随机分析及其应用(第3版影印版)

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