　4.6.1 Multiple Brownian Motions

　　4.6.2 Ito-Doeblin Formula for Multiple Processes

　　4.6.3 Recognizing a Brownian Motion

　4.7 Brownian Bridge

　　4.7.1 Gaussian Processes

　　4.7.2 Brownian Bridge as a Gaussian Process

　　4.7.3 Brownian Bridge as a Scaled Stochastic Integral

　　4.7.4 Multidimensional Distribution of the　Brownian Bridge

　　4.7.5 Brownian Bridge as a Conditioned Brownian Motion

　4.8 Summary

　4.9 Notes

　4.10 Exercises

5　Risk-Neutral Pricing

　5.1 Introduction

　5.2 Risk-Neutral Measure

　　5.2.1 Girsanov's Theorem for a Single Brownian Motion..

　　5.2.2 Stock Under the Risk-Neutral Measure

　　5.2.3 Value of Portfolio Process Under the　Risk-Neutral Measure

　　5.2.4 Pricing Under the Risk-Neutral Measure

　　5.2.5 Deriving the Black-Scholes-Merton Formula

　5.3 Martingale Representation Theorem

　　5.3.1 Martingale Representation with One　Brownian Motion

　　5.3.2 Hedging with One Stock

　5.4 Fundamental Theorems of Asset Pricing

　　5.4.1 Girsanov and Martingale Representation Theorems.

　　5.4.2 Multidimensional Market Model

5.4.3 Existence of the Risk-Neutral Measure

5.4.4 Uniqueness of the Risk-Neutral Measure

　5.5 Dividend-Paying Stocks

5.5.1 Continuously Paying Dividend

5.5.2 Continuously Paying Dividend with　Constant Coefficients

5.5.3 Lump Payments of Dividends

5.5.4 Lump Payments of Dividends with　Constant Coefficients

　5.6 Forwards and Futures

5.6.1 Forward Contracts

5.6.2 Futures Contracts

5.6.3 Forward-Futures Spread

　5.7 Summary

　5.8 Notes

　5.9 Exercises

6　Connections with Partial Differential Equations

　6.1 Introduction

　6.2 Stochastic Differential Equations

　6.3 The Markov Property

 6.4 Partial Differential Equations

 6.5 Interest Rate Models

 6.6 Multidimensional Fcynman-Kac Theorems

 6.7 Summary

 6.8 Notes

 6.9 Exercises

7　Exotic Options

　7.1 Introduction

　7.2 Maximum of Brownian Motion with Drift

　7.3 Knock-out Barrier Options

　　7.3.1 Up-and-Out Call

　　7.3.2 Black-Scholes-Merton Equation

　　7.3.3 Computation of the Price of the Up-and-Out Call

 7.4 Lookback Options

　　7.4.1 Floating Strike Lookback Option

　　7.4.2 Black-Scholes-Merton Equation

　　7.4.3 Reduction of Dimension

　　7.4.4 Computation of the Price of the Lookback Option

　7.5 Asian Options

　　7.5.1 Fixed-Strike Asian Call

　　7.5.2 Augmentation of the State

　　7.5.3 Change of Numeraire

 7.6 Summary

 7.7 Notes

 7.8 Exercises

8  American Derivative Securities

　8.1 Introduction

　8.2 Stopping Times

　8.3 Perpetual American Put

　　8.3.1 Price Under Arbitrary Exercise

　　8.3.2 Price Under Optimal Exercise

　　8.3.3 Analytical Characterization of the Put Price

　　8.3.4 Probabilistic Characterization of the Put Price

　8.4 Finite-Expiration American Put

　　8.4.1 Analytical Characterization of the Put Price

　　8.4.2 Probabilistic Characterization of the Put Price

　8.5 American Call

　　8.5.1 Underlying Asset Pays No Dividends

　　8.5.2 Underlying Asset Pays Dividends

　8.6 Summary

　8.7 Notes

　8.8 Exercises

9　Change of Numeraire

　9.1 Introduction

　9.2 Numeraire

　9.3 Foreign and Domestic Risk-Neutral Measures

9.3.1 The Basic Processes

9.3.2 Domestic Risk-Neutral Measure

9.3.3 Foreign Risk-Neutral Measure

9.3.4 Siegel's Exchange Rate Paradox

9.3.5 Forward Exchange Rates

9.3.6 Garman-Kohlhagen Formula

9.3.7 Exchange Rate Put-Call Duality

　9.4 Forward Measures

9.4.1 Forward Price

9.4.2 Zero-Coupon Bond as Numeraire

9.4.3 Option Pricing with a Random Interest Rate

　9.5 Summary

　9.6 Notes

　9.7 Exercises

10　Term-Structure Models

　10.1 Introduction

　10.2 Affine-Yield Models

10.2.1 Two-Factor Vasicek Model

10.2.2 Two-Factor CIR Model

10.2.3 Mixed Model

　10.3 Heath-Jarrow-Morton Model

10.3.1 Forward Rates

10.3.2 Dynamics of Forward Rates and Bond Prices

10.3.3 No-Arbitrage Condition

10.3.4 HJM Under Risk-Neutral Meaaure

10.3.5 Relation to Affine-Yield Models

10.3.6 Implementation of HJM

　10.4 Forward LIBOR Model

10.4.1 The Problem with Forward Rates

10.4.2 LIBOR and Forward LIBOR

10.4.3 Pricing a Backset LIBOR Contract

10.4.4 Black Caplet Formula

10.4.5 Forward LIBOR and Zero-Coupon Bond Volatilities ..

10.4.6 A Forward LIBOR Term-Structure Model

　10.5 Summary

　10.6 Notes

　10.7 Exercises

11 Introduction to Jump Processes

　11.1 Introduction

　11.2 Poisson Process

　　11.2.1 Exponential Random Variables

　　11.2.2 Construction of a Poisson Process

　　11.2.3 Distribution of Poisson Process Increments

　　11.2.4 Mean and Variance of Poisson Increments

　　11.2.5 Martingale Property

　11.3 Compound Poisson Process

　　11.3.1 Construction of a Compound Poisson Process

　　11.3.2 Moment-Generating Function

　11.4 Jump Processes and Their Integrals

　　11.4.1 Jump Processes

　　11.4.2 Quadratic Variation

　11.5 Stochastic Calculus for Jump Processes

　　11.5.1 Ito-Doeblin Fermula for One Jump Process

　　11.5.2 Ith-Doeblin Formula for Multiple Jump Processes

　11.6 Change of Measure

　　11.6.1 Change of Measure for a Poisson Process

　　11.6.2 Change of Measure for a Compound Poisson Process

　　11.6.3 Change of Measure for a Compound Poisson Process　and a Brownian Motion

　11.7 Pricing a European Call in a Jump Model

　　11.7.1 Asset Driven by a Poisson Process

　　11.7.2 Asset Driven by a Brownian Motion and a Compound　Poisson Process

　11.8 Summary

　11.9 Notes

　11.10 Exercises

A Advanced Topics in Probability Theory

　A.1 Countable Additivity

　A.2 Generating a-algebras

　A.3 Random Variable with Neither Density nor Probability MassFunction

B Existence of Conditional Expectations

C Completion of the Proof of the Second Fundamental

Theorem of Asset Pricing

References

Index