Proceedings of the London Mathematical Society Advance Access originally published online on November 27, 2006
Proceedings of the London Mathematical Society 2007 94(2):386-420; doi:10.1112/plms/pdl011
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© 2006 London Mathematical Society
Finiteness of integrals of functions of Lévy processes
1 Department 5 of Mathematics
University of Washington Seattle
WA 98195
USA
erickson{at}math.washington.edu
2 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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, 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 
t g(a(t) + |Xt|) dt, t g(St) dt, and t 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 
, etc.