Sharp bounds for transition probability densities of a class of diffusions

Let p<inf>b</inf>(x, t, y) be the transition probability density of the one dimensional diffusion process dX<inf>t</inf> = dW<inf>t</inf> + b(X<inf>t</inf>) dt, where b(.) ∞ ≤ 1. We show that the upper and lower bounds of p<inf>b</inf>(x, t, y) are achieved for fixed (x, t, y) when b(z) = sgn(y - z) and b(z) = sgn(z -y) respectively. Moreover, the precise bounds are given. © 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.