- #1
mathy_girl
- 22
- 0
Hi all,
For my thesis I would like to solve the following second order nonlinear PDE for [tex]V(x,\sigma,t)[/tex]:
[tex]\frac{1}{2}\sigma^2\frac{\partial^2 V}{\partial x^2}+\frac{1}{2}B^2\frac{\partial^2 V}{\partial \sigma^2}+a\frac{\partial V}{\partial \sigma}=0,[/tex]
subject to the following boundary conditions:
[tex]V(x,0,t)=\epsilon^{3/2}K \max(x,0)e^{-r(T-t)}[/tex] and
[tex]V(x,\infty,t)=K (1+\epsilon^{3/2}x)[/tex]
and with terminal condition
[tex]
V(x,\sigma,T)=\epsilon^{3/2} K \max(x,0).
[/tex]
I've tried separation of variables, by writing [tex]V=X(x)Y(\sigma)[/tex]
which gives 2 ODEs
[tex]X''(x)=kX(x)[/tex] (which is nicely solvable) and
[tex]B^2Y''(\sigma)+2aY'(\sigma)+k\sigma^2Y(\sigma)=0[/tex], which is quite difficult, because it's still nonlinear.
Can anyone help me solve this? I don't know if this is the right way to do it, so other suggestions are also welcome!
Thanks!
Mathy_girl
PS: Help, I don't get the tex-code in my post in the right way... how does this work?
For my thesis I would like to solve the following second order nonlinear PDE for [tex]V(x,\sigma,t)[/tex]:
[tex]\frac{1}{2}\sigma^2\frac{\partial^2 V}{\partial x^2}+\frac{1}{2}B^2\frac{\partial^2 V}{\partial \sigma^2}+a\frac{\partial V}{\partial \sigma}=0,[/tex]
subject to the following boundary conditions:
[tex]V(x,0,t)=\epsilon^{3/2}K \max(x,0)e^{-r(T-t)}[/tex] and
[tex]V(x,\infty,t)=K (1+\epsilon^{3/2}x)[/tex]
and with terminal condition
[tex]
V(x,\sigma,T)=\epsilon^{3/2} K \max(x,0).
[/tex]
I've tried separation of variables, by writing [tex]V=X(x)Y(\sigma)[/tex]
which gives 2 ODEs
[tex]X''(x)=kX(x)[/tex] (which is nicely solvable) and
[tex]B^2Y''(\sigma)+2aY'(\sigma)+k\sigma^2Y(\sigma)=0[/tex], which is quite difficult, because it's still nonlinear.
Can anyone help me solve this? I don't know if this is the right way to do it, so other suggestions are also welcome!
Thanks!
Mathy_girl
PS: Help, I don't get the tex-code in my post in the right way... how does this work?
Last edited: