We show that, apart from deterministic processes, compound Poisson processes with exponential jumps are the only (shifted) non-decreasing Lévy processes whose inverses are also (shifted) non-decreasing Lévy processes

Ferreira, J.A.

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Centrum voor Wiskunde en InformaticaREPORTRAPPORTA note on inverses of non-decreasing Lévy processesJ.A. FerreiraProbability, Networks and Algorithms (PNA)PNA-R0110 July 31, 2001Report PNA-R0110ISSN 1386-3711CWIP.O. Box 940791090 GB  AmsterdamThe NetherlandsCWI is the National Research Institute for Mathematicsand Computer Science. CWI is part of the StichtingMathematisch Centrum (SMC), the Dutch foundationfor promotion of mathematics and computer scienceand their applications.SMC is sponsored by the Netherlands Organization forScientific Research (NWO). CWI is a member ofERCIM, the European Research Consortium forInformatics and Mathematics.Copyright © Stichting Mathematisch CentrumP.O. Box 94079, 1090 GB  Amsterdam (NL)Kruislaan 413, 1098 SJ  Amsterdam (NL)Telephone +31 20 592 9333Telefax +31 20 592 4199A Note on Inverses of Non-decreasing Levy ProcessesJ.A. FerreiraCWIP.O. Box 94079, 1090 GB Amsterdam, The Netherlandsjose.ferreira@cwi.nlABSTRACTWe show that, apart from deterministic processes, compound Poisson processes with exponential jumps are theonly (shifted) non-decreasing Levy processes whose inverses are also (shifted) non-decreasing Levy processes.2000 Mathematics Subject Classication: 60G51, 60G17.Keywords and Phrases: Non-decreasing Levy processes, compound Poisson processes, inverse of a process.Note: Work carried out under the project PNA3.3, `Stochastic Processes and Applications', and funded by TheFifth Framework Programme of the European Commission through the Dynstoch research network.1 IntroductionIt is known (see Bertoin (1996), p. 72, or Sato (1999), pp. 137-8) that a Levy process X = fXt:t  0g is non-decreasing (i.e., its realizations are a.s. non-decreasing functions) if and only if theLaplace transform of Xtis of the formLXt(u) = e t	(u); (1.1)where 	(u) =   R10(e ux  1)(dx),   0, and  is a measure on (0;1) satisfyingR10x(dx) <1. A non-decreasing Levy process is also called a subordinator.Example AA non-decreasing compound Poisson process X is dened in terms of a Poisson pro-cess N = fNt: t  0g with rate  and of an independent, non-negative i.i.d. sequence1; 2; : : : by the formula Xt=PNti=1i, t  0. Clearly, X has independent increments andLXt(u) = expf t(1  L(u))g, where Lis the Laplace transform of the i's. Let us dene a process X 1whose realizations work as inverses of those of X by puttingX 1x= infft : Xt> xg; x  0:It is well known (e.g. Revuz and Yor (1991), p. 7) that X 1is non-decreasing and right-continuous if X is, and that X 1x = infft : Xt xg. Moreover, Xt x if and only if X 1x > t,soP (X 1x > t) = P (Xt x); x; t  0: (1.2)1This shows that if X 1is continuous in probability then P (X 1x> t) = P (Xt x) for almostall t.If we think of X as describing the growth in time of some physical quantity, then an eventsuch as fX 1x> tg corresponds to a statement about the time it takes for X to exceed thethreshold x. This suggests that the inverse process X 1may occur in some practical problems.Since one usually hopes to work with processes which are in some sense amenable or have anice structure, it is natural to ask, for example, whether X 1can itself be a (non-decreasing)Levy process. The following example suggests a slightly dierent form of this question:Example BConsider a compound Poisson process with 1; 2; : : : exponentially distributed with mean 1=.ThenLXt(u) = exp tu+ u; u >  ; (1.3)which corresponds to (1.1) with  = 0 and (dx) = e xdx.Let  1;  2; : : : be the lengths of the successive constancy intervals of N , and denote by Mthe Poisson process determined by 1; 2; : : : We can see thatX 1(x) =  1, 0  x < 1, Mx= 0;X 1(x) =  1+  2, 1 x < 1+ 2, Mx= 1;and so on, so that X 1(x) =  1+PMxi=1 i+1, x  0. It follows that~X 1:= X 1  1is a Levyprocess withL~X 1t(u) = exp tu+ u; u >  ;which is the same as (1.3) with  and  interchanged. In other words, ifX is a compound Poissonprocess with exponential jumps thenX 1is a compound Poisson process with exponential jumpsplus an additional independent exponential jump.It can be checked that the inverse of X 1is X itself. Thus we have the converse statementthat if X 1is a compound Poisson process with exponential jumps plus an additional inde-pendent exponential jump then its inverse X is a compound Poisson process with exponentialjumps. Let us say that Y is a shifted Levy process if Y is the sum of a Levy process and anindependent, non-negative random variable, and that Y is deterministic if, with probability1, Yt/ t for all t  0. We next show that, apart from deterministic processes, the onlynon-decreasing, shifted Levy processes whose inverses are also shifted Levy processes are thecompound Poisson processes with exponential jumps and their inverses, which are compoundPoisson processes with exponential jumps shifted by an additional independent exponentialjump.In view of the representation (1.1) this result is almost trivial, but it may be of somecuriosity value. We do not know whether such models have been, or can be, successfully usedin statistical modelling, but the result suggests an obvious model checking procedure.22 Proof of the resultWe know that if Y and Y 1are non-decreasing, shifted Levy processes thenLYt(u) = '(u)e t	(u)and LY 1t(u) = ~'(u)e t~	(u);where ' and ~' are Laplace transforms of some random variables, and	(u) =   Z10(e ux  1)(dx);~	(u) = ~  Z10(e ux  1)~(dx);withR10x(dx);R10x~(dx) <1, ; ~  0.Put^LY(u; v) :=Z10e vtLYt(u)dt ='(u)	(u) + v;^LY 1(u; v) :=Z10e vtLY 1t(u)dt =~'(u)~	(u) + v:Noting LY 1t(u) = uR10e usP (Y 1t s)ds  P (Y 1t= 0), recalling that P (Y 1t s) = P (Ys>t) for almost all s from the discussion around (1.2), using the denition of^LY(u; v), and writingp(v) =R10e vtP (Y 1t= 0)dt, we see that^LY 1(u; v) =Z10e vtuZ10e usP (Y 1t s) ds dt  p(v)=Z10e vtuZ10e usP (Ys> t) ds dt  p(v)=Z10ue usZ10e vtP (Ys> t) dt ds  p(v)=Z10ue us1v 1vLYs(v)ds  p(v)=1v uv^LY(v; u)  p(v);which givesv ~'(u)~	(u) + v= 1 u'(v)	(v) + u  p(v);for all u and v for which the Laplace transforms involved are dened. But with v = 0 this reads0 = 1 u=(+u) p(0), implying p(0) = 0 and hence p  0. Thus we have the simpler equationv ~'(u)~	(u) + v= 1 u'(v)	(v) + u: (2.1)It can be checked that (2.1) is satised by ~'(u) = =(+u),~	(u) = u=(+u), '(u) = 1, 	(u) =u=(+u), which correspond to the shifted compound Poisson processes with exponential jumpsof Example B.3To solve (2.1), assume rst that '  1, i.e., that ' is the Laplace transform of a randomvariable concentrated at 0. Then (2.1) becomesv ~'(u)~	(u) + v=	(v)	(v) + u: (2.2)Letting v ! 1 in this equation we get ~'(u) = =( + u), where  := ~ + ~(0;1). If  =1 = ~(0;1), so that ~'  1, (2.2) reduces to~	(u)=v = u=	(v), which implies 	(v) = v=~	(1),~	(u) = u=	(1) and shows that Yt/ t, t  0, with probability 1, i.e., that Y is deterministic.If  <1, (2.2) is equivalent tov+ u1~	(u) + v=	(v)	(v) + u: (2.3)Taking u = 1 here and solving for 	(v), we nd	(v) =v+ v; v >  ;where  := (+ 1)~	(1); thus Y must be a compound Poisson process with exponential jumps.Now suppose ' 6 1. Letting v ! +1 in (2.1) we nd ~'  1 andv~	(u) + v= 1 u'(v)	(v) + u; oru'(v)	(v) + u=~	(u)~	(u) + v:But this last equation is the same as (2.2) with u and 	 interchanged respectively with v and~	 and ~' replaced by '; therefore, the same arguments as before show that Y 1is either adeterministic process or a compound Poisson process with exponential jumps, or, equivalently,that Y is either a deterministic process or the inverse of a compound Poisson process withexponential jumps.AknowledgementsI am grateful to Kacha Dzhaparidze and Peter Spreij for useful suggestions.ReferencesBertoin, J. (1996). Levy processes. Cambridge University Press, Cambridge.Revuz, D., and Yor, M. (1991). Continuous martingales and brownian motion. Springer-Verlag,Berlin.Sato, K. (1999). Levy processes and innitely divisible distributions. Cambridge UniversityPress, Cambridge.4

A note on inverses of non-decreasing Lévy processes

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