This textbook, now in its fourth edition, offers a rigorous and self-contained introduction to the theory of continuous-time stochastic processes, stochastic integrals, and stochastic differential equations. Expertly balancing theory and applications, it features concrete examples of modeling real-world problems from biology, medicine, finance, and insurance using stochastic methods. No previous knowledge of stochastic processes is required. Unlike other books on stochastic methods that specialize in a specific field of applications, this volume examines the ways in which similar stochastic methods can be applied across different fields.
Beginning with the fundamentals of probability, the authors go on to introduce the theory of stochastic processes, the Itô Integral, and stochastic differential equations. The following chapters then explore stability, stationarity, and ergodicity. The second half of the book is dedicated to applications to a variety of fields, including finance, biology, and medicine. Some highlights of this fourth edition include a more rigorous introduction to Gaussian white noise, additional material on the stability of stochastic semigroups used in models of population dynamics and epidemic systems, and the expansion of methods of analysis of one-dimensional stochastic differential equations.
An Introduction to Continuous-Time Stochastic Processes, Fourth Edition is intended for graduate students taking an introductory course on stochastic processes, applied probability, stochastic calculus, mathematical finance, or mathematical biology. Prerequisites include knowledge of calculus and some analysis; exposure to probability would be helpful but not required since the necessary fundamentals of measure and integration are provided. Researchers and practitioners in mathematical finance, biomathematics, biotechnology, and engineering will also find this volume to be of interest, particularly the applications explored in the second half of the book.
Author(s): Vincenzo Capasso, David Bakstein
Series: Modeling and Simulation in Science, Engineering and Technology
Edition: 4
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 560
Tags: Stochastic Processes, Ito Integral, Stochastic Differential Equations, Stability, Stationarity, Ergodicity
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
