推荐理由
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。
本书内容简洁，阐述透彻，包含丰富的例子，并有精彩解答。
——Robert Liptser教授，以色列特拉维夫大学
在讲述随机分析的著作中，像本书这样涵盖广泛而又具有很强可读性的实属罕见。
——Mathematical Reviews
内容简介
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。
本书作者
Fima C Klebaner，澳夫利亚Monash(莫纳什)大学教授，IMS(国际数理统计学会)会士，著名数理统计和金融数学家。主要研究领域有：随饥过程、概率应用、随机分析、金融数学、动态系统的随机扰动等。
目录
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
4 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
5 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
