导语

  

内容提要

  

    F.C.克莱巴纳著的《随机分析及其应用（第3版）（英文版）》是随机分析方面的名著之一，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。本书主题广泛丰富，论述简洁易懂而又不失严密著称。
    本书读者对象：数学分析及金融数学专业的高年级本科生、研究生和研究人员。

Preface
1.Preliminaries From Calculus
  1.1  Fhnctions 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 It5 Integral
  4.2  It5 Integral Process
  4.3  It5 Integral and Gaussian Processes
  4.4  ItS's Formula for Brownian Motion
  4.5  It5 Processes and Stochastic Differentials
  4.6  ItS's Formula for It5 Processes
  4.7  It5 Processes in Higher Dimensions
  4.8  Exercises
5.Stochastic Differential Equations
  5.1  Definition of Stochastic Differential
  Equations (SDEs)
  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  Stratonovich 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  It6'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  Ito'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  Generators and Dynkin's Formula
  9.8  Explosions in Markov Jump Processes
  9.9  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 Derivatives 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 Options
  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  Growth of Birth-Death Processes
  13.5  Extinction, Probability, and Time to Exit
  13.6  Processes in Genetics
  13.7  Birth-Death Processes in Many Dimensions
  13.8  Cancer Models
  13.9  Branching Processes
  13.10  Stochastic Lotka-Volterra Model
  13.11  Exercises
14.Applications in Engineering and Physics
  14.1  Filtering
  14.2  Random Oscillators
  14.3  Exercises
Solutions to Selected Exercises
References
Index