Note 11: How to solve parital integro differential equation(PIDE)

In BS formalism, we know V_t=LV which is a partial differential equation. However, if we included jump
as Merton model, our equation changes to V_t=LV+I(V),
I(V) means an integration term. So now it is PIDE.

1) How to deal with the integration term
* FFT evaluation
* a) Interpolate the discrete values to equal space (Or using unequal space FFT evaluation, it is costly) b) after the result, interpolate back...
The reason is that we usually use unequally spaced gird in S coordinates as near the strikes and barriers we use small mesh.
* For avoid “wrap around pollution”, use linear extrapolation to extend the Smax, Smin.

2) For each step, after getting the value of integration, we can boost the calculation using Fix point interation et al. to use the PDE in this step.