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Journal of Zhejiang University SCIENCE A
ISSN 1673-565X(Print), 1862-1775(Online), Monthly
2007 Vol.8 No.1 P.158-163
Lp-estimates on a ratio involving a Bessel process
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.
Key words: Bessel processes, Diffusion process, Itô’s formula, Domination relation
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DOI:
10.1631/jzus.2007.A0158
CLC number:
O211
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2024-08-27
Received:
2023-10-17
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2024-05-08
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