Weighted moment inequalities for bessel processes

Assume that φ(·) is a non-negative, continuously differentiable weight function and φ′(·) is nondecreasing on [0,∞), and let 0<θ<1. For any stopping time τ of a Bessel process X=(Xt)t≥0 of dimension α≥1 and starting at zero such that E[τμ]<∞, where 1<μ<∞ and (1−1μ)−1θ≤1, we establish an upper estimate for E[φ(sup0≤t≤τXt)(sup0≤t≤τ|Xt2−αt|)θ]. Our result extends the special case φ≡1 and θ≡1 proved by Graversen and Peškir (Citation1998).