Hello I need some help in a problem set I'm having for my class in Stochastic calculus.Hints, suggestions, and help are much appreciated.Here's the question:We consider a diffusion dXt = A(Xt)dWt + B(Xt)dt. We assume that B is bounded and that there exists E and M such that 0<E<=A(x)<M<=infinity, for any x element of Real. 1) Find a function s (continuously twice differentiable function) such that (s(Xt)) is a martingale. This function is the so-called "scaling function".2) For any a<Xo<b. We set Ta = inf{t>=0 : Xt = a} and Tb = inf{t>=o: Xt = b}. Admitting that the equality Exp(s(X sub Ta^Tb)) = s(Xo) holds true, show that P(Tb < Ta) = [s(Xo) - s(a)] / [s(b) - s(a)].3) Assume that lim x->infinity s(x) < infinity. Show that in this case limb->infinity P(Tb<Ta) > 0. Derive that with positive probabliry the diffusion X will never reach a.Thanks for all your help!

i think the answer to b in is in michael steele's probability book, the title of which escapes me at the moment ( justgo to amazon and type in his name ) . the rest may be in there somewhere also.

thanks for the reply, I looked at Steele's book, Stochastic Calculus and Financial Applications, but I couldn't find the answers. I was able to find a similar question to 1), but it was part of the exercises portion (exercise 9.4), with no solution.

Karlin & Taylor's 'A second course...'