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| Lp-estimates on a ratio involving a Bessel process |
| LU Li-gang, YAN Li-tan, XIANG Li-chi |
| Basic College, Zhejiang Wanli University, Ningbo 315101, China; Department of Mathematics, College of Science, Donghua University, Shanghai 200051, China |
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Abstract Let Z=(Zt)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 Lp-norm of log1/2(1+δJτ) and Lp-norm of sup Zt[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 ||supZt∏[1+Lj(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 Lm(t) (t≥0) is inductively defined by Lm+1(t)=log[1+Lm(t)] with L0(t)=1.
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Received: 03 March 2006
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