Numerical Methods in Computational Finance - Daniel J. Duffy

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alt="d 2 left-parenthesis f left-parenthesis x right-parenthesis comma f left-parenthesis y right-parenthesis right-parenthesis less-than epsilon whenever x comma y element-of upper E and d 1 left-parenthesis x comma y right-parenthesis less-than delta period"/>

      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 f colon left-bracket a comma b right-bracket right-arrow normal double struck upper R be a real-valued function and suppose that we can find two constants M and alpha such that StartAbsoluteValue f left-parenthesis x right-parenthesis minus f left-parenthesis y right-parenthesis EndAbsoluteValue less-than-or-equal-to upper M StartAbsoluteValue x minus y EndAbsoluteValue Superscript alpha Baseline comma for-all x comma y element-of left-bracket a comma b right-bracket. Then we say that f satisfies a Lipschitz condition of order alpha, and we write f element-of italic upper L i p left-parenthesis alpha right-parenthesis.

      We take an example. Let f left-parenthesis x right-parenthesis equals x squared on the interval [a, b].

      Then:

StartLayout 1st Row 1st Column Blank 2nd Column StartAbsoluteValue f left-parenthesis x right-parenthesis minus f left-parenthesis y right-parenthesis EndAbsoluteValue equals StartAbsoluteValue x squared minus y squared EndAbsoluteValue equals StartAbsoluteValue left-parenthesis x plus y right-parenthesis left-parenthesis x minus y right-parenthesis EndAbsoluteValue less-than-or-equal-to left-parenthesis StartAbsoluteValue x EndAbsoluteValue plus StartAbsoluteValue y EndAbsoluteValue right-parenthesis StartAbsoluteValue x minus y EndAbsoluteValue 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to upper M StartAbsoluteValue x minus y EndAbsoluteValue comma where upper M equals 2 max left-parenthesis StartAbsoluteValue a EndAbsoluteValue comma StartAbsoluteValue b EndAbsoluteValue right-parenthesis period EndLayout

      Hence f element-of italic upper L i p left-parenthesis 1 right-parenthesis.

      A concept related to Lipschitz continuity is called a contraction.

      Definition 1.10 Let left-parenthesis upper X comma d 1 right-parenthesis and left-parenthesis upper Y comma d 2 right-parenthesis be metric spaces. A transformation T from X into Y is called a contraction if there exists a number normal lamda element-of left-parenthesis 0 comma 1 right-parenthesis such that:

d 2 left-parenthesis upper T left-parenthesis x right-parenthesis comma upper T left-parenthesis y right-parenthesis right-parenthesis less-than-or-equal-to normal lamda d 1 left-parenthesis x comma y right-parenthesis for a l l x comma y element-of upper X period

      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.

StartLayout 1st Row 1st Column d left-parenthesis upper T left-parenthesis x right-parenthesis comma upper T left-parenthesis y right-parenthesis right-parenthesis less-than-or-equal-to normal lamda d left-parenthesis x comma y right-parenthesis comma 2nd Column lamda element-of left-parenthesis 0 comma 1 right-parenthesis period EndLayout

      Then T has a unique fixed-point x overbar. Moreover, if x 0 is any point in X and the sequence left-parenthesis x Subscript n Baseline right-parenthesis is defined recursively by the formula x Subscript n Baseline equals upper T left-parenthesis x Subscript n minus 1 Baseline right-parenthesis comma n equals 1 comma 2 comma ellipsis, then limit x Subscript n Baseline equals x overbar and:

d left-parenthesis x overbar comma x Subscript n Baseline right-parenthesis less-than-or-equal-to StartFraction normal lamda Over 1 minus normal lamda EndFraction d left-parenthesis x Subscript n minus 1 Baseline comma x Subscript n Baseline right-parenthesis less-than-or-equal-to StartFraction normal lamda Superscript n Baseline Over 1 minus normal lamda EndFraction d left-parenthesis x 0 comma x 1 right-parenthesis period

      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 italic upper T x equals x.

      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.

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

      George Pólya.

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