Introduction to stochastic calculus with applications

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Introduction to stochastic calculus with applications / Fima C. Klebaner

版:: 3rd ed
資料種別:: 図書
出版情報:: London : Imperial College Press
Singapore : distributed by World Scientific Pub., c2012
形態:: xiv, 438 p. ; 24 cm
著者名:: Klebaner, Fima C <DA11758332>
ISBN:: 9781848168312 [1848168314] (: hbk)
9781848168329 [1848168322] (: pbk)
書誌ID:: BB08940780

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Introduction to stochastic calculus with applications. 2nd ed. (: ; : pbk)
Fima C. Klebaner , Imperial College Press , c2005

書誌詳細

注記:

Includes bibliographical references (p. 429-434) and index

主題:

Calculus; Stochastic analysis

言語:

英語

目次情報:

Preface

Preliminaries From Calculus / 1：

Functions in Calculus / 1.1：

Variation of a Function / 1.2：

Riemann Integral and Stieltjes Integral / 1.3：

Lebesgue's Method of Integration / 1.4：

Differentials and Integrals / 1.5：

Taylor's Formula and Other Results / 1.6：

Concepts of Probability Theory / 2：

Discrete Probability Model / 2.1：

Continuous Probability Model / 2.2：

Expectation and Lebesgue Integral / 2.3：

Transforms and Convergence / 2.4：

Independence and Covariance / 2.5：

Normal (Gaussian) Distributions / 2.6：

Conditional Expectation / 2.7：

Stochastic Processes in Continuous Time / 2.8：

Basic Stochastic Processes / 3：

Brownian Motion / 3.1：

Properties of Brownian Motion Paths / 3.2：

Three Martingales of Brownian Motion / 3.3：

Markov Property of Brownian Motion / 3.4：

Hitting Times and Exit Times / 3.5：

Maximum and Minimum of Brownian Motion / 3.6：

Distribution of Hitting Times / 3.7：

Reflection Principle and Joint Distributions / 3.8：

Zeros of Brownian Motion - Arcsine Law / 3.9：

Size of Increments of Brownian Motion / 3.10：

Brownian Motion in Higher Dimensions / 3.11：

Random Walk / 3.12：

Stochastic Integral in Discrete Time / 3.13：

Poisson Process / 3.14：

Exercises / 3.15：

Brownian Motion Calculus / 4：

Definition of Itô Integral / 4.1：

Itô Integral Process / 4.2：

Itô Integral and Gaussian Processes / 4.3：

Itô's Formula for Brownian Motion / 4.4：

Itô Processes and Stochastic Differentials / 4.5：

Itô's Formula for Ito Processes / 4.6：

Itô Processes in Higher Dimensions / 4.7：

Stochastic Differential Equations / 4.8：

Definition of Stochastic Differential Equations (SDEs) / 5.1：

Stochastic Exponential and Logarithm / 5.2：

Solutions to Linear SDEs / 5.3：

Existence and Uniqueness of Strong Solutions / 5.4：

Markov Property of Solutions / 5.5：

Weak Solutions to SDEs / 5.6：

Construction of Weak Solutions / 5.7：

Backward and Forward Equations / 5.8：

Stratonovich Stochastic Calculus / 5.9：

Diffusion Processes / 5.10：

Martingales and Dynkin's Formula / 6.1：

Calculation of Expectations and PDEs / 6.2：

Time-Homogeneous Diffusions / 6.3：

Exit Times from an Interval / 6.4：

Representation of Solutions of ODES / 6.5：

Explosion / 6.6：

Recurrence and Transience / 6.7：

Diffusion on an Interval / 6.8：

Stationary Distributions / 6.9：

Multi-dimensional SDEs / 6.10：

Martingales / 6.11：

Definitions / 7.1：

Uniform Integrability / 7.2：

Martingale Convergence / 7.3：

Optional Stopping / 7.4：

Localization and Local Martingales / 7.5：

Quadratic Variation of Martingales / 7.6：

Martingale Inequalities / 7.7：

Continuous Martingales - Change of Time / 7.8：

Calculus For Semimartingales / 7.9：

Semimartingales / 8.1：

Predictable Processes / 8.2：

Doob-Meyer Decomposition / 8.3：

Integrals with Respect to Semimartingales / 8.4：

Quadratic Variation and Covariation / 8.5：

Ito's Formula for Continuous Semimartingales / 8.6：

Local Times / 8.7：

Stochastic Exponential / 8.8：

Compensators and Sharp Bracket Process / 8.9：

Ito's Formula for Semimartingales / 8.10：

Martingale (Predictable) Representations / 8.11：

Elements of the General Theory / 8.13：

Random Measures and Canonical Decomposition / 8.14：

Pure Jump Processes / 8.15：

Pure Jump Process Filtration / 9.1：

Itô's Formula for Processes of Finite Variation / 9.3：

Counting Processes / 9.4：

Markov Jump Processes / 9.5：

Stochastic Equation for Jump Processes / 9.6：

Generators and Dynkin's Formula / 9.7：

Explosions in Markov Jump Processes / 9.8：

Change of Probability Measure / 9.9：

Change of Measure for Random Variables / 10.1：

Change of Measure on a General Space / 10.2：

Change of Measure for Processes / 10.3：

Change of Wiener Measure / 10.4：

Change of Measure for Point Processes / 10.5：

Likelihood Functions / 10.6：

Applications in Finance: Stock and FX Options / 10.7：

Financial Derivatives and Arbitrage / 11.1：

A Finite Market Model / 11.2：

Semimartingale Market Model / 11.3：

Diffusion and the Black-Scholes Model / 11.4：

Change of Numeraire / 11.5：

Currency (FX) Options / 11.6：

Asian, Lookback, and Barrier Options / 11.7：

Applications in Finance: Bonds, Rates, and Options / 11.8：

Bonds and the Yield Curve / 12.1：

Models Adapted to Brownian Motion / 12.2：

Models Based on the Spot Rate / 12.3：

Merton's Model and Vasicek's Model / 12.4：

Heath-Jarrow-Morton (HJM) Model / 12.5：

Forward Measures - Bond as a Numeraire / 12.6：

Options, Caps, and Floors / 12.7：

Brace-Gatarek-Musiela (BGM) Model / 12.8：

Swaps and Swaptions / 12.9：

Applications in Biology / 12.10：

Feller's Branching Diffusion / 13.1：

Wright-Fisher Diffusion / 13.2：

Birth-Death Processes / 13.3：

Growth of Birth-Death Processes / 13.4：

Extinction, Probability, and Time to Exit / 13.5：

Processes in Genetics / 13.6：

Birth-Death Processes in Many Dimensions / 13.7：

Cancer Models / 13.8：

Branching Processes / 13.9：

Stochastic Lotka-Volterra Model / 13.10：

Applications in Engineering and Physics / 13.11：

Filtering / 14.1：

Random Oscillators / 14.2：

Solutions to Selected Exercises / 14.3：

References

Index

Definition of Ito Integral

Ito Integral Process

Ito Integral and Gaussian Processes

Ito's Formula for Brownian Motion

Ito Processes and Stochastic Differentials

Ito's Formula for Ito Processes

Ito Processes in Higher Dimensions

Itô's Formula for Semimartingales

Ito's Formula for Processes of Finite Variation

Preface

Preliminaries From Calculus / 1：

Functions in Calculus / 1.1：

Variation of a Function / 1.2：

Riemann Integral and Stieltjes Integral / 1.3：

Lebesgue's Method of Integration / 1.4：

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