Numerical Methods in Computational Finance - Daniel J. Duffy

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element-of left-bracket 0 comma upper T right-bracket period"/>

      Roughly speaking, this lemma states that you cannot get a negative solution from positive input.

      The following result gives bounds on the growth of u left-parenthesis t right-parenthesis.

      Theorem 2.1 Let u left-parenthesis t right-parenthesis be the solution of Equation (2.1). Then:

StartLayout 1st Row 1st Column Blank 2nd Column StartAbsoluteValue u left-parenthesis t right-parenthesis EndAbsoluteValue less-than-or-equal-to StartFraction upper N Over alpha EndFraction plus StartAbsoluteValue upper A EndAbsoluteValue for-all t element-of left-bracket 0 comma upper T right-bracket 2nd Row 1st Column Blank 2nd Column where 3rd Row 1st Column Blank 2nd Column StartAbsoluteValue f left-parenthesis t right-parenthesis EndAbsoluteValue less-than-or-equal-to upper N for-all t element-of left-bracket 0 comma upper T right-bracket period EndLayout

      This result states that the value of the solution is bounded by the input data. In other words, it is a well-posed problem.

      We wish to replicate these properties in our difference schemes for Equation (2.1).

      For completeness, we show the steps to be executed in order to produce the result in Equation (2.2).

      (2.3)upper L e t upper I left-parenthesis t right-parenthesis equals exp left-parenthesis integral Subscript 0 Superscript t Baseline a left-parenthesis s right-parenthesis italic d s right-parenthesis comma upper I Superscript negative 1 Baseline left-parenthesis t right-parenthesis equals exp left-parenthesis minus integral Subscript 0 Superscript t Baseline a left-parenthesis s right-parenthesis italic d s right-parenthesis period

      Then from Equation (2.1) we see:

upper I left-parenthesis t right-parenthesis left-parenthesis StartFraction italic d u Over italic d t EndFraction plus italic a u right-parenthesis equals upper I left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis

      or:

StartFraction d Over italic d t EndFraction left-parenthesis upper I left-parenthesis t right-parenthesis u right-parenthesis equals upper I left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis period

      Integrating this equation between t equals 0 and t equals xi gives:

StartLayout 1st Row 1st Column Blank 2nd Column integral Subscript 0 Superscript xi Baseline StartFraction d Over italic d t EndFraction left-parenthesis upper I left-parenthesis t right-parenthesis u right-parenthesis italic d t equals integral Subscript 0 Superscript xi Baseline upper I left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis italic d t left-parenthesis and using the fact that upper I left-parenthesis 0 right-parenthesis equals 1 right-parenthesis 2nd Row 1st Column Blank 2nd Column upper I left-parenthesis xi right-parenthesis u left-parenthesis xi right-parenthesis equals u left-parenthesis 0 right-parenthesis plus integral Subscript 0 Superscript xi Baseline upper I left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis italic d t 3rd Row 1st Column Blank 2nd Column u left-parenthesis xi right-parenthesis equals u left-parenthesis 0 right-parenthesis upper I Superscript negative 1 Baseline left-parenthesis xi right-parenthesis plus upper I Superscript negative 1 Baseline left-parenthesis xi right-parenthesis integral Subscript 0 Superscript xi Baseline upper I left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis italic d t 4th Row 1st Column Blank 2nd Column equals exp left-parenthesis minus integral Subscript 0 Superscript xi Baseline a left-parenthesis s right-parenthesis italic d s right-parenthesis u left-parenthesis 0 right-parenthesis plus exp left-parenthesis minus integral Subscript 0 Superscript xi Baseline a left-parenthesis s right-parenthesis italic d s right-parenthesis integral Subscript 0 Superscript xi Baseline upper I left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis italic d t period EndLayout

      2.2.2 Rationale and Generalisations

      The IVP Equation (2.1) is a model for all the linear time-dependent differential equations that we encounter in this book. We no longer think in terms of scalar problems in which the functions in Equation (2.1) are scalar-valued, but we can view an ODE at different levels of abstraction. To this end, we focus on the generic homogeneous ODE with solution u left-parenthesis t right-parenthesis:

      This equation subsumes several special cases:

      1 The variable is a square matrix, and then Equation (2.4) represents a system of ODEs. This is a very important area of research having many applications in science, engineering, and finance.

      2 The variable is an ordinary or partial differential operator, and then Equation (2.4) represents an ODE in a Hilbert or Banach space.

      3 The variable is a tridiagonal or block tridiagonal matrix that originates from a semi-discretisation in space of a time-dependent partial differential equation (PDE) using the Method of Lines (MOL) as discussed in Chapter 20.

      4 The formal solution of (2.4) is:(2.5) In other words, we express the solution in terms of the exponential function of a matrix or of a differential operator. In the former case, there are many ways to compute the exponential of a matrix (see Moler and Van Loan (2003)).

      5 The solution of Equation (2.4) can be simplified by matrix or operator splitting of the operator :(2.6) For example, we can split a matrix into two simpler matrices, or we can split an operator into its convection and diffusion components. In other words, we solve Equation (2.4) as a sequence of simpler problems in (2.6). These topics will be discussed in Chapters 18, Скачать книгу