Introduction to Stochastic Calculus with Applications

ISBN-10: 1848168322
ISBN-13: 9781848168329
Edition: 3rd 2011
Authors: Fima C. Klebaner
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Description: This text presents a concise and rigorous treatment of stochastic calculus. It also gives its main applications in finance, biology and engineering.

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Book details

List price: $58.00
Edition: 3rd
Copyright year: 2011
Publisher: Imperial College Press
Publication date: 12/15/2011
Binding: Paperback
Pages: 470
Size: 6.00" wide x 9.00" long x 1.25" tall
Weight: 1.386
Language: English

This text presents a concise and rigorous treatment of stochastic calculus. It also gives its main applications in finance, biology and engineering.

Preface
Preliminaries From Calculus
Functions in Calculus
Variation of a Function
Riemann Integral and Stieltjes Integral
Lebesgue's Method of Integration
Differentials and Integrals
Taylor's Formula and Other Results
Concepts of Probability Theory
Discrete Probability Model
Continuous Probability Model
Expectation and Lebesgue Integral
Transforms and Convergence
Independence and Covariance
Normal (Gaussian) Distributions
Conditional Expectation
Stochastic Processes in Continuous Time
Basic Stochastic Processes
Brownian Motion
Properties of Brownian Motion Paths
Three Martingales of Brownian Motion
Markov Property of Brownian Motion
Hitting Times and Exit Times
Maximum and Minimum of Brownian Motion
Distribution of Hitting Times
Reflection Principle and Joint Distributions
Zeros of Brownian Motion - Arcsine Law
Size of Increments of Brownian Motion
Brownian Motion in Higher Dimensions
Random Walk
Stochastic Integral in Discrete Time
Poisson Process
Exercises
Brownian Motion Calculus
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
Exercises
Stochastic Differential Equations
Definition of Stochastic Differential Equations (SDEs)
Stochastic Exponential and Logarithm
Solutions to Linear SDEs
Existence and Uniqueness of Strong Solutions
Markov Property of Solutions
Weak Solutions to SDEs
Construction of Weak Solutions
Backward and Forward Equations
Stratonovich Stochastic Calculus
Exercises
Diffusion Processes
Martingales and Dynkin's Formula
Calculation of Expectations and PDEs
Time-Homogeneous Diffusions
Exit Times from an Interval
Representation of Solutions of ODES
Explosion
Recurrence and Transience
Diffusion on an Interval
Stationary Distributions
Multi-dimensional SDEs
Exercises
Martingales
Definitions
Uniform Integrability
Martingale Convergence
Optional Stopping
Localization and Local Martingales
Quadratic Variation of Martingales
Martingale Inequalities
Continuous Martingales - Change of Time
Exercises
Calculus For Semimartingales
Semimartingales
Predictable Processes
Doob-Meyer Decomposition
Integrals with Respect to Semimartingales
Quadratic Variation and Covariation
Ito's Formula for Continuous Semimartingales
Local Times
Stochastic Exponential
Compensators and Sharp Bracket Process
It�'s Formula for Semimartingales
Stochastic Exponential and Logarithm
Martingale (Predictable) Representations
Elements of the General Theory
Random Measures and Canonical Decomposition
Exercises
Pure Jump Processes
Definitions
Pure Jump Process Filtration
Ito's Formula for Processes of Finite Variation
Counting Processes
Markov Jump Processes
Stochastic Equation for Jump Processes
Generators and Dynkin's Formula
Explosions in Markov Jump Processes
Exercises
Change of Probability Measure
Change of Measure for Random Variables
Change of Measure on a General Space
Change of Measure for Processes
Change of Wiener Measure
Change of Measure for Point Processes
Likelihood Functions
Exercises
Applications in Finance: Stock and FX Options
Financial Derivatives and Arbitrage
A Finite Market Model
Semimartingale Market Model
Diffusion and the Black-Scholes Model
Change of Numeraire
Currency (FX) Options
Asian, Lookback, and Barrier Options
Exercises
Applications in Finance: Bonds, Rates, and Options
Bonds and the Yield Curve
Models Adapted to Brownian Motion
Models Based on the Spot Rate
Merton's Model and Vasicek's Model
Heath-Jarrow-Morton (HJM) Model
Forward Measures - Bond as a Numeraire
Options, Caps, and Floors
Brace-Gatarek-Musiela (BGM) Model
Swaps and Swaptions
Exercises
Applications in Biology
Feller's Branching Diffusion
Wright-Fisher Diffusion
Birth-Death Processes
Growth of Birth-Death Processes
Extinction, Probability, and Time to Exit
Processes in Genetics
Birth-Death Processes in Many Dimensions
Cancer Models
Branching Processes
Stochastic Lotka-Volterra Model
Exercises
Applications in Engineering and Physics
Filtering
Random Oscillators
Exercises
Solutions to Selected Exercises
References
Index

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