本书是随机分析方面的名著之一。以主题广泛丰富，论述简洁易懂而又不失严密著称。书中阐述了各领域的典型应用，包括数理金融、生物学、工程学中的模型。还提供了很多示例和习题，并附有解答。

第2版增加了讲述证券，利率及其期权的一章，并在全书增加了许多新内容，以反映随机分析研究和应用的最新成果。

本书可作为高年级本科生和研究生的随机分析和金融数学的教材，也非常适合各领域专业人士自学。

本书介绍了随机分析的理论和应用两方面的知识。内容涉及积分和概率论的基础知识、基本的随机过程，布朗运动和伊藤过程的积分、随机微分方程、半鞅积分、纯离散过程，以及随机分析在金融、生物、工程和物理等方面的应用。书中有大量的例题和习题，并附有答案，便于读者进行深层次的学习。

本书非常适合初学者阅读，可作为高等院校经管、理工和社科类各专业高年级本科生和研究生随机分析和金融数学的教材，也可供相关领域的科研人员参考。

1 Preliminaries From Calculus

1.1 Functions in Calculus

1.2 Variation of a Function

1.3 Riemann Integral and Stieltjes Integral

1.4 Lebesgue's Method of Integration

1.5 Differentials and Integrals

1.6 Taylor's Formula and Other Results

2 Concepts of Probability Theory

2.1 Discrete Probability Model

2.2 Continuous Probability Model

2.3 Expectation and Lebesgue Integral

2.4 Transforms and Convergence

2.5 Independence and Covariance

2.6 Normal (Gaussian) Distributions

2.7 Conditional Expectation

2.8 Stochastic Processes in Continuous Time

3 Basic Stochastic Processes

  3.1 Brownian Motion

  3.2 Properties of Brownian Motion Paths

  3.3 Three Martingales of Brownian Motion

  3.4 Markov Property of Brownian Motion

  3.5 Hitting Times and Exit Times

  3.6 Maximum and Minimum of Brownian Motion

  3.7 Distribution of Hitting Times

  3.8 Reflection Principle and Joint Distributions

  3.9 Zeros of Brownian Motion. Arcsine Law

  3.10 Size of Increments of Brownian Motion

  3.11 Brownian Motion in Higher Dimensions

3.12 Random Walk

3.13 Stochastic Integral in Discrete Time

3.14 Poisson Process

3.15 Exercises

Brownian Motion Calculus

4.1 Definition of It6 Integral

4.2 Ito Integral Process

4.3 Ito Integral and Gaussian Processes

4.4 Ito's Formula for Brownian Motion

4.5 Ito Processes and Stochastic Differentials

4.6 Ito's Formula for It6 Processes

4.7 Ito Processes in Higher Dimensions

4.8 Exercises

Stochastic Differential Equations

5.1 Definition of Stochastic Differential Equations

5.2 Stochastic Exponential and Logarithm

5.3 Solutions to Linear SDEs

5.4 Existence and Uniqueness of Strong Solutions

5.5 Markov Property of Solutions

5.6 Weak Solutions to SDEs

5.7 Construction of Weak Solutions

5.8 Backward and Forward Equations

5.9 Stratanovich Stochastic Calculus

5.10 Exercises

6 Diffusion Processes

6.1 Martingales and Dynkin's Formula

6.2 Calculation of Expectations and PDEs

6.3 Time Homogeneous Diffusions

6.4 Exit Times from an Interval

6.5 Representation of Solutions of ODEs

6.6 Explosion

6.7 Recurrence and Transience

6.8 Diffusion on an Interval

6.9 Stationary Distributions

6.10 Multi-Dimensional SDEs

6.11 Exercises

7 Martingales

7.1 Definitions

7.2 Uniform Integrability

7.3 Martingale Convergence

7.4 Optional Stopping

7.5 Localization and Local Martingales

7.6 Quadratic Variation of Martingales

7.7 Martingale Inequalities

7.8 Continuous Martingales. Change of Time

7.9 Exercises

8 Calculus For Semimartingales

8.1 Semimartingales

8.2 Predictable Processes

8.3 Doob-Meyer Decomposition

8.4 Integrals with respect to Semimartingales

8.5 Quadratic Variation and Covariation

8.6 ItS's Formula for Continuous Semimartingales

8.7 Local Times

8.8 Stochastic Exponential

8.9 Compensators and Sharp Bracket Process

8.10 ItS's Formula for Semimartingales

8.11 Stochastic Exponential and Logarithm

8.12 Martingale (Predictable) Representations

8.13 Elements of the General Theory

8.14 Random Measures and Canonical Decomposition

8.15 Exercises

9 Pure Jump Processes

9.1 Definitions

9.2 Pure Jump Process Filtration

9.3 ItS's Formula for Processes of Finite Variation

9.4 Counting Processes

9.5 Markov Jump Processes

9.6 Stochastic Equation for Jump Processes

9.7 Explosions in Markov Jump Processes

9.8 Exercises

10 Change of Probability Measure

10.1 Change of Measure for Random Variables

10.2 Change of Measure on a General Space

10.3 Change of Measure for Processes

10.4 Change of Wiener Measure

10.5 Change of Measure for Point Processes

10.6 Likelihood Functions

10.7 Exercises

11 Applications in Finance: Stock and FX Options

11.1 Financial Deriwtives and Arbitrage

11.2 A Finite Market Model

11.3 Semimartingale Market Model

11.4 Diffusion and the Black-Scholes Model

11.5 Change of Numeraire

11.6 Currency (FX) Options

11.7 Asian, Lookback and Barrier Options

11.8 Exercises

12 Applications in Finance: Bonds, Rates and Option

12.1 Bonds and the Yield Curve

12.2 Models Adapted to Brownian Motion

12.3 Models Based on the Spot Rate

12.4 Merton's Model and Vasicek's Model

12.5 Heath-Jarrow-Morton (HJM) Model

12.6 Forward Measures. Bond as a Numeraire

12.7 Options, Caps and Floors

12.8 Brace-Gatarek-Musiela (BGM) Model

12.9 Swaps and Swaptions

12.10 Exercises

13 Applications in Biology

13.1 Feller's Branching Diffusion

13.2 Wright-Fisher Diffusion

13.3 Birth-Death Processes

13.4 Branching Processes

13.5 Stochastic Lotka-Volterra Model

13.6 Exercises

14 Applications in Engineering and Physics

14.1 Filtering

14.2 Random Oscillators

14.3 Exercises

Solutions to Selected Exercises

References

Index