An Introduction to Continuous-Time Stochastic Processes

Foreword
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
Part I Theory of Stochastic Processes
1 Fundamentals of Probability
	1.1 Probability and Conditional Probability
	1.2 Random Variables and Distributions
		1.2.1 Random Vectors
	1.3 Independence
	1.4 Expectations
		1.4.1 Mixing inequalities
		1.4.2 Characteristic Functions
	1.5 Gaussian Random Vectors
	1.6 Conditional Expectations
	1.7 Conditional and Joint Distributions
	1.8 Convergence of Random Variables
	1.9 Infinitely Divisible Distributions
		1.9.1 Examples
	1.10 Stable Laws
	1.11 Martingales
	1.12 Exercises and Additions
2 Stochastic Processes
	2.1 Definition
	2.2 Stopping Times
	2.3 Canonical Form of a Process
	2.4 L2 Processes
		2.4.1 Gaussian Processes
		2.4.2 Karhunen–Loève Expansion
	2.5 Markov Processes
		2.5.1 Markov Diffusion Processes
	2.6 Processes with Independent Increments
	2.7 Martingales
		2.7.1 The martingale property of Markov processes
		2.7.2 The martingale problem for Markov processes
	2.8 Brownian Motion and the Wiener Process
	2.9 Counting and Poisson Processes
	2.10 Random Measures
		2.10.1 Poisson random measures
	2.11 Marked Counting Processes
		2.11.1 Counting Processes
		2.11.2 Marked Counting Processes
		2.11.3 The Marked Poisson Process
		2.11.4  Time-space Poisson Random Measures
	2.12 White Noise
		2.12.1 Gaussian white noise
		2.12.2 Poissonian white noise
	2.13 Lévy Processes
	2.14 Exercises and Additions
3 The Itô Integral
	3.1 Definition and Properties
	3.2 Stochastic Integrals as Martingales
	3.3 Itô Integrals of Multidimensional Wiener Processes
	3.4 The Stochastic Differential
	3.5 Itô's Formula
	3.6 Martingale Representation Theorem
	3.7 Multidimensional Stochastic Differentials
	3.8 The Itô Integral with Respect to Lévy Processes
	3.9 The Itô–Lévy Stochastic Differential  and the Generalized Itô Formula
	3.10 Fractional Brownian Motion
		3.10.1 Integral with respect to a fBm
	3.11 Exercises and Additions
4 Stochastic Differential Equations
	4.1 Existence and Uniqueness of Solutions
	4.2 Markov Property of Solutions
	4.3 Girsanov Theorem
	4.4 Kolmogorov Equations
	4.5 Multidimensional Stochastic Differential Equations
		4.5.1 Multidimensional diffusion processes
		4.5.2 The time-homogeneous case
	4.6 Applications of Itô's Formula
		4.6.1 First Hitting Times
		4.6.2 Exit Probabilities
	4.7 Itô–Lévy Stochastic Differential Equations
		4.7.1 Markov Property of Solutions of Itô–Lévy Stochastic Differential Equations
	4.8 Exercises and Additions
5 Stability, Stationarity, Ergodicity
	5.1 Time of explosion and regularity
		5.1.1 Application: A Stochastic Predator-Prey model
		5.1.2 Recurrence and transience
	5.2 Stability of Equilibria
	5.3 Stationary distributions
		5.3.1 Existence of a stationary distribution—Ergodic theorems
	5.4 Stability of invariant measures
	5.5 The one-dimensional case
		5.5.1 Invariant distributions
		5.5.2 First passage times
		5.5.3 Ergodic theorems
	5.6 Exercises and Additions
Part II Applications of Stochastic Processes
6 Applications to Finance and Insurance
	6.1 Arbitrage-Free Markets
	6.2 The Standard Black–Scholes Model
	6.3 Models of Interest Rates
	6.4 Extensions and Alternatives to Black–Scholes
	6.5 Insurance Risk
	6.6 Exercises and Additions
7 Applications to Biology and Medicine
	7.1 Population Dynamics: Discrete-in-Space–Continuous-in-Time Models
		7.1.1 Inference for Multiplicative Intensity Processes
	7.2 Population Dynamics: Continuous Approximation of Jump Models
		7.2.1 Deterministic Approximation: Law of Large Numbers
		7.2.2 Diffusion Approximation: Central Limit Theorem
	7.3 Population Dynamics: Individual-Based Models
		7.3.1 A Mathematical Detour
		7.3.2 A ``Moderate'' Repulsion Model
		7.3.3 Ant Colonies
		7.3.4 Price Herding
	7.4 Tumor-driven angiogenesis
		7.4.1 The capillary network
	7.5 Neurosciences
	7.6 Evolutionary biology
	7.7 Stochastic Population models
		7.7.1 Logistic population growth
		7.7.2 Stochastic Prey–Predator Models
		7.7.3 An SIS Epidemic Model
		7.7.4 A stochastic SIS Epidemic model with two correlated environmental noises
		7.7.5 A vector-borne epidemic system
		7.7.6 Stochastically perturbed SIR and SEIR epidemic models
		7.7.7 Environmental noise models
	7.8 Exercises and Additions
A Measure and Integration
	A.1  Rings and σ-Algebras
	A.2  Measurable Functions and Measures
	A.3  Lebesgue Integration
	A.4  Lebesgue–Stieltjes Measure and Distributions
	A.5  Radon Measures
	A.6  Signed measures
	A.7  Stochastic Stieltjes Integration
B Convergence of Probability Measures on Metric Spaces
	B.1  Metric Spaces
	B.2  Prohorov's Theorem
	B.3  Donsker's Theorem
C Diffusion Approximation of a Langevin System
D Elliptic and Parabolic Equations
	D.1  Elliptic Equations
	D.2  The Cauchy Problem and Fundamental Solutions for Parabolic Equations
E Semigroups of Linear Operators
	E.1  Markov transition kernels
		E.1.1 Feller semigroups
		E.1.2 Hille–Yosida Theorem for Feller semigroups
F Stability of Ordinary Differential Equations
References
Index