Numerical Methods in Computational Finance. Daniel J. Duffy

CHAPTER 3 Ordinary Differential Equations (ODEs), Part 2

      The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.

      Paul Halmos

3.1 INTRODUCTION AND OBJECTIVES

      In Chapter 2 we discussed both systems of ODEs and scalar ODEs. The focus was mainly concerned with notation, the structure of ODEs and finite difference schemes to approximate them. We implicitly assumed that the solution of the corresponding initial value problem existed in an otherwise unspecified time interval and that the solution was unique. These assumptions constitute a huge leap of faith. In this chapter we discuss existence and uniqueness results for ODEs and stochastic differential equation (SDEs). We also introduce several important numerical schemes and code in C++ and Python.

3.2 EXISTENCE AND UNIQUENESS RESULTS

      We turn our attention to a more general initial value problem for a non-linear system of ODEs:

(3.1)

      where:

Some of the important questions to be answered are:

       Does System (3.1) have a unique solution?

       In which interval does this solution exist?

       What is the asymptotic behaviour of the solution as ?

      To this end, let B be a region of dimensional space, and let f be continuously differentiable with respect to t and with respect to all the components of y at all points of B. We assume that the following inequalities hold:

(3.2)

      and:

(3.3)

Theorem 3.1 Let f and be continuous in the box where a and b are positive numbers and satisfying the bounds (3.2) and (3.3) for (t, y) in B. Let be the smaller of the numbers a and b/M and define the successive approximations:

(3.4)

      Then the sequence of successive approximations converges (uniformly) in the interval to a solution of (3.1) that satisfies the initial condition .

Method (3.4) is called the Picard iterative method and it is used to prove the existence of the solution of systems of ODE (3.1). It is mainly of theoretical value, as it should not necessarily be seen as a practical way to construct a numerical solution. However, it does give us insights into the qualitative properties of the solution. On the other hand, it is a useful exercise to construct the sequence of iterates in Equation (3.4) for some simple cases.

      We note that the IVP (3.1) can be written as an integral equation as follows:

(3.5)

It