February 19th, 2020, 12:08 pm
Volume 3, page 1040, section 64.4 "Stochastic Dividends"
The principal part (i.e. the 2nd order derivatives in [$]S,D[$]) in its current form is parabolic and not elliptic (discriminant is zero) because the correlation term [$]\rho[$] is not present in the PDE. Should it not be
[$]\rho \sigma q S \frac{\partial^2V}{\partial S \partial D}[$]?
BTW it is possible to remove the mixed derivatives by the canonical transformation
[$]\xi = \alpha logx - y, \eta = \beta log x[$]
where [$]\alpha = \rho q \sigma[$] and [$]\beta = \sqrt{1-\rho^2} q \sigma[$]
resulting in
[$]A \frac{\partial^2V}{\partial \xi^2} + B \frac{\partial^2V}{\partial \eta^2}[$] + lower-order terms
and with
[$]A = B = \frac{1}{2} q^2 (1 - \rho^2)[$].
This is also an elliptic PDE since its discriminant is negative.
Last edited by
Cuchulainn on January 19th, 2021, 6:57 pm, edited 1 time in total.