Stochastic Differential Equations: An Introduction with Applications

Citation

Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications (6th ed.). Springer.

Chapter Summary

The book develops the theory of stochastic differential equations driven by Brownian motion, with complete proofs, and then applies it to a series of problems: filtering, boundary value problems for partial differential equations, optimal stopping, stochastic control and mathematical finance. It went through six editions, each reviewed in zbMATH as it grew; the sixth edition has twelve chapters, appendices, solutions and additional hints to some of the exercises, references and a list of notation. The first eight chapters form the theoretical core and Chapters 9 to 12 are applications.

Chapter 1: Introduction

A motivating chapter of five pages that introduces the problems the book will solve. The examples include population growth in a randomly varying environment, electrical circuits with noisy input and option pricing.

Chapter 2: Some Mathematical Preliminaries

Probability spaces, random variables and stochastic processes, and the construction and basic properties of Brownian motion. At fourteen pages it is a concise review, not a course in measure-theoretic probability.

Chapter 3: Itô Integrals

The construction of the Itô integral with respect to Brownian motion, its properties and its extensions. It is the foundation for everything that follows.

Chapter 4: The Itô Formula and the Martingale Representation Theorem

The Itô formula, the stochastic counterpart of the chain rule, in one and several dimensions, followed by the Martingale Representation Theorem.

Chapter 5: Stochastic Differential Equations

Worked examples and solution methods, existence and uniqueness theorems, and the distinction between strong and weak solutions.

Chapter 6: The Filtering Problem

Estimating the state of a system from noisy observations. The chapter treats the linear filtering problem and the Kalman–Bucy filter.

Chapter 7: Diffusions: Basic Properties

The Markov and strong Markov properties of Itô diffusions, their generators and the Dynkin formula.

Chapter 8: Other Topics in Diffusion Theory

Further tools for diffusions: Kolmogorov's backward equation, the Feynman–Kac formula, random time change and the Girsanov transformation.

Chapters 9–10: Applications to Boundary Value Problems; Application to Optimal Stopping

Chapter 9 gives stochastic solutions of the Dirichlet and Poisson problems for partial differential equations. Chapter 10 treats optimal stopping problems, including their treatment by variational inequalities.

Chapters 11–12: Application to Stochastic Control; Application to Mathematical Finance

Chapter 11 formulates control problems in terms of stochastic differential equations and uses the Hamilton–Jacobi–Bellman equation. Chapter 12 applies the theory to a financial market driven by Brownian motion, with option pricing for complete markets and attainable options.

Appendices

The appendices cover normal random variables, conditional expectation, uniform integrability and martingale convergence, and an approximation result.

Key Concepts

Chapters 2–5: Stochastic Calculus

• Brownian motion: the continuous random path that drives the equations of the book.
• Itô integral: an integral with respect to Brownian motion, defined for suitably non-anticipating integrands.
• Itô formula: the change-of-variables rule for functions of Itô processes.
• Strong and weak solutions: two notions of what it means to solve a stochastic differential equation.

Chapters 6–8: Filtering and Diffusions

• Kalman–Bucy filter: the optimal estimator of a linear system from noisy observations.
• Generator and Dynkin formula: tools linking diffusions to partial differential equations.
• Girsanov transformation: a change of measure that alters the drift of a diffusion.

Chapters 9–12: Applications

• Dirichlet and Poisson problems: classical boundary value problems represented by diffusions.
• Optimal stopping and stochastic control: choosing a stopping time or a control to optimise an expected reward, with the Hamilton–Jacobi–Bellman equation.
• Option pricing: valuation of financial claims in a Brownian market.

Critical Analysis

Strengths

• Rigour and accessibility together. Evelyn Buckwar, reviewing the sixth edition in zbMATH, says the presentation is successfully balanced between being easily accessible to a broad audience and being mathematically rigorous, with complete proofs, and calls the book a first choice for graduate courses in applied stochastic differential equations.
• A wide range of applications. One book takes the reader from the Itô integral to filtering, boundary value problems, optimal stopping, control and finance. Earlier reviews describe the same combination: Manfred Breger called the third edition an excellent elementary introduction that follows its stated programme.
• Support for self-study. The book includes solutions and additional hints to some of the exercises (about thirty pages), and Buckwar notes that the detailed solutions to many exercises make the sixth edition useful for independent study.
• Refined over six editions. Reviewers recorded the growth of the book: the later editions added martingale representation, optimal stopping by variational inequalities and a chapter on mathematical finance.

Limitations

• Prerequisites are significant. The preliminaries chapter is short, and the appendices on conditional expectation and martingale convergence exist to supply what the main text uses. A reader without a measure-theoretic probability course will need to prepare first.
• Brownian-motion setting. The theory is built on Brownian motion; the book is not a general treatment of semimartingales or jump processes.
• Finance is one chapter. Mathematical finance appears in Chapter 12 only, so a reader who wants a finance-focused course will need a dedicated text.
• Compact treatment of applications. Each application chapter is a short introduction to a large field (control, optimal stopping, partial differential equations), so further reading is needed to go deep.

Where It Fits in the Course

This is the main text for the stochastic-calculus end of the stochastic processes course. Sheldon Ross's Introduction to Probability Models and Robert Dobrow's Introduction to Stochastic Processes with R come first, covering Markov chains and Poisson processes, and Dobrow ends with a gentle look at stochastic calculus. Steven Shreve's Stochastic Calculus for Finance I builds the ideas in discrete time using the binomial model without measure theory, a useful bridge before Øksendal. Sidney Resnick's Adventures in Stochastic Processes treats Markov chains, renewal processes, point processes, Brownian motion and random walks.

For finance, Tomas Björk's Arbitrage Theory in Continuous Time and Lamberton and Lapeyre's Introduction to Stochastic Calculus Applied to Finance spend much more time on pricing and arbitrage than Øksendal's single chapter, and are natural companions or follow-ups. David Williams's Probability with Martingales is the sensible preparation in measure-theoretic probability.

How to Use This Book

The book is for graduate students in probability, statistics and related fields, and for others with a strong probability background. It needs real analysis, probability at the level of conditional expectation and martingales, and some familiarity with differential equations; Chapter 9 also draws on partial differential equations.

Read Chapters 1 and 2 quickly, then Chapters 3 to 5 carefully, since the Itô integral, the Itô formula and the existence and uniqueness theorems are the core. Chapters 7 and 8 introduce the machinery of diffusions that the applications use. After that, choose among Chapters 6 and 9 to 12 according to interest: filtering and control, boundary value problems, or Chapter 12 for finance. Consult the appendices when conditional expectation or martingale convergence is needed.

Do the exercises and use the solutions and hints at the back to check them.