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
0, where Cp and cp are some positive constants depending only
on p, and Hμ,hμ are the inverses of x →(e2μx −2μx −1)/2μ2 and x →(e−2μx +2μx −1)/2μ2
on (0,∞), respectively.
2004 Elsevier Inc. All rights reserved.
