Daniel J. Duffy - Numerical Methods in Computational Finance

We define a complete metric space X as one in which every Cauchy sequence converges to an element in X . Examples of complete metric spaces are:

Euclidean space .

The metric space C[a, b] of continuous functions on the interval [a, b].

By definition, Banach spaces are complete normed linear spaces. A normed linear space has a norm based on a metric, as follows .

is the Banach space of functions defined by the norm for .

Definition 1.6An open cover of a set E in a metric space X is a collection of open subsets of X such that .

Finally, we say that a subset K of a metric space X is compact if every open cover of K contains a finite subcover , that is for some finite N .

We now examine functions that map one metric space into another one. In particular, we discuss the concepts of continuity and Lipschitz continuity .

It is convenient to discuss these concepts in the context of metric spaces.

Definition 1.7Let and be two metric spaces. A function f from X into Y is said to be continuous at the point if for each there exists a such that:

This is a generalisation of the concept of continuity in Section 1.2( Definition 1.1). We should note that this definition refers to the continuity of a function at a single point. Thus, a function can be continuous at some points and discontinuous at other points.

Definition 1.8A function f from a metric space into a metric space is said to be a uniformly continuous on a set if for each there exists a such that:

If the function f is uniformly continuous, then it is continuous, but the converse is not necessarily true. Uniform continuity holds for all points in the set E , whereas normal continuity is only defined at a single point.

Definition 1.9Let be a real-valued function and suppose that we can find two constants M and such that . Then we say that f satisfies a Lipschitz condition of order , and we write .

We take an example. Let on the interval [ a , b ].

Then:

Hence .

A concept related to Lipschitz continuity is called a contraction .

Definition 1.10Let and be metric spaces. A transformation T from X into Y is called a contraction if there exists a number such that:

In general, a contraction maps a pair of points into another pair of points that are closer together. A contraction is always continuous.

The ability to discover and apply contraction mappings has considerable theoretical and numerical value. For example, it is possible to prove that stochastic differential equations (SDEs) have unique solutions by the application of fixed point theorems :

Brouwer's fixed point theorem

Kakutani's fixed point theorem

Banach's fixed point theorem

Schauder's fixed point theorem

Our interest here lies in the following fixed point theorem.

Theorem 1.2(Banach Fixed Point Theorem) Let T be a contraction of a complete metric space ( X , d ) into itself: