Numerical Methods in Computational Finance - Daniel J. Duffy

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alt="normal upper Delta t"/>, then there exist constants upper K 0 comma upper K 1 comma upper K 2 and normal upper Delta t 0 such that for all t Subscript n Baseline element-of left-bracket a comma b right-bracket comma normal upper Delta t less-than-or-equal-to normal upper Delta t 0:

double-vertical-bar upper X Subscript n Baseline minus y left-parenthesis t Subscript n Baseline right-parenthesis double-vertical-bar less-than-or-equal-to upper K 0 normal upper Delta t Superscript p Baseline left-parenthesis t Subscript n Baseline minus a right-parenthesis upper K 0 plus sigma-summation Underscript j equals 0 Overscript n Endscripts double-vertical-bar normal epsilon Subscript j Baseline double-vertical-bar upper K 1 e Superscript upper K 2 left-parenthesis t Super Subscript n Superscript minus a right-parenthesis Baseline period double-vertical-bar z double-vertical-bar equals left-parenthesis sigma-summation Underscript j equals 1 Overscript n Endscripts StartAbsoluteValue z Subscript j Baseline EndAbsoluteValue squared right-parenthesis right-parenthesis Superscript 1 slash 2 Baseline or double-vertical-bar z double-vertical-bar equals max Underscript j equals 1 comma ellipsis comma n Endscripts StartAbsoluteValue z Subscript j Baseline EndAbsoluteValue period

      We state this theorem in more general terms: consistency and stability of a multistep scheme are sufficient for convergence.

      Finally, the discussion in this section is also applicable to systems of ODEs. For more discussions, we recommend Henrici (1962) and Lambert (1991).

      Finally, we present four finite difference schemes for the IVP (2.31) and their generating polynomials as defined by Equations (2.34):

StartLayout 1st Row 1st Column Blank 2nd Column Leapfrog Scheme colon 2nd Row 1st Column Blank 2nd Column StartFraction upper X Subscript n plus 1 Baseline minus upper X Subscript n minus 1 Baseline Over 2 normal upper Delta t EndFraction equals mu left-parenthesis t Subscript n Baseline comma upper X Subscript n Baseline right-parenthesis comma 1 less-than-or-equal-to n less-than-or-equal-to upper N minus 1 3rd Row 1st Column Blank 2nd Column rho left-parenthesis zeta right-parenthesis equals zeta squared minus 1 comma sigma left-parenthesis zeta right-parenthesis equals 2 zeta period 4th Row 1st Column Blank 2nd Column Trapezoidal Scheme colon 5th Row 1st Column Blank 2nd Column StartFraction upper X Subscript n plus 1 Baseline minus upper X Subscript n Baseline Over normal upper Delta t EndFraction equals one half left-bracket mu left-parenthesis t Subscript n Baseline comma upper X Subscript n Baseline right-parenthesis plus mu left-parenthesis t Subscript n plus 1 Baseline comma upper X Subscript n plus 1 Baseline right-parenthesis right-bracket comma 0 less-than-or-equal-to n less-than-or-equal-to upper N minus 1 6th Row 1st Column Blank 2nd Column rho left-parenthesis zeta right-parenthesis equals zeta minus 1 comma sigma left-parenthesis zeta right-parenthesis equals one half left-parenthesis zeta plus 1 right-parenthesis period 7th Row 1st Column Blank 2nd Column Implicit Euler Scheme colon 8th Row 1st Column Blank 2nd Column StartFraction upper X Subscript n plus 1 Baseline minus upper X Subscript n Baseline Over normal upper Delta t EndFraction equals mu left-parenthesis t Subscript n plus 1 Baseline comma upper X Subscript n plus 1 Baseline right-parenthesis comma 0 less-than-or-equal-to n less-than-or-equal-to upper N minus 1 9th Row 1st Column Blank 2nd Column rho left-parenthesis zeta right-parenthesis equals zeta minus 1 comma sigma left-parenthesis zeta right-parenthesis equals zeta period 10th Row 1st Column Blank 2nd Column Adams en-dash Bashforth Scheme colon 11th Row 1st Column Blank 2nd Column StartFraction upper X Subscript n plus 1 Baseline minus upper X Subscript n Baseline Over normal upper Delta t EndFraction equals one twelfth left-bracket 23 mu left-parenthesis t Subscript n Baseline comma upper X Subscript n Baseline right-parenthesis minus 16 mu left-parenthesis t Subscript n minus 1 Baseline comma upper X Subscript n minus 1 Baseline right-parenthesis plus 5 mu left-parenthesis normal t Subscript n minus 2 Baseline comma upper X Subscript n minus 2 Baseline right-parenthesis right-bracket comma 2 less-than-or-equal-to n less-than-or-equal-to upper N 12th Row 1st Column Blank 2nd Column rho left-parenthesis zeta right-parenthesis equals zeta minus 1 13th Row 1st Column Blank 2nd Column sigma left-parenthesis zeta right-parenthesis equals StartFraction 23 Over 12 EndFraction zeta squared minus StartFraction 16 Over 12 EndFraction zeta plus five twelfths period EndLayout

      We recommend that you verify the results using the forms of the generating polynomials for one-step and two-step methods, respectively. The general forms are:

StartLayout 1st Row 1st Column rho left-parenthesis zeta right-parenthesis 2nd Column equals alpha 0 zeta plus alpha 1 2nd Row 1st Column sigma left-parenthesis zeta right-parenthesis 2nd Column equals beta 0 zeta plus beta 1 3rd Row 1st Column rho left-parenthesis zeta right-parenthesis 2nd Column equals alpha 0 zeta squared plus alpha 1 zeta plus alpha 2 4th Row 1st Column sigma left-parenthesis zeta right-parenthesis 2nd Column equals beta 0 zeta squared plus beta 1 zeta plus beta 2 period EndLayout

      (2.40)StartLayout Enlarged left-brace 1st Row StartFraction d y Over italic d t EndFraction plus italic a y equals 1 comma t element-of left-parenthesis 0 comma upper T right-bracket comma a greater-than 0 is normal a constant 2nd Row y left-parenthesis 0 right-parenthesis equals upper A EndLayout

      whose exact solution is given by:

y left-parenthesis t right-parenthesis equals italic upper A e Superscript negative italic a t Baseline plus StartFraction 1 Over a EndFraction left-bracket 1 minus e Superscript negative italic a t Baseline right-bracket equals left-parenthesis upper A minus StartFraction 1 Over a EndFraction right-parenthesis e Superscript negative italic a t Baseline plus StartFraction 1 Over a EndFraction period

      In this case the transient solution is the exponential term, and this decays very fast (especially when the constant a is large) for increasing t. The steady-state solution is a constant, and this is the value of the solution when t is infinity. The transient solution is called the complementary function, and the steady-state solution is called the particular integral (when StartFraction italic d y Over italic d y EndFraction equals 0), the latter including no arbitrary constant. The stiffness in the above example is caused when the value a is large; in this case traditional finite difference schemes can produce unstable and highly oscillating solutions. One remedy is to define very small time steps. Special finite difference techniques have been developed that remain stable even when the parameter a is large. These are the exponentially fitted schemes, and they have a number of variants. The variant described in Liniger and Willoughby (1970) is motivated by finding a fitting factor for a general

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