Daniel J. Duffy - Numerical Methods in Computational Finance

Then T has a unique fixed-point . Moreover, if is any point in X and the sequence is defined recursively by the formula , then and:

In general, we assume that X is a Banach space and that T is a linear or non-linear mapping of X into itself. We then say that x is a fixed point of T if .

In this chapter we gave an introduction to a number of relevant mathematical concepts from real analysis that are used throughout this book, directly or indirectly. We also have introduced other relevant topics in other chapters. To summarise:

Chapter 1: Real analysis

Chapter 4: Finite dimensional vector spaces

Chapter 5: Numerical linear algebra

Chapter 16: Complex analysis

In this way we hope that this book becomes more self-contained than otherwise.

It is better to solve one problem five different ways, than to solve five problems one way .

George Pólya.

In this chapter we introduce a class of differential equations in which the highest order derivative is one. Furthermore, these equations have a single independent variable (which in nearly all applications plays the role of time). In short, these are termed ordinary differential equations (ODEs) precisely because of the dependence on a single variable.

ODEs crop up in many application areas, such as mechanics, biology, engineering, dynamical systems, economics and finance, to name just a few. It is for this reason that we devote two dedicated chapters to them.

The following topics are discussed in this chapter:

Motivational examples of ODEs

Qualitative properties of ODEs

Common finite difference schemes for initial value problems for ODEs

Some theoretical foundations.

In Chapter 3we continue with our discussion of ODEs, including code examples in C++ and Python.

In this section we introduce the very first differential equation of this book. It is a scalar first-order linear ordinary differential equation (ODE), and we shall analyse it from several qualitative and quantitative viewpoints.

Consider a bounded interval where . This interval could represent time or distance, for example. In most cases we shall view this interval as representing time values. In the interval we define the initial value problem (IVP) for an ODE:

(2.1)

where is a first-order linear differential operator involving the derivative with respect to the time variable and is a strictly positive function in . The term is called the inhomogeneous forcing term , and it is independent of . Finally, the solution to the IVP must be specified at ; this is the so-called initial condition .

In general, the problem (2.1)has a unique solution given by:

(2.2)

(See Hochstadt (1964), where the so-called integration factor is used to determine a solution.)

A special case of (2.1)is when the right-hand term is zero and is constant; in this case the solution becomes a simple exponential term without any integrals, and this will be used later when we examine difference schemes to determine their feasibility. In particular, a scheme that behaves badly for the above special case will be unsuitable for more general or more complex problems unless some modifications are introduced.

Before we introduce difference schemes for (2.1), we discuss a number of results that allow us to describe how the solution behaves. First, we wish to conclude that if the initial value and inhomogeneous term are positive, then the solution should also be positive for any value in . This so-called positivity or monotonicity result should be reflected in our difference schemes (not all schemes possess this property). Second, we wish to know how the solution grows or decreases as a function of time. The following two results deal with these issues.