Introduction To Stochastic Calculus With Applications (3Rd Edition)

Contents
Preface
	Preface to the Third Edition
	Preface to the Second Edition
	Preface to the First Edition
	Acknowledgments
1. Preliminaries From Calculus
	1.1 Functions in Calculus
		Right- and Left-Continuous Functions
	1.2 Variation of a Function
		Quadratic Variation
	1.3 Riemann Integral and Stieltjes Integral
		Riemann Integral
		Stieltjes Integral
		Stieltjes Integral with Respect to Monotone Functions
		Particular Cases
		Impossibility of a Direct Definition of an Integral with Respect to Functions of Infinite Variation
		Integration by Parts
		Change of Variables
	1.4 Lebesgue’s Method of Integration
	1.5 Differentials and Integrals
	1.6 Taylor’s Formula and Other Results
		Taylor’s Formula for Functions of One Variable
		Taylor’s Formula for Functions of Several Variables
		Lipschitz and Holder Conditions
		Growth Conditions
		Solution of First Order Linear Differential Equations
		Further Results on Functions and Integration
2. Concepts of Probability Theory
	2.1 Discrete Probability Model
		Filtered Probability Space
		Sample Space
		Fields of Events
		Filtration
		Stochastic Processes
		Field Generated by a Random Variable
		Filtration Generated by a Stochastic Process
		Predictable Processes
		Stopping Times
		Probability
		Distribution of a Random Variable
		Expectation
		Conditional Probabilities and Expectations
		Conditional Expectation
	2.2 Continuous Probability Model
		σ-Fields
		Borel σ-Field
		Probability
		Lebesgue Measure
		Random Variables
		σ-Field Generated by a Random Variable
		Distribution of a Random Variable
		Joint Distribution
		Transformation of Densities
	2.3 Expectation and Lebesgue Integral
		Lebesgue–Stieltjes Integral
		Lebesgue Integral on the Line
		Properties of Expectation (Lebesgue Integral)
		Jumps and Probability Densities
		Decomposition of Distributions and FV Functions
	2.4 Transforms and Convergence
		Convergence of Random Variables
		Convergence of Expectations
	2.5 Independence and Covariance
		Independence
		Covariance
	2.6 Normal (Gaussian) Distributions
	2.7 Conditional Expectation
		Conditional Expectation and Conditional Distribution
		General Conditional Expectation
		Properties of Conditional Expectation
	2.8 Stochastic Processes in Continuous Time
		Continuity and Regularity of Paths
		σ-Field Generated by a Stochastic Process
		Filtered Probability Space and Adapted Processes
		The Usual Conditions
		Stopping Times
		Fubini’s Theorem
3. Basic Stochastic Processes
	Introduction
	3.1 Brownian Motion
		Defining Properties of Brownian Motion
		Transition Probability Functions
		Space Homogeneity
		Brownian Motion as a Gaussian Process
		Brownian Motion as a Random Series
	3.2 Properties of Brownian Motion Paths
		Quadratic Variation of Brownian Motion
		Properties of Brownian Paths
	3.3 Three Martingales of Brownian Motion
	3.4 Markov Property of Brownian Motion
		Stopping Times and Strong Markov Property
	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
		Graphs of Some Functions of Brownian Motion
	3.11 Brownian Motion in Higher Dimensions
	3.12 Random Walk
		Martingales in Random Walks
	3.13 Stochastic Integral in Discrete Time
		Stopped Martingales
	3.14 Poisson Process
		Defining Properties of Poisson Process
		Variation and Quadratic Variation of the Poisson Process
		Poisson Process Martingales
	3.15 Exercises
4. Brownian Motion Calculus
	4.1 Definition of Ito Integral
		Ito Integral of Simple Processes
		Properties of the Ito Integral of Simple Adapted Processes
		Ito Integral of Adapted Processes
	4.2 Ito Integral Process
		Martingale Property of the Ito Integral
		Quadratic Variation and Covariation of Ito Integrals
	4.3 Ito Integral and Gaussian Processes
	4.4 Ito’s Formula for Brownian Motion
	4.5 Ito Processes and Stochastic Differentials
		Definition of Ito Processes
		Quadratic Variation of Ito Processes
