Numerical Methods in Computational Finance. Daniel J. Duffy

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Let us assume that we wish to find dx and dy, given that all other quantities are known. Some arithmetic applied to Equation (1.13) (two equations in two unknowns!) results in:

      where J is the Jacobian determinant defined by:

      We can thus conclude the following result.

      Theorem 1.1 The functions and exist if:

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

      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).

1.6 METRIC SPACES AND CAUCHY SEQUENCES

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

      1.6.1 Metric Spaces

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.

      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 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: