Abstract: Let Z=(Z_t)_t≥0 be a Bessel process of dimension δ (δ>0) starting at zero and let K(t) be a differentiable function on [0, ∞) with K(t)>0 (∀t≥0). Then we establish the relationship between L^p-norm of log^1/2(1+δJ_τ) and L^p-norm of sup Z_t[t+k(t)]^–1/2 (0≤t≤τ) for all stopping times τ and all 0<p<+∞. As an interesting example, we show that ||log^1/2(1+δL_m+1(τ))||_p and ||supZ_t∏[1+L_j(t)]^–1/2||_p (0≤j≤m, j∈Ζ; 0≤t≤τ) are equivalent with 0<p<+∞ for all stopping times τ and all integer numbers m, where the function L_m(t) (t≥0) is inductively defined by L_m+1(t)=log[1+L_m(t)] with L₀(t)=1.

Key words: Bessel processes, Diffusion process, Itô’s formula, Domination relation

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