Numerical Methods in Computational Finance - Daniel J. Duffy

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StartFraction partial-differential u Over partial-differential y EndFraction italic d y 2nd Row 1st Column d v 2nd Column equals StartFraction partial-differential v Over partial-differential x EndFraction italic d x plus StartFraction partial-differential v Over partial-differential y EndFraction italic d y period EndLayout"/>

StartLayout 1st Row italic d x equals left-parenthesis StartFraction partial-differential v Over partial-differential y EndFraction italic d u minus StartFraction partial-differential u Over partial-differential y EndFraction d v right-parenthesis slash upper J 2nd Row Blank 3rd Row italic d y equals left-parenthesis minus StartFraction partial-differential v Over partial-differential x EndFraction italic d u plus StartFraction partial-differential u Over partial-differential x EndFraction d v right-parenthesis slash upper J EndLayout

      where J is the Jacobian determinant defined by:

upper J equals Start 2 By 2 Determinant 1st Row 1st Column StartFraction partial-differential u Over partial-differential x EndFraction 2nd Column StartFraction partial-differential u Over partial-differential y EndFraction 2nd Row 1st Column StartFraction partial-differential v Over partial-differential x EndFraction 2nd Column StartFraction partial-differential v Over partial-differential y EndFraction EndDeterminant equals StartFraction partial-differential left-parenthesis u comma v right-parenthesis Over partial-differential left-parenthesis x comma y right-parenthesis EndFraction period

      We can thus conclude the following result.

      Theorem 1.1 The functions x equals upper F left-parenthesis u comma v right-parenthesis and y equals upper G left-parenthesis u comma v right-parenthesis exist if:

StartFraction partial-differential u Over partial-differential x EndFraction comma StartFraction partial-differential u Over partial-differential y EndFraction comma StartFraction partial-differential v Over partial-differential x EndFraction comma StartFraction partial-differential v Over partial-differential y EndFraction

      are continuous at (a, b) and if the Jacobian determinant is non-zero at (a, b).

      Let us take the example:

u equals x squared slash y comma v equals y squared slash x period

      You can check that the Jacobian is given by:

StartFraction partial-differential left-parenthesis u comma v right-parenthesis Over partial-differential left-parenthesis x comma y right-parenthesis EndFraction equals Start 3 By 2 Matrix 1st Row 1st Column 2 x slash y 2nd Column minus x squared slash y squared 2nd Row Blank 3rd Row 1st Column minus y squared slash x squared 2nd Column 2 y slash x EndMatrix equals 3 not-equals 0

      Solving for x and y gives:

x equals u Superscript 2 slash 3 Baseline v Superscript 1 slash 3 Baseline comma y equals u Superscript 1 slash 3 Baseline v Superscript 2 slash 3 Baseline period

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

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

      1.6.1 Metric Spaces

StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column upper D Baseline 1 colon d left-parenthesis x comma y right-parenthesis greater-than-or-equal-to 0 semicolon d left-parenthesis x comma y right-parenthesis equals 0 if and only if x equals y 2nd Row 1st Column Blank 2nd Column Blank 3rd Column upper D Baseline 2 colon d left-parenthesis x comma y right-parenthesis equals d left-parenthesis y comma x right-parenthesis 3rd Row 1st Column Blank 2nd Column Blank 3rd Column upper D Baseline 3 colon d left-parenthesis x comma y right-parenthesis less-than-or-equal-to d left-parenthesis x comma z right-parenthesis plus d left-parenthesis z comma y right-parenthesis where x comma y comma z element-of upper X period EndLayout

      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.

      1.6.2 Cauchy Sequences

      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 1.4 A sequence left-parenthesis a Subscript n Baseline right-parenthesis of elements on the real line normal double struck upper R is said to be convergent if there exists an element a element-of normal double struck upper R such that for each positive element epsilon in normal double struck upper R there exists a positive integer n 0 such that:

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