Numerical Methods in Computational Finance. Daniel J. Duffy

only if and this implies (recall that is a complex number) where means ‘real part of ’.

This example leads us to our first encounter with the concept of stability of finite difference schemes. In particular, we define the region of absolute stability of a numerical method for an IVP as the set of complex values of for which all discrete approximations to the test problem (2.37) remain bounded when tends to infinity. For example, for the trapezoidal method (2.39) the left-half plane is the stability region.

Theorem 2.1 (The Root Condition). A necessary and sufficient condition for the stability of a linear multistep method (2.32) is that all the roots of the polynomial defined by Equation (2.34) are located inside or on the unit circle and that the roots of modulus 1 are simple.

      We now discuss convergence issues. We say that a difference scheme has order of accuracy if:

      where approximate solution of (2.32), exact solution of (2.31), and is independent of .

      We conclude this section by stating a convergence result that allows us to estimate the error between the exact solution of an initial value problem and the solution of a multistep scheme that approximates it. To this end, we consider the -dimensional autonomous initial value problem:

      By autonomous we mean that is a function of the dependent variable only and is thus not of the form . The latter form is called non-autonomous.

      We approximate this IVP using the multistep method (2.32). We recall:

      Theorem 2.2 Assume that the solution of the is times differentiable with and assume that is differentiable for all .

      Suppose furthermore that the sequence is defined by the equations:

      If the multistep method is stable and satisfies:

      where C is a positive constant independent of