Hello everyone,I'm writing up my thesis and i'm having a bit of trouble with a few very technical points, relating to valuation/calibration methods of the single-factor Cheyette model with displaced diffusion and stochastic volatilitywhere W1 and W2 are independent brownians and, say, nu = short rate = f(t) + X.My questions are:1. I'd like to prove that the associated Feynman-Kac PDE has a solution, which i belive is necessary to actually apply the theorem. The generator attached to this process isn't elliptic though, because (among other things) there's no Y diffusion term; therefore it's non-negative definite rather than positive definite, as seems to be required by the standard PDE literature. Is this a real problem, or is there some trick to get around it?2. I also discuss the Markov semigroup attached to a similar process (perhaps with the drift taken out), and I'd like to apply it to functions such as (x-K)+, which is neither C^2 nor bounded, again, as is required by the standard literature. The boundedness is less of a problem, because I think it can be replaced with a polynomial growth condition. As for the regularity, I imagine that any reasonable function f inherits regularity from the transition density once you convolute it. Therefore I ask; is there a standard theory regarding the existence and regularity of the transition densities of SDEs?I realise these questions are very specific so I don't expect perfect answers; any suggestion is welcome though.Thanks a lot!

If you could write out the PDE this would be very helpful in attempting an answer.BTW did you see the book by Peter Langraf on Cheyette? (books forum). Your SDE is more general. Quote there's no Y diffusion term; therefore it's non-negative definite rather than positive definite, as seems to be required by the standard PDE literatureIn principle, a PDE with one vol = 0 does not have to be awkward. 'Standard' PDE literature tends to examine ideal equations. In partucular, diffusion equations are rife but the interesting ones are convecton-diffusion. BTW there is a theory for degenerate elliptic PDE but let's wait.

Yes, I have read Landgraf's book, in fact it was the starting point for my thesis work. As far as I can tell he handwaved a few mathematical issues here and there (which is fair enough, I'm not criticising him). Of course I could do that too, but I'd rather not!In its entirety, the PDE iswhere L, epsilon, beta, kappa are constants, sigma and alpha vary over time (though let's just assume they're constant) and nu = x let's say. Terminal condition u(T,x,y,z) = f(x,y).I've put it in an FD solver and it works. A couple of side issues, since we're having fun:3. The CIR process V and also Y are almost surely positive, so the domain of the pde is V>0, Y>0. Intuitively it seems to me that the boundary conditions at V=0 and Y=0 shouldn't really influence the theoretical solution. Maybe?4. Cuchulainn: since you're the expert on FD methods, is Craig-Sneyd a good one to use in this case, do you think? Or is there something better?Thanks again.

QuoteOriginally posted by: ehremoYes, I have read Landgraf's book, in fact it was the starting point for my thesis work. As far as I can tell he handwaved a few mathematical issues here and there (which is fair enough, I'm not criticising him). Of course I could do that too, but I'd rather not!In its entirety, the PDE isIn fairness, the theory is not well known although some finance people do know it. Another common myth is that the von Neumann stability analysis proves stability of finance PDE. Again, people hand wave it's just correct maths that is being applied outside its intended scope, aka hand-waving.Now, your PDE is degenerate (coefficient of second derivative == 0 in 1) some part of the boundary and 2) everywhere). Then use Fichera theory, see discussion and PPT slide Get the PDE book by OLeinik and Radkevic, might be the solution.QuoteI've put it in an FD solver and it works. A couple of side issues, since we're having fun:Which scheme? you are obviously amost finished, no sweat? Quote3. The CIR process V and also Y are almost surely positive, so the domain of the pde is V>0, Y>0. Intuitively it seems to me that the boundary conditions at V=0 and Y=0 shouldn't really influence the theoretical solution. Maybe?You mean TS in the interior of the domain?On v = 0 no BC (check Fichera function) and the pde is there U_t + AU_x + BU_y + C = 0 and on y = o it depend on the sign of E (> 0, < 0) in pde U_t + EU_y = 0(characteristics). Quote..is Craig-Sneyd a good one to use in this case, do you think? Or is there something better?If you it as a recipe, no. The problem is you have to take upwinding in U_y or maybe Lax-Wendroff as Peter L. had suggested. Other option if you have time at IMEX, and possibly Strang splitting. If you use CN and centred difference in t and y respectively ==> But with a workaround OK! I have solved this by Artificial BC.Your Pde has no mixed derivatives BTW Craig-Sneyd method is a Soviet Split and was discovered in 1964.