Weighted moment inequalities for bessel processes

Abstract

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)−𝟏⁢𝛉≤𝟏, we establish an upper estimate for 𝐄⁡[𝜑⁡ ( sup 0≤𝑡≤𝜏𝑋𝑡)⁢(sup 0≤𝑡≤𝜏 ∣ 𝑋2 𝑡−𝛼⁢𝑡∣)𝜃]. Our result extends the special case 𝜑≡1 and 𝜃≡1 proved by Graversen and Peškir (1998).