Numerical Methods in Computational Finance. Daniel J. Duffy

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Название Numerical Methods in Computational Finance
Автор произведения Daniel J. Duffy
Жанр Ценные бумаги, инвестиции
Серия
Издательство Ценные бумаги, инвестиции
Год выпуска 0
isbn 9781119719724
Numerical Methods in Computational Finance - Daniel J. Duffy
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right-parenthesis period 3rd Row 1st Column Blank 2nd Column upper V equals upper K cubed comma upper W equals upper K Superscript 4 Baseline 4th Row 1st Column Blank 2nd Column upper T left-parenthesis x 1 comma x 2 comma x 3 right-parenthesis equals left-parenthesis x 1 plus x 2 comma x 3 comma x 3 comma 0 right-parenthesis period EndLayout"/>

      A more general linear transformation (in fact, a vector-valued transformation) is:

StartLayout 1st Row 1st Column Blank 2nd Column upper T colon upper K Superscript n Baseline right-arrow upper K Superscript m Baseline 2nd Row 1st Column Blank 2nd Column upper T left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis equals left-parenthesis f 1 left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis comma ellipsis comma f Subscript m Baseline left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis right-parenthesis period EndLayout

      Another example of a mapping is upper T colon upper V left-parenthesis upper K right-parenthesis right-arrow upper V left-parenthesis upper K right-parenthesis comma italic upper T x equals normal lamda x comma normal lamda element-of upper K fixed, and this is called a magnification or a dilation.

      Theorem 4.2 For any given r greater-than-or-equal-to 2:

StartLayout 1st Row 1st Column Blank 2nd Column upper T left-parenthesis normal lamda 1 x 1 plus ellipsis plus normal lamda Subscript r Baseline x Subscript r Baseline right-parenthesis equals normal lamda 1 italic upper T x 1 plus ellipsis plus normal lamda Subscript r Baseline italic upper T x Subscript r Baseline 2nd Row 1st Column Blank 2nd Column normal lamda Subscript j Baseline element-of upper K comma j equals 1 comma ellipsis comma r comma x Subscript j Baseline element-of upper V left-parenthesis upper K right-parenthesis comma j equals 1 comma ellipsis r EndLayout

      is a necessary and sufficient condition for a mapping to be a linear transformation. From this result we can conclude that a linear transformation upper T colon upper V left-parenthesis upper K right-parenthesis right-arrow upper W left-parenthesis upper K right-parenthesis is determined by the images upper T phi 1 comma ellipsis upper T phi Subscript n Baseline in upper W left-parenthesis upper K right-parenthesis of a basis StartSet phi 1 comma ellipsis comma phi Subscript n Baseline EndSet in upper V left-parenthesis upper K right-parenthesis.

      4.5.1 Invariant Subspaces

      We adopt the notation L(V; W) to denote the set of linear transformations from the vector space V to the vector space W.

      Definition 4.6 Let upper T element-of upper L left-parenthesis upper V semicolon upper V right-parenthesis and let U be a subspace of V such that italic upper T upper U subset-of upper U where italic upper T upper U equals left-brace italic upper T x semicolon x element-of upper U right-brace. Then U is called an invariant subspace of V under T, or more briefly, U is T-invariant.

      We take some examples. Each subspace is invariant with respect to the following operators:

StartLayout 1st Row 1st Column Blank 2nd Column upper T 1 comma upper T 2 comma upper T 3 element-of upper L left-parenthesis upper V semicolon upper V right-parenthesis 2nd Row 1st Column Blank 2nd Column upper T 1 x equals 0 for-all x element-of upper V left-parenthesis zero operator right-parenthesis 3rd Row 1st Column Blank 2nd Column upper T 2 x equals x for-all x element-of upper V left-parenthesis identity operator right-parenthesis 4th Row 1st Column Blank 2nd Column upper T 3 x equals normal lamda x for-all x element-of upper V comma normal lamda element-of upper K left-parenthesis fixed right-parenthesis left-parenthesis similarity operator right-parenthesis period EndLayout x equals sigma-summation Underscript j equals 1 Overscript n Endscripts xi Subscript j Baseline e Subscript j

      We define the vector left-parenthesis m less-than n right-parenthesis:

italic upper P x equals sigma-summation Underscript j equals 1 Overscript m Endscripts xi Subscript j Baseline e Subscript j Baseline period

      Then P is a linear operator (called the projection operator) on to the subspace upper K Superscript m spanned by the vectors StartSet e 1 comma ellipsis comma e Subscript m Baseline EndSet.

      The projection operator has the following invariant subspaces:

        which remain unchanged and

        that are carried into zero.

      4.5.2 Rank and Nullity

      Definition 4.7 Let upper T element-of upper L left-parenthesis upper V semicolon upper W right-parenthesis. The rank of T is the dimension of the range TV of T. It is denoted by r(T).

      Definition 4.8 Let upper T element-of upper L left-parenthesis upper V semicolon upper W right-parenthesis. The null space (or kernel) of T is the set of vectors x such that upper T left-parenthesis x right-parenthesis equals 0. It is denoted by N(T).

      Definition 4.9 The dimension of the subspace N(T) is called the nullity and is denoted by n(T).

      For example: the zero operator ω has r left-parenthesis omega right-parenthesis equals 0 comma n left-parenthesis omega right-parenthesis equals dimension upper V and the identity operator i has r left-parenthesis i right-parenthesis equals dimension upper V comma n left-parenthesis i right-parenthesis equals 0.

      Theorem 4.3 Let upper T element-of upper L left-parenthesis </div> </div> </div> <hr> <div class=

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