Daniel J. Duffy - Numerical Methods in Computational Finance

Paul Halmos

In Chapter 2we 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.

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 3.1Let 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 can be proved that the solution of (3.1)is also the solution of (3.5)and vice versa. We see then that Picard iteration is based on (3.5)and that we wish to have the iterates converging to a solution of (3.5).

We note that the Picard method is used primarily to prove the existence of solutions and it is not a numerical method as such.

We take a simple autonomous non-linear scalar ODE to show how to calculate Picard iterates:

(3.6)

whose solution is given by:

We now compute the Picard iterates (3.4)for this ODE in order to determine the values of t for which the ODE has a solution. For convenience, let us take . Some simple integration shows that:

(3.7)

We can see that the series is beginning to look like . We know that this series is convergent for . A nice exercise is to compute the Picard iterates in the most general case (that is, ) and to determine under which circumstances the ODE (3.6)has a solution. In this case we have represented the solution of an ODE as a series, and we then analysed this series for which there are many convergence results, such as the root test and the ratio test .

We take some model ODEs for motivation.

The Bernoulli ODE is named after Jacob Bernoulli. It is special in the sense that it is a non-linear equation having an exact solution: