Numerical Methods in Computational Finance. Daniel J. Duffy
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href="#fb3_img_img_11ac2821-40c1-5f51-bd44-0f38e9b738a7.png" alt="beta Subscript j"/> are constants,
Since this is a
(2.33)
We note that the first initial condition is known from the continuous problem (2.31) while the determination of the other
We discuss consistency of scheme (2.32). This is a measure of how well the exact solution of (2.31) satisfies (2.32). Consistency states that the difference Equation (2.32) formally converges to the differential equation in (2.31) when
It can be shown that consistency (see Henrici (1962), Dahlquist and Björck (1974)) is equivalent to the following conditions:
Let us take the explicit Euler method applied to IVP (2.31):
The reader can check the following:
(2.36)
from which we deduce that the explicit Euler scheme is consistent with the IVP (2.31) by checking with Equation (2.35).
The class of difference schemes (2.32) subsumes well-known specific schemes, for example:
The one-step () explicit and implicit Euler schemes.
The two-step () leapfrog scheme.
The three-step () Adams–Bashforth scheme.
The one-step trapezoidal () scheme.
Each of these schemes is consistent with the IVP (2.31), as can be checked by calculating their generating polynomials.
We now discuss what is meant by the stability of a finite difference scheme. To take a simple counterexample, a scheme whose solution is exponentially increasing or oscillating in time while the exact solution is decreasing in time cannot be stable. In order to define stability, it is common practice to examine model problems (whose solutions are known) and apply various finite difference schemes to them. We then examine the stability properties of the schemes. The model problem in this case is the constant-coefficient scalar IVP in which the coefficient
and whose solution is given by:
(2.38)
Thus, the solution is increasing when
The solution of Equation (2.39) is given by:
We see that