but what about alpha =0 and beta <0Is it still a martingale ?

<t>In one working paper by rubinstein and Jackeert, Unresctricted CEV models are introduced. They correspond to processes like dS=alpha*dt + sigma*S^(beta) dWwith beta <0it seems that these processes are not martingales.Is it true ?I am not sure whether we can define them on negative values of S or ...

Andrew,The problem is that we want to compute : delta ( ((x-X)^2 +(y-Y)^2)/(2yY) ) and the jacobian of {X,Y} - > { ((x-X)^2 +(y-Y)^2)/(2yY) , y } is 0 at (x=X,y=Y)

<t>point1)If I understand, small z is the starting point, and capital Z is the final point, but the Kernels KZ(s,z) represent in fact functions of Z ? Anyway . How do you deduce the fact that delta(Cosh d(z,Z) -1) = Y^2 delta(z-Z) ?It also seems that after the pull back , the initial condition of th...

oops, a small mistake in the formula. It does not change what I suggested as the derivative. I also joined a pdf, if you cannot open a mathematica 5 file

<t>Hi Andrew,point1)We are looking at the measure z -> delta(z-Z) Y^2 aren't we. Y^2 behave like a constant and delta(z-Z) = delta(x-X) delta(y-Y)so I do not see how Y^2 delta(z-Z) can be invariant by the transformation of SL(2,R) . There is something I do not see..point2) here is a very simple math...

<t>Hi Andrew.There are a couple of unclear points in your paper:1) page 23 , you say that Y^2 delta(z-Z)= delta(cosh d(z,Z)-1) /Piit is not clear what it means. On the left you have a 2 dimensional distribution because z is complex.but on the right you have a one dimensional distribution because d(z...

Sorry about the preceding remark 1) . I am wrong, it is effectively Xt=X0+Y0{...}

<t>Thanks Andrew,A remarkable paper indeed !When do you plan to have this paper finished ?A few remarks:1) The equation (54) : it seems to me that it should be Xt=X0+Yt Integral{...} 2)end of paragraph 5.4 : The handling of shifted log model is missing3) You managed to get an asymptotic development ...

<r>An interesting site for hyperbolic geometry is :<URL url="http://www.ics.uci.edu/~eppstein/junkyard/hyper.htmlit">http://www.ics.uci.edu/~eppstein/junkyard/hyper.htmlit</URL> seems that the definition of hyperbolic metric is a metric whose total curvature is negative,may be locally. The curvature...

Thanks Elan, maybe that is the point. What is the Poincare metric on the upper half plane?

<r>I agree with everything you say, but the thing I do not understand is why the metric is hyperbolicwhen the the metric should be given by the leading terms of the generator which is elliptic.Or may be it should not...I rely for that on the monograph available here : <URL url="http://arxiv.org/abs/...

I agree , but it seems to me that to indroduce the correlation, we just have to make the metric :ds^2= x^2alpha dx^2 + 2rho x^alpha dx dy + dy^2 (alpha=0 for a normal model)In that case , the equation is still elliptic, not hyperbolicI think , there a subtility that I do not see ...

<t>I just got the abstract of the A. Lesniewski lecture. It looks very interesting. The only thing , I have difficulties to grasp is why is it a hyperbolic space. Because when I look at the PdE of the SABR model, The generator ((non including the time derivative) is an elliptic equation. Knowing tha...

<t>SABR is a very famous formula , but when I look at it, it seems a kind of "tour de force" managing to reach a usable closed form with a couple of approximations.Nevertheless, the formula look complicated and is obtained after a couple of pages of computationsHas someone heard about a way to get i...