		Integrals With Respect to Ito Processes
	4.6 Ito’s Formula for Ito Processes
		Integration by Parts
		Ito’s Formula for Functions of Two Variables
	4.7 Ito Processes in Higher Dimensions
		Ito’s Formula for Functions of Several Variables
	4.8 Exercises
5. Stochastic Differential Equations
	5.1 Definition of Stochastic Differential Equations (SDEs)
		Ordinary Differential Equations (ODEs)
		White Noise and SDEs
		A Physical Model of Diffusion and SDEs
		Stochastic Differential Equations
		Stochastic and Random Ordinary Differential Equations (ODEs)
	5.2 Stochastic Exponential and Logarithm
		Stochastic Logarithm
	5.3 Solutions to Linear SDEs
		Stochastic Exponential SDEs
		General Linear SDEs
		Langevin-Type SDE
		Brownian Bridge
	5.4 Existence and Uniqueness of Strong Solutions
		Less Stringent Conditions for Strong Solutions
	5.5 Markov Property of Solutions
		Transition Function
	5.6 Weak Solutions to SDEs
	5.7 Construction of Weak Solutions
		Canonical Space for Diffusions
		Probability Space ( , F, IF)
		Probability Measure
		Transition Function
		SDE on the Canonical Space is Satisfied
		Weak Solutions and the Martingale Problem
	5.8 Backward and Forward Equations
	5.9 Stratonovich Stochastic Calculus
		Integration by Parts: Stratonovich Product Rule
		Change of Variables: Stratonovich Chain Rule
		Conversion of Stratonovich SDEs into Ito SDEs
	5.10 Exercises
6. Diffusion Processes
	6.1 Martingales and Dynkin’s Formula
	6.2 Calculation of Expectations and PDEs
		Backward PDE and E g(X(T ))|X(t) = x
		Feynman–Kac Formula
	6.3 Time-Homogeneous Diffusions
		Ito’s Formula and Martingales
	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
		Invariant Measures
	6.10 Multi-dimensional SDEs
		Bessel Process
		Ito’s Formula and Dynkin’s Formula
		Higher Order Random Differential Equations
	6.11 Exercises
7. Martingales
	7.1 Definitions
		Square Integrable Martingales
	7.2 Uniform Integrability
	7.3 Martingale Convergence
	7.4 Optional Stopping
		Optional Stopping of Discrete Time Martingales
		Gambler’s Ruin
		Hitting Times in Random Walks
	7.5 Localization and Local Martingales
	7.6 Quadratic Variation of Martingales
	7.7 Martingale Inequalities
		Application to Itˆo Integrals
	7.8 Continuous Martingales — Change of Time
		Levy’s Characterization of Brownian Motion
		Change of Time for Martingales
		Change of Time in SDEs
	7.9 Exercises
8. Calculus For Semimartingales
	8.1 Semimartingales
	8.2 Predictable Processes
	8.3 Doob–Meyer Decomposition
		Doob’s Decomposition
	8.4 Integrals with Respect to Semimartingales
		Stochastic Integral With Respect to Martingales
		Properties of Stochastic Integrals With Respect to Martingales
		Stochastic Integrals With Respect to Semimartingales
		Properties of Stochastic Integrals With Respect to Semimartingales
	8.5 Quadratic Variation and Covariation
		Properties of Quadratic Variation
		Quadratic Variation of Stochastic Integrals
	8.6 Ito’s Formula for Continuous Semimartingales
		Ito’s Formula for Functions of Several Variables
	8.7 Local Times
	8.8 Stochastic Exponential
		Stochastic Exponential of Martingales
	8.9 Compensators and Sharp Bracket Process
		Sharp Bracket for Square Integrable Martingales
		Continuous Martingale Component of a Semimartingale
		Conditions for Existence of a Stochastic Integral
		Properties of the Predictable Quadratic Variation
	8.10 Ito’s Formula for Semimartingales
	8.11 Stochastic Exponential and Logarithm
	8.12 Martingale (Predictable) Representations
	8.13 Elements of the General Theory
		Remarks:
		Stochastic Sets
		Classification of Stopping Times
	8.14 Random Measures and Canonical Decomposition
		Random Measure for a Single Jump
		Random Measure of Jumps and its Compensator in Discrete Time
		Random Measure of Jumps and its Compensator
	8.15 Exercises
9. Pure Jump Processes
	9.1 Definitions
	9.2 Pure Jump Process Filtration
		Assumptions
	9.3 Ito’s Formula for Processes of Finite Variation
