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Numerical Methods in Computational Finance. Daniel J. Duffy
Читать онлайн.Название Numerical Methods in Computational Finance
Год выпуска 0
isbn 9781119719724
Автор произведения Daniel J. Duffy
Жанр Ценные бумаги, инвестиции
Издательство John Wiley & Sons Limited
2.1 INTRODUCTION AND OBJECTIVES
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 3 we continue with our discussion of ODEs, including code examples in C++ and Python.
2.2 BACKGROUND AND PROBLEM STATEMENT
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
In general, the problem (2.1) has a unique solution given by:
(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
2.2.1 Qualitative Properties of the Solution and Maximum Principle
Before we introduce difference schemes for (2.1), we discuss a number of results that allow us to describe how the solution
Lemma 2.1 (Positivity). Let the operator
Then the following result holds true: