aa r X i v : . [ m a t h . P R ] J a n PERPETUAL INTEGRALS FOR L´EVY PROCESSES
LEIF D ¨ORING AND ANDREAS E. KYPRIANOU
Abstract.
Given a L´evy process ξ we ask for necessary and sufficient conditions for almost surefiniteness of the perpetual integral R ∞ f ( ξ s ) ds , where f is a positive locally integrable function. If µ = E [ ξ ] ∈ (0 , ∞ ) and ξ has local times we prove the 0-1 law P (cid:16) Z ∞ f ( ξ s ) ds < ∞ (cid:17) ∈ { , } with the exact characterization P (cid:16) Z ∞ f ( ξ s ) ds < ∞ (cid:17) = 0 ⇐⇒ Z ∞ f ( x ) dx = ∞ . The proof uses spatially stationary L´evy processes, local time calculations, Jeulin’s lemma and theHewitt-Savage 0-1 law.
The study of perpetual integrals R f ( X s ) ds with finite or infinite horizon for diffusion processes X has a long history partially because of their use in the analysis of stochastic differential equations andinsurance, financial mathematics as the present value of a continuous stream of perpetuities.The main result of the present article is a characterization of finiteness for perpetual integrals of L´evyprocesses: Theorem 1.
Suppose that ξ is a L´evy process that has strictly positive mean µ < ∞ , local times andis not a compound Poisson process. If f is a measurable locally integrable positive function, then thefollowing 0-1 law holds: P (cid:16) Z ∞ f ( ξ s ) ds < ∞ (cid:17) = 1 ⇐⇒ Z ∞ f ( x ) dx < ∞ (T1) and P (cid:16) Z ∞ f ( ξ s ) ds < ∞ (cid:17) = 0 ⇐⇒ Z ∞ f ( x ) dx = ∞ . (T2)Let us briefly compare the theorem with the existing literature:(i) If ξ is a Brownian motion with positive drift, then results were obtained through the Ray-Knighttheorem, Jeulin’s lemma and Khashminkii’s lemma by Salminen/Yor [11], [12].(ii) For spectrally negative ξ , i.e. ξ only jumps downwards, the equivalence was obtained in Khosh-nevisan/Salminen/Yor [12], see also Example 3.9 of Schilling/Voncracek [9]. The spectrally negativecase also turns out to be easier in our proof.(iii) If f is (ultimately) decreasing, results for general L´evy processes have been proved in Erick-son/Maller [5]. In this case the result stated in Theorem 1 follows easily from the law of large numbersby estimating 2 µt > ξ t > µt for t big enough. The very same argument also shows that for µ = + ∞ the integral test (T) fails in general. For a result in this case we again refer to [5]. Remark 2.
It is not clear whether or not the assumption ξ having local time plays a role. For(ultimately) decreasing f the existence of local time is clearly not needed, whereas we have no conjecturefor general f . Proof of Theorem 1
Before going into the proof let us fix some notation and facts needed below. For more definitions andbackground we refer for instance to [1] or [8]. The law of ξ issued from x ∈ R will be denoted by P x , abbreviating P = P , and the characteristic exponent is defined asΨ( λ ) := − log E (cid:2) exp( iλξ ) (cid:3) , λ ∈ R . We recall from Theorem V.1 of [1] that ξ has local times (cid:0) L t ( x ) (cid:1) t ≥ ,x ∈ R if and only if Z ∞−∞ R (cid:18)
11 + Ψ( r ) (cid:19) dr < ∞ . (1)This means that for any bounded measureable function f : R → [0 , ∞ ) the occupation time formula Z t f ( ξ s ) ds = Z R f ( x ) L t ( x ) dx, t ≥ , holds almost surely.A consequence of (1) is also that points are non-polar. More precisely, a Theorem of Kesten andBretagnolle states that P ( τ x < ∞ ) > x > τ x = inf { t : ξ t = x } , see for instance Theorem7.12 of [8].Throughout we assume ξ is transient so that R ε − ε R ( ψ ( r ) ) dr < ∞ and, consequently, (1) implies R ∞−∞ R (cid:16) r ) (cid:17) dr < ∞ . But then Theorem II.16 of [1] implies that the potential measure U ( dx ) = Z ∞ P (cid:0) ξ s ∈ dx (cid:1) ds has a bounded density u ( x ) with respect to the Lebesgue measure.We start with the easy direction of Theorem 1: Proof of Theorem 1, Sufficiency of Integral Test.
Suppose that R R f ( x ) dx < ∞ . Since we assume that ξ is transient and has a local time we can use the existence and boundedness of the potential densityto obtain E h Z ∞ f ( ξ s ) ds i = Z R f ( x ) Z ∞ P ( ξ s ∈ dx ) ds = Z R f ( x ) u ( x ) dx ≤ sup x ∈ R u ( x ) Z R f ( x ) dx < ∞ . Since finiteness of the expectation implies almost sure finiteness the sufficiency of the integral test foralmost sure finiteness of the perpetual integral is proved. (cid:3)
For the reverse direction we use Jeulin’s lemma, here is a simple version:
Lemma 3.
Suppose ( X x ) x ∈ R are non-negative, non-trivial and identically distributed random variableson some probability space (Ω , F , P ) , then P (cid:16) Z R f ( x ) X x dx < ∞ (cid:17) = 1 = ⇒ Z R f ( x ) dx < ∞ . Proof.
Since X x are identically distributed, we may choose ε > P ( X x > ε ) = δ > x ∈ R . Since R R f ( x ) X x dx is almost surely finite, there is some N ∈ N so that P ( A N ) > − δ/ A N = { R R f ( x ) X x dx > N } . Hence, we have E (cid:2) X x A N (cid:3) ≥ εP ( { X x > ε } ∩ A N ) > εδ/ > . But then N ≥ N P ( A N ) ≥ E h Z R f ( x ) X x dx A N i = Z R f ( x ) E [ X x A N ] dx ≥ εδ/ Z R f ( x ) dx. The proof is now complete. (cid:3)
Note that there are different versions of Jeulin’s lemma (see for instance [10]). Most commonly, onerefers to Jeulin’s lemma if P ( R R f ( x ) X x dx < ∞ ) > X x are posed.Those extra assumptions on X x are not satisfied in our setting but we can employ a 0-1 law thatallows us to work with P ( R R f ( x ) X x dx < ∞ ) = 1. ERPETUAL INTEGRALS FOR L´EVY PROCESSES 3
We would like to apply Jeulin’s lemma via the occupation time formula Z ∞ f ( ξ s ) ds = lim t ↑∞ Z t f ( ξ s ) ds = lim t ↑∞ Z ∞ f ( x ) L t ( x ) dx = Z ∞ f ( x ) L ∞ ( x ) dx with L ∞ ( x ) := lim t ↑∞ L t ( x ). The argument is too simplistic because the distribution of L ∞ ( x ) dependson x . Only if ξ is spectrally negative the laws L ∞ ( x ) are independent of x by the strong Markovproperty. To make this idea work we work with randomized initial conditions instead. In what followswe chose a particularly convenient initial distribution motivated by a result from fluctuation theory(see Lemma 3 of [2]). Our assumption E [ ξ ] < ∞ implies that P (cid:0) ξ T z − z ∈ dy (cid:1) z →∞ = ⇒ ρ ( dy ) , (2)where T z = inf { t ≥ ξ t ≥ z } and ρ is a non-degenerate probability law, called the stationaryovershoot distribution. The convergence in (2) is a consequence of the quintuple law of [8] and thedistribution ρ can be written down explicitly.Since ρ is the stationary overshoot distribution we have P ρ (cid:0) ξ T a − a ∈ dy (cid:1) := Z P x (cid:0) ξ T a − a ∈ dy (cid:1) ρ ( dx ) = ρ ( dy ) , ∀ a > , so that spatial stationarity holds due to the strong Markov property: under P ρ ( ξ ( a ) t ) t ≥ := ( ξ T a + t − a ) t ≥ (3)has law P ρ for all a >
0. The stationarity property (3) will be the key to apply Jeulin’s lemma.
Lemma 4.
For any x > we have P ρ ( L ∞ ( x ) ∈ dy ) = P ρ ( L ∞ (1) ∈ dy ) . Proof.
First note that P ρ ( T x < ∞ ) = 1 for all x > L · ( x ) only starts to increase at some time ator after T x . Then P ρ ( L ∞ ( x ) ∈ dy ) = Z ∞ P z ( L ∞ ( x ) ∈ dy ) P ρ ( ξ T x ∈ dz )= Z ∞ P z + x ( L ∞ ( x ) ∈ dy ) P ρ ( ξ T x − x ∈ dz )= Z ∞ P z + x ( L ∞ ( x ) ∈ dy ) ρ ( dz )= Z ∞ P z ( L ∞ (0) ∈ dy ) ρ ( dz ) , using the strong Markov property, the spatial stationarity of P ρ and spatial homogeneity of ξ . Sincethe righthand side is independent of x the proof is complete. (cid:3) Next, we use the Hewitt-Savage 0-1 law (see [4]) in order to get the weak version of Jeulin’s lemmagoing. If X , X , ... denotes a sequence of random variables taking values in some measurable space,then an event A ∈ σ ( X , X , ... ) is called exchangeable if it is invariant under finite permutations (i.e.only finitely many indices are changed) of the sequence X , X , ... . The Hewitt-Savage 0-1 law statesthat any exchangeable event of an iid sequence has probability 0 or 1. Lemma 5. P (cid:0) R ∞ f ( ξ s ) ds < ∞ (cid:1) ∈ { , } .Proof. The idea of the proof is to write Λ := { R ∞ f ( ξ s ) ds < ∞} as an exchangeable event withrespect to the iid increments of ξ on intervals [ n, n + 1] so that P (Λ) ∈ { , } . Let D denote the RCLLfunctions w : [0 , → R . If ξ is the given L´evy process, then define the increment processes as( ξ nt ) t ∈ [0 , = ( ξ n + t − ξ n ) t ∈ [0 , . LEIF D ¨ORING AND ANDREAS E. KYPRIANOU
The L´evy property implies that the sequence ξ , ξ , ... is iid on D . Furthermore, note that ξ can bereconstructed from the ξ n through ξ r = ξ nr − n + n − X i =0 ξ i ∀ r ∈ [ n, n + 1) . Using that g : ( w t ) t ∈ [0 , ( w ) t ∈ [0 , , g : ( w, w ′ ) t ∈ [0 , ( w t + w ′ t ) t ∈ [0 , and g : ( w t ) t ∈ [0 , R f ( w s ) ds are measurable mappings, there are measurable mappings g n : D n → R such that Z f (cid:16) ξ nr + n − X i =0 ξ i (cid:17) dr = g n ( ξ , ..., ξ n ) . As a consequence we find that n Z ∞ f ( ξ s ) ds < ∞ o = n ∞ X n =0 Z n +1 n f ( ξ s ) ds < ∞ o = n ∞ X n =0 Z f (cid:16) ξ nr + n − X i =0 ξ i (cid:17) dr < ∞ o = n ∞ X n =0 g n ( ξ , ..., ξ n ) < ∞ o ∈ σ ( ξ , ξ , ... ) . Since clearly Λ is exchangeable for ξ , ξ , ... the Hewitt-Savage 0-1 law implies the claim. (cid:3) Lemma 6.
Suppose P ( R ∞ f ( ξ s ) ds < ∞ ) = 1 , then P ρ ( R ∞ f ( ξ s ) ds < ∞ ) = 1 .Proof. The statement is obvious if ξ is a subordinator, so we assume it is not.Next we show that P x ( R ∞ f ( ξ s ) ds < ∞ ) = 1 for any x >
0. To see this we use the strong Markovproperty at τ = inf { t : ξ t = 0 } which is finite with positive probabillity since points in R arenon-polar: P x (cid:16) Z ∞ f ( ξ s ) ds < ∞ (cid:17) ≥ P x (cid:16) Z ∞ τ f ( ξ s ) ds < ∞ , τ < ∞ (cid:17) = P x (cid:16) Z ∞ f ( ξ s + τ − ξ τ ) ds < ∞ , τ < ∞ (cid:17) = P (cid:16) Z ∞ f ( ξ s ) ds < ∞ (cid:17) P x ( τ < ∞ ) > . But then the 0-1 law of Lemma 5 implies that P x (cid:16) R ∞ f ( ξ s ) ds < ∞ (cid:17) = 1. Finally, we obtain P ρ (cid:16) Z ∞ f ( ξ s ) ds < ∞ (cid:17) = Z R P x (cid:16) Z ∞ f ( ξ s ) ds < ∞ (cid:17) ρ ( dx ) = Z R ρ ( dx ) = 1and the proof is complete. (cid:3) Now we are ready to prove the more delicate part of Theorem 1.
Proof of Theorem 1, Necessity of Integral Test.
Suppose P ( R ∞ f ( ξ s ) ds < ∞ ) > P ( R ∞ f ( ξ s ) ds < ∞ ) = 1 by Lemma 5. Hence, P ρ ( R ∞ f ( ξ s ) ds < ∞ ) = 1 by Lemma 6. Using theoccupation time formula we get Z ∞ f ( ξ s ) ds = lim t →∞ Z t f ( ξ s ) ds = lim t →∞ Z R f ( x ) L t ( x ) dx = Z R f ( x ) L ∞ ( x ) dx P ρ -a.s.In Lemma 4 we proved that L ∞ ( x ) is independent of x under P ρ so that Jeulin’s Lemma implies R R f ( x ) dx < ∞ . (cid:3) ERPETUAL INTEGRALS FOR L´EVY PROCESSES 5
Acknowledgement.
The authors thank Jean Bertoin for several discussions on the topic.
References [1] J. Bertoin: ”L´evy processes.” Cambridge Tracts in Mathematics 121, Cambridge University Press, 1996.[2] J. Bertoin, M. Savov: ”Some applications of duality for L´evy processes in a half-line.” Bull. London Math. Soc. 43,pp. 97-110, 2011.[3] D. Dufresne: ”The distribution of a perpetuity, with applications to risk theory and pension funding.” Scand.Actuarial J., pp. 39-79, 1990.[4] E. Hewitt, L. J. Savage: “Symmetric measures on Cartesian products.” Trans. Amer. Math. Soc., 80, pp. 470-501(1955).[5] K.B. Erickson, R.A. Maller: ”Generalised Ornstein-Uhlenbeck processes and the convergence of L´evy integrals.”S´eminaire de Probabilit´es XXXVIII, 2005.[6] D. Khoshnevisan, P. Salminen, M. Yor: ”A note on a.s. finiteness of perpetual integral functionals of diffusions.”Electr. Comm. Probab., Vol. 11, pp. 108-117.[7] A. Kyprianou, J.C. Pardo, V. Rivero: ”Exact and asymptotic n-tuple laws at first and last passage.” Ann. Appl.Probab. Volume 20, pp. 522-564, 2010.[8] A. Kyprianou: “Fluctuations of L´evy Processes with Applications.” Springer (2013).[9] R. Schilling, Z. Vondraˇcek: “Absolute continuity and singularity of probability measures induced by a purelydiscontinuous Girsanov transform of a stable process.” arXiv:1403.7364[10] A. Matsumoto, K. Yano: ”On a zero-one law for the norm process of transient random walk.” S´eminaire deProbabilit´es XLIII, pp. 105-126, 2011.[11] P. Salminen, M. Yor: ”Properties of perpetual integral functionals of Brownian motion with drift.” Ann. I.H.P., 41,pp. 335-347, 2005.[12] P. Salminen, M. Yor: ”Perpetual integral fuctionals as hitting and occupation times.” Electr. J. Probab., Vol. 10,p. 371-419, 2005.[1] J. Bertoin: ”L´evy processes.” Cambridge Tracts in Mathematics 121, Cambridge University Press, 1996.[2] J. Bertoin, M. Savov: ”Some applications of duality for L´evy processes in a half-line.” Bull. London Math. Soc. 43,pp. 97-110, 2011.[3] D. Dufresne: ”The distribution of a perpetuity, with applications to risk theory and pension funding.” Scand.Actuarial J., pp. 39-79, 1990.[4] E. Hewitt, L. J. Savage: “Symmetric measures on Cartesian products.” Trans. Amer. Math. Soc., 80, pp. 470-501(1955).[5] K.B. Erickson, R.A. Maller: ”Generalised Ornstein-Uhlenbeck processes and the convergence of L´evy integrals.”S´eminaire de Probabilit´es XXXVIII, 2005.[6] D. Khoshnevisan, P. Salminen, M. Yor: ”A note on a.s. finiteness of perpetual integral functionals of diffusions.”Electr. Comm. Probab., Vol. 11, pp. 108-117.[7] A. Kyprianou, J.C. Pardo, V. Rivero: ”Exact and asymptotic n-tuple laws at first and last passage.” Ann. Appl.Probab. Volume 20, pp. 522-564, 2010.[8] A. Kyprianou: “Fluctuations of L´evy Processes with Applications.” Springer (2013).[9] R. Schilling, Z. Vondraˇcek: “Absolute continuity and singularity of probability measures induced by a purelydiscontinuous Girsanov transform of a stable process.” arXiv:1403.7364[10] A. Matsumoto, K. Yano: ”On a zero-one law for the norm process of transient random walk.” S´eminaire deProbabilit´es XLIII, pp. 105-126, 2011.[11] P. Salminen, M. Yor: ”Properties of perpetual integral functionals of Brownian motion with drift.” Ann. I.H.P., 41,pp. 335-347, 2005.[12] P. Salminen, M. Yor: ”Perpetual integral fuctionals as hitting and occupation times.” Electr. J. Probab., Vol. 10,p. 371-419, 2005.