隨機分析及應(yīng)用

內(nèi)容概要

本書介紹了隨機分析的理論和應(yīng)用兩方面的知識。內(nèi)容涉及積分和概率論的基礎(chǔ)知識、基本的隨機過程，布朗運動和伊藤過程的積分、隨機微分方程、半鞅積分、純離散過程，以及隨機分析在金融、生物、工程和物理等方面的應(yīng)用。書中有大量的例題和習(xí)題，并附有答案，便于讀者進行深層次的學(xué)習(xí)。    本書非常適合初學(xué)者閱讀，可作為高等院校經(jīng)管、理工和社科類各專業(yè)高年級本科生和研究生隨機分析和金融數(shù)學(xué)的教材，也可供相關(guān)領(lǐng)域的科研人員參考。

作者簡介

Fima C Klebaner，澳夫利亞Monash(莫納什)大學(xué)教授，IMS(國際數(shù)理統(tǒng)計學(xué)會)會士，著名數(shù)理統(tǒng)計和金融數(shù)學(xué)家。主要研究領(lǐng)域有：隨饑過程、概率應(yīng)用、隨機分析、金融數(shù)學(xué)、動態(tài)系統(tǒng)的隨機擾動等。

書籍目錄

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 Results2  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 Time3  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  Exercises4  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  Exercises5  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  Exercises6  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 Exercises7  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  Exercises8  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  Exercises9  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  Exercises10  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  Exercises11  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  Exercises12  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  Exercises13  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  Exercises14  Applications in Engineering and Physics    14.1  Filtering    14.2  Random Oscillators    14.3  ExercisesSolutions to Selected ExercisesReferencesIndex

編輯推薦

《隨機分析及應(yīng)用(英文版)(第2版)》是隨機分析方面的名著之一。以主題廣泛豐富，論述簡潔易懂而又不失嚴(yán)密著稱。書中闡述了各領(lǐng)域的典型應(yīng)用，包括數(shù)理金融、生物學(xué)、工程學(xué)中的模型。還提供了很多示例和習(xí)題，并附有解答。第2版增加了講述證券，利率及其期權(quán)的一章，并在全書增加了許多新內(nèi)容，以反映隨機分析研究和應(yīng)用的最新成果。《隨機分析及應(yīng)用(英文版)(第2版)》可作為高年級本科生和研究生的隨機分析和金融數(shù)學(xué)的教材，也非常適合各領(lǐng)域?qū)I(yè)人士自學(xué)?！峨S機分析及應(yīng)用(英文版)(第2版)》內(nèi)容簡潔，闡述透徹，包含豐富的例子，并有精彩解答。——Robert Liptser教授，以色列特拉維夫大學(xué)在講述隨機分析的著作中，像《隨機分析及應(yīng)用(英文版)(第2版)》這樣涵蓋廣泛而又具有很強可讀性的實屬罕見?！狹athematical Reviews

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