An Introduction to the Numerical Simulation of Stochastic Differential Equations

The study of stochastic differential equations (SDEs) has developed over the last several years from a specialty to a subject of more general interest. The current book is designed to present a self-contained accessible introduction for undergraduate and beginning graduate students that teaches the fundamentals of the numerical solution and simulation of SDEs as succinctly as possible. The authors’ reach is broad: they hope the book will appeal to students in mathematics, statistics, finance and and the sciences as well as researchers in those fields.

 

In principle only competence with first-year calculus is required, but the authors suggest that a basic acquaintance with numerical analysis and probability would be desirable. This is not a rigorous text and the authors are careful to make readers aware that they’re not getting the full story.  Nonetheless, sketches of proofs for some of the major theorems are included.  The authors also provide many pointers to the technical literature and an extensive bibliography. Throughout the book a number of applications are addressed. There is a considerable variety, ranging from financial analysis to the physics of phase transitions and chemical kinetics.

 

A stochastic differential equation can be represented symbolically as

 

\( dX(t) =f (X(t))dt + g(X(t))dW (t) \)

 

where \( f \) and \( g \) are real functions, \( W(t) \) is an input stochastic process and \( X(t) \) is an output stochastic process. \( X(t) \) is a solution of this SDE if it solves the integral equation

 

\( X(t)-X(0)= \int_{0}^{t} f(X(s)) ds + \int_{0}^{t} g(X(s)) dW(s) \).  

 

The second integral is an Itô stochastic integral.  The function \( f \) is is commonly called the ”drift” coefficient, and \( g \) is called the ”diffusion” coefficient. Since \( W (t) \) is a stochastic process such as Brownian motion, it is not differentiable and \( \frac{dW(t)}{dt} \) has no meaning. So the symbolic representation of the SDE is really only a shorthand for the integral equation.  A simple example of an SDE is linear with multiplicative noise:

 

\( dX(t) = μX(t)dt + σX(t)dW (t) \)

 

where \( \mu \) and \( \sigma \) are real constants. Versions of this equation are used to model the evolution of stock prices in the Black-Scholes theory of stock option valuation.

 

Much of the preliminary work that the authors must do is to fill in the background with discussions of random variables, stochastic processes, Brownian motion and the Itô integral. The Itô integral is a generalization of the Riemann integral to stochastic functions. It is the most commonly used integral for stochastic functions, but the authors also briefly discuss an alternative, the Stratonovich integral, and later compare the two.

 

The hard work begins with a development of the Euler-Maruyama method for simulating SDEs. This is the simplest and most widely-used technique.  The authors then introduce the concepts of weak and strong convergence and begin discussing issues of numerical stability. From here on things get more complicated and more challenging. More advanced topics include computing mean exit times for a process in a Monte Carlo setting using SDE approximation, the use of higher order methods, and systems of SDEs.

 

This is one of the first books to appear that aims to bring the subject of stochastic differential equations to undergraduate students and researchers in other fields. It is a worthy attempt, but it poses serious challenges to readers new to the subject. The authors provide exercises, both theoretical and computational, supporting MATLAB code, and extensive notes and references.  The early chapters are more accessible to novices than the later ones. Overall, the book is likely to be most appropriate for well-prepared undergraduates, graduate students, and researchers in related fields who have some prior experience.

 

A related book is An Introduction to Stochastic Differential Equations by Lawrence C. Evans.  Its focus is more on development of the theory of SDEs and it does not consider any computational or numerical questions.  It is aimed at a similar set of readers, but it is no less challenging.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films, material science and the odd bit of high performance computing. He did his PhD work in dynamical systems and celestial mechanics.