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 log1/2(1+δJτ) 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 ‖log1/2(1+δLm+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 Lm+1(t)=log[1+L m (t)] with L0(t)=1.