Bounding the Maximal Height of a Diffusion by the Time Elapsed

Let X=(X t ) t=> 0 be a one-dimensional time-homogeneous diffusion process associated with the infinitesimal generator L(x)=(mu)(partial defferential)/(partial defferential)x + (sigma)^2(x)/2 (partial defferential)^2/(partial defferential)x^2 where x-(mu) (x) and x-(sigma) (x)>0 are continuous. We show how the question of finding a function x-H(x) such that c1E(H(tau)) <= E(max |Xt|) <= c2E(H(tau)) holds for all stopping times (tau) of X relates to solutions of the equation: Lx(F)=1 Explicit expressions for H are derived in terms of (mu) and (sigma). The method of proof relies upon a domination principle established by Lenglart and Ito calculus.