		Stochastic Exponential
		Integration by Parts for Processes of Finite Variation
	9.4 Counting Processes
		Point Process of a Single Jump
		Compensators of Counting Processes
		Renewal Process
		Stochastic Intensity
		Non-Homogeneous Poisson Processes
		Compensators of Pure Jump Processes
	9.5 Markov Jump Processes
		Definitions
		The Compensator and the Martingale
	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
		Change of Measure on a Discrete Probability Space
		Change of Measure for Normal Random Variables
	10.2 Change of Measure on a General Space
	10.3 Change of Measure for Processes
		Change of Drift in Diffusions
	10.4 Change of Wiener Measure
	10.5 Change of Measure for Point Processes
	10.6 Likelihood Functions
		Likelihood for Discrete Observations
		Likelihood Ratios for Diffusions
	10.7 Exercises
11. Applications in Finance: Stock and FX Options
	11.1 Financial Derivatives and Arbitrage
		Equivalence Portfolio. Pricing by No Arbitrage
		Binomial Model
		Pricing by No Arbitrage
	11.2 A Finite Market Model
	11.3 Semimartingale Market Model
		Arbitrage in Continuous Time Models
		EMM Assumption
		Admissible Strategies
		Pricing of Claims
		Completeness of a Market Model
	11.4 Diffusion and the Black–Scholes Model
		Black–Scholes Model
		Pricing a Call Option
		Pricing of Claims by a PDE. Replicating Portfolio
		Validity of the Assumptions
		Implied Volatility
		Stochastic Volatility Models
	11.5 Change of Numeraire
		A General Option Pricing Formula
		SDEs Under a Change of Numeraire
	11.6 Currency (FX) Options
		Options on Foreign Currency
		Options on Foreign Assets Struck in Foreign Currency
		Guaranteed Exchanged Rate (Quanto) Options
	11.7 Asian, Lookback, and Barrier Options
		Asian Options
		Lookback Options
		Barrier Options
	11.8 Exercises
12. Applications in Finance: Bonds, Rates, and Options
	12.1 Bonds and the Yield Curve
		EMM Assumption
	12.2 Models Adapted to Brownian Motion
	12.3 Models Based on the Spot Rate
	12.4 Merton’s Model and Vasicek’s Model
		Vasicek’s Model
	12.5 Heath–Jarrow–Morton (HJM) Model
		EMM Assumption
		Bonds and Rates Under Q and the No-Arbitrage Condition
	12.6 Forward Measures — Bond as a Numeraire
		Options on a Bond
		Forward Measures
		Forward Measure in HJM
		Distributions of the Bond in HJM with Deterministic Volatilities
	12.7 Options, Caps, and Floors
		Cap and Caplets
		Caplet as a Put Option on the Bond
		Caplet Pricing in the HJM Model
	12.8 Brace–Gatarek–Musiela (BGM) Model
		LIBOR
		Caplet in BGM
		SDEs for Forward LIBOR Under Different Measures
		Choice of Bond Volatilities
	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
		Definition
		Stochastic Equation
		Generator of a Birth–Death Process
		Forward Kolmogorov Equations
		Stationary Distribution
	13.4 Growth of Birth–Death Processes
		Growth of Birth–Death Processes with Linear Rates
		Growth of Processes with Stabilizing Reproduction
	13.5 Extinction, Probability, and Time to Exit
		Probability of Extinction
		Mean Exit Time
		Exit Probabilities
	13.6 Processes in Genetics
		Wright–Fisher Model
		Moran Model in Discrete Time
		Neutral drift
		Mutation
		Moran Model in Continuous Time
		Stochastic Equation and Its Limit
		Moran Model with Selective Advantage
	13.7 Birth–Death Processes in Many Dimensions
	13.8 Cancer Models
		Cancer Development
		Oncogene Model
		Two-Hit Model
		Tumor–Host Interaction
	13.9 Branching Processes
	13.10 Stochastic Lotka–Volterra Model
		Deterministic Lotka–Volterra System
		Stochastic Lotka–Volterra System
		Existence
		Deterministic (Fluid) Approximation
	13.11 Exercises
14. Applications in Engineering and Physics
	14.1 Filtering
		General Non-linear Filtering Model
		Filtering of Diffusions
		Kalman–Bucy Filter
	14.2 Random Oscillators
		Non-linear Systems
		Duffing Equation
		Random Oscillator
		A System with a Cylindric Phase Plane
	14.3 Exercises
Solutions to Selected Exercises
References
Index