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 Eφ(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).
Cloud Makasu (Wed,) studied this question.