Numerical Methods in Computational Finance. Daniel J. Duffy

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      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 1.9 Let 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 1.10 Let 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:

      Then T has a unique fixed-point . Moreover, if is any point in X and the sequence is defined recursively by the formula , then and:

      In general, we assume that X is a Banach space and that T is a linear or non-linear mapping of X into itself. We then say that x is a fixed point of T if .

1.7 SUMMARY AND CONCLUSIONS

      In this chapter we gave an introduction to a number of relevant mathematical concepts from real analysis that are used throughout this book, directly or indirectly. We also have introduced other relevant topics in other chapters. To summarise:

       Chapter 1: Real analysis

       Chapter 4: Finite dimensional vector spaces

       Chapter 5: Numerical linear algebra

       Chapter 16: Complex analysis

      In this way we hope that this book becomes more self-contained than otherwise.

CHAPTER 2 Ordinary Differential Equations (ODEs), Part 1

      It is better to solve one problem five different ways, than to solve five problems one way.

      George Pólya.