The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 264 Wednesday, February 4, 2004 12:51 PM

264 The Mathematics of Financial Modeling and Investment Management

∂f ft( + ∆t, x) – ftx ( , )

-----≈ -------------------------------------------------

∂t ∆t

∂^2 f ftx ( , + ∆x) – 2 ftx ( , ) + ftx ( , – ∆x)

---------≈ -----------------------------------------------------------------------------------------

∂x

2

(∆x)

2

In the case of the Cauchy problem, this approximation scheme

defines the forward recursive algorithm. It can be proved that the algo-

rithm is stable only if the Courant-Friedrichs-Lewy (CFL) conditions

(∆x)

2

∆t< --------------

2 a

2

are satisfied.

Different approximation schemes can be used. In particular, the for-

ward approximation to the derivative used above could be replaced by

centered approximations. Exhibit 9.5 illustrates the solution of a Cauchy

problem for initial conditions that vanish outside of a finite interval. The

simulation shows that solutions diffuse in the entire half space.

EXHIBIT 9.5 Solution of the Cauchy Problem by the Finite Difference Method