## Fichera in 'Unconditionally Stable+ 2nd Order......

### Fichera in 'Unconditionally Stable+ 2nd Order......

Daniel,

I have finished reading your paper 'Unconditionally Stable & Second order ...' at SSRN. Thank you it was a great read. I particularly liked the section on gridding - very useful stuff.

I have a question regarding the Fichera theory. I had not heard of Fichera prior to reading this and it looks very useful. I have a question regarding your definitions however, which I cannot look up because I don't have access to Fichera's originals. When I apply the definitions given in this paper all of the signs in well known inequalities (e.g. Feller's inequality) seem backwards. Please let me know what I am doing wrong.

Take for instance the Feller process

dx = k*(r-x)*dt + s*sqrt(x)*dW

with Fokker-Planck Equation (with subscript notation for derivatives)

f_{t} = 0.5*x*s^2*f_{xx} + [s - k*(r-x)]*f_{x} + k*f,

which is in the form of Equation (22) in your paper with operator

Lf = 0.5*x*s^2*f_{xx} + [s - k*(r-x)]*f_{x} + k*f.

The two functions for Equation (20) (i.e. for the Fichera function) are

a = 0.5*x*s^2

and

b = [s - k*(r-x)],

so that the Fichera function is

Fichera = [0.5*s^2 - k*(r-x)]v

where v is the outward normal. Now at x=0 the inward normal is v=1

and so

Fichera = [0.5*s^2 - kr].

No BCs are allowed at zero when this is positive, which gives

2*k*r < s^2

which is of course the wrong way round!!! Are my readings of the definition of the operator L wrong?

Posts: 1
Joined: Thu May 02, 2013 8:11 am

Some discussions can be found here; it seems like an open issue

Cuchulainn

Posts: 677
Joined: Mon Dec 18, 2006 2:48 pm
Location: Amsterdam, the Netherlands

### Re: Fichera in 'Unconditionally Stable+ 2nd Order......

Cuchulainn

Posts: 677
Joined: Mon Dec 18, 2006 2:48 pm
Location: Amsterdam, the Netherlands