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=∫0tφ(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−p4−p)1/p‖τ‖p⩽‖Zτ∗‖p⩽δ(4−p2−p)1/p‖τ‖p hold for all 00, 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.