Daniel J. Duffy - Numerical Methods in Computational Finance

Let us take the example:

You can check that the Jacobian is given by:

Solving for x and y gives:

You need to be comfortable with partial derivatives. A good reference is Widder (1989).

Section 1.6may be skipped on a first reading without loss of continuity.

We work with sets and other mathematical structures in which it is possible to assign a so-called distance function or metric between any two of their elements. Let us suppose that X is a set, and let x , y and z be elements of X . Then a metric d on X is a non-negative real-valued function of two variables having the following properties:

The concept of distance is a generalisation of the difference between two real numbers or the distance between two points in n -dimensional Euclidean space, for example.

Having defined a metric d on a set X , we then say that the pair ( X , d ) is a metric space . We give some examples of metrics and metric spaces:

1 We define the set X of all continuous real-valued functions of one variable on the interval [a, b] (we denote this space by C[a, b])), and we define the metric:Then (X, d) is a metric space.

2 n-dimensional Euclidean space, consisting of vectors of real or complex numbers of the form:with metric:

3 Let be the space of all square-integrable functions on the interval [a, b]:We can then define the distance between two functions f and g in this space by the metric:This metric space is important in many branches of mathematics, including probability theory and stochastic calculus.

4 Let X be a non-empty set and let the metric d be defined by:Then (X, d) is a metric space.

Many of the results and theorems in mathematics are valid for metric spaces, and this fact means that the same results are valid for all specialisations of these spaces.

We define the concept of convergence of a sequence of elements of a metric space X to some element that may or may not be in X . We introduce some definitions that we state for the set of real numbers, but they are valid for any ordered field , which is basically a set of numbers for which every non-zero element has a multiplicative inverse and there is a certain ordering between the numbers in the field.

Definition 1.4A sequence of elements on the real line is said to be convergent if there exists an element such that for each positive element in there exists a positive integer such that:

A simple example is to show that the sequence converges to 0. To this end, let be a positive real number. Then there exists a positive integer such that whenever .

Definition 1.5A sequence of elements of an ordered field F is called a Cauchy sequence if for each in F there exists a positive integer such that:

In other words, the terms in a Cauchy sequence get close to each other while the terms of a convergent sequence get close to some fixed element. A convergent sequence is always a Cauchy sequence, but a Cauchy sequence whose elements belong to a field F does not necessarily converge to an element in F . To give an example, let us suppose that F is the set of rational numbers; consider the sequence of integers defined by the Fibonacci recurrence relation :

It can be shown that:

(1.14)

Now define the sequence of rational numbers by:

We can show that:

and this limit is not a rational number. The Fibonacci numbers are useful in many kinds of applications, such as optimisation (finding the minimum or maximum of a function) and random number generation.