Proceedings of the London Mathematical Society Advance Access published online on November 27, 2006
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdl011
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© 2006 London Mathematical Society
FINITENESS OF INTEGRALS OF FUNCTIONS OF LÉVY PROCESSES
Department 5 of Mathematics, University of Washington, Seattle, WA 98195, USA, erickson{at}math.washington.edu
Centre for Financial Mathematics, MSI, and School of Finance and Applied Statistics, Australian National University, Canberra, ACT 0200, Australia, Ross.Maller{at}anu.edu.au
Received 7 May 2005. Revision received 17 January 2006.
We prove necessary and sufficient conditions for the almost sure convergence of the integrals
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0
g(a(t) + Mt)df(t), where Mt=sup{|Xs| : s
t} is the two-sided maximum process corresponding to a Lévy process (Xt)t
0, a(·) is a non-decreasing function on [0, ) with a(0)=0, g(·) is a positive non-increasing function on (0, ), possibly with g(0+)=, and f(·) is a positive non-decreasing function on [0, ) with f(0)=0. The conditions are expressed in terms of the canonical measure, 
(·), of the process Xt. The special case when a(x)=0, f(x)=x and g(·) is equivalent to the tail of (at zero or infinity) leads to an interesting comparison of Mt with the largest jump of Xt in (0, t].
Some results concerning the convergence at zero and infinity of integrals like g(a(t)+|Xt|) dt, g(St) dt, and g(Rt) dt, where St is the supremum process and Rt=St–Xt is the process reflected in its supremum, are also given. We also consider the convergence of integrals such as 0Eg(a(t)+Mt)df(t), etc.