Lp-estimates on diffusion processes

Let X = (Xt ,Ft )t 0 be a diffusion process on R given by dXt = μ(Xt)dt + σ(Xt)dBt, X0 = x0, where B = (Bt )t 0 is a standard Brownian motion starting at zero and μ, σ are two continuous functions on R, and σ(x) > 0 if x = 0. For a nonnegative continuous function ϕ we define the functional J = (Jt ,Ft )t 0 by Jt = t 0 ϕ(Xs)ds, t 0. Then under suitable conditions we establish the relationship between Lp-norm of sup0 t τ |Xt | and Lp-norm of Jτ for all stopping times τ. In particular, for a Bessel process Z of dimension δ >0 starting at zero, we show that the inequalities √δ 2− p 4− p 1/p √τ p Z∗τ p √δ 4− p 2− p 1/p √τ p hold for all 0