L~p-estimates on a ratio involving a Bessel process

摘要

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 (any ≥ 0). Then we establish the relationship between L~p-norm of log~(1/2)(1+δJ_τ) and L~p-norm of supZ_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_tΠ[1+L_j(t)]~(-1/2)‖_p (0 ≤ j ≤ m, j ∈ Z; 0 ≤ t ≤ τ) are equivalent with 0 < p< + ∞ for all stopping times rand 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 L_0(t)=1.