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

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 Π[1+L _j (t]^−1/2‖ _p (0≤j≤m,j∈ —; 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 L ₀(t)=1.