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| 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 |
| 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。 |
| 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 |
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