内容简介

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。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。

作者简介

Fima C. Klebaner （F. C. 克莱巴纳）是世界百强名校，澳大利亚学府，莫纳什大学（Monash University）知名教授。

章节目录

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

随机分析及其应用 第3版是2018年由世界图书出版公司出版,作者[澳]F.。

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