1 Introduction

Lévy processes conditioned on having a large height process

Abstract.

In the present work, we consider spectrally positive Lévy processes not drifting to and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with ) before hitting .

This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law of this conditioned process is defined as a Doob -transform via a martingale. For Lévy processes with infinite variation paths, this martingale is for some and where is the past infimum process of , where is the so-called exploration process defined in [10] and where is the hitting time of 0 for . Under , we also obtain a path decomposition of at its minimum, which enables us to prove the convergence of as .

When the process is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of . The computations are easier in this case because can be viewed as the contour process of a (sub)critical splitting tree. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.

Key words and phrases:
Lévy process, height process, Doob harmonic transform, splitting tree, spine decomposition, size-biased distribution, queueing theory
2000 Mathematics Subject Classification:
Primary: 60G51, 60J80; Secondary: 60J85, 60G44, 60K25, 60G07, 60G57

1. Introduction

In this paper, we consider Lévy processes with no negative jumps (or spectrally positive), not drifting to and conditioned to reach arbitrarily large heights (in the sense of the height process defined in [10]) before hitting 0. Let be the law of conditional on and be its natural filtration.

Many papers deal with conditioning Lévy processes in the literature. In seminal works by L. Chaumont [6, 7] and then in [8], for general Lévy processes, L. Chaumont and R. Doney construct a family of measures of Lévy processes starting from and conditioned to stay positive defined via a -transform and it can be obtained as the limit

for , and for an exponential r.v. with parameter 1 independent from the process and where is the killing time of . In the spectrally positive case, when , is a sub-probability while, if , it is a probability. In [14], K. Hirano considers Lévy processes drifting to conditioned to stay positive. More precisely, under exponential moment assumption, he is interested in two types of conditioning events: either the process is conditioned to reach after time or to reach level before . Then, at the limit , in both cases, he defines two different conditioned Lévy processes which can be described via -transforms. In [5, ch. VII], J. Bertoin considers spectrally negative Lévy processes, i.e. with no positive jumps, and also constructs a family of conditioned processes to stay positive via the scale function associated with .

Here, we restrict ourselves to study spectrally positive Lévy processes and consider a new way to obtain a Lévy process conditioned to stay positive without additional assumptions and contrary to [8], the law of the conditioned process is honest. The process is conditioned to reach arbitrarily large heights before . The term height should not be confused with the level used in the previously mentioned conditioning of Hirano. It has to be understood in the sense of the height process associated with and defined below. More precisely, for , we are interested in the limit

(1)

In the following, we will consider three different cases for the Lévy process : a Lévy process with finite variation and infinite Lévy measure, a Lévy process with finite variation and finite Lévy measure and finally a Lévy process with infinite variation.

In the first case, as it is stated in Theorem 2.3, the conditioning in (1) is trivial because for all positive .

In the second case, is simply a compensated compound Poisson process whose Laplace exponent can be written as

where is a finite measure on such that (without loss of generality, we suppose that the drift is ). Thus, is either recurrent or drifts to and its hitting time of is finite a.s. In this finite variation case, the height at time is the (finite) number of records of the future infimum, that is, the number of times such that

The process is then conditioned to reach height before . In the limit in (1), we get a new probability , the law of the conditioned process. It is defined as a -transform, via a martingale which depends on the jumps of the future infimum. In the particular case , this martingale is and we recover the same -transform as the one obtained in [8]. The key result used in our proof is due to A. Lambert [19]. Indeed, the process can be seen as a contour process of a splitting tree [13]. These random trees are genealogical trees where each individual lives independently of other individuals, gives birth at rate to individuals whose life-lengths are distributed as . Then, to consider conditioned to reach height before is equivalent to look at a splitting tree conditional on having alive descendance at generation .

Notice that we only consider the case when the drift of equals 1. However, the case can be treated in the same way because is still the contour process of a splitting tree but visited at speed .

We also obtain a more precise result about conditional subcritical and critical splitting trees. For , set the law of a splitting tree conditional on where denotes the number of extant individuals in the splitting tree belonging to generation . In fact, is a Galton-Watson process. We are interested in the law of the tree under as . We obtain that under a -condition on the measure , the limiting tree has a unique infinite spine where individuals have the size-biased lifelength distribution and typical finite subtrees are grafted on this spine.

The spine decomposition with a size-biased spine that we obtain is similar to the construction of size-biased splitting trees marked with a uniformly chosen point in [11] where all individuals on the line of descent between root and this marked individual have size-biased lifelengths. It is also analogous to the construction of size-biased Galton-Watson trees in Lyons et al. [23]. These trees arise by conditioning subcritical or critical GW-trees on non-extinction. See also [2, 12, 17]. In [9], T. Duquesne studied the so-called sin-trees that were introduced by D. Aldous in [2]. These trees are infinite trees with a unique infinite line of descent. He also considers the analogous problem for continuous trees and continuous state branching processes as made by other authors in [16, 17, 21].

We finally consider the case where has paths with infinite variation. Its associated Laplace exponent is specified by the Lévy-Khintchine formula

where , and either or . In order to compute the limit (1) in that case, we use the height process defined in [10, 20] which is the analogue of the discrete-space height process in the finite variation case. We set . Then, since 0 is regular for itself for , is defined through local time. Indeed, for , is the value at time of the local time at level 0 of where is the time-reversed process of at

(with the convention ) and is its past supremum.

Under the additional hypothesis

(2)

which implies that has paths with infinite variation and that the height process is locally bounded, we obtain a similar result to the finite variation case: the limit in (1) allows us to define a family of (honest) probabilities of conditioned Lévy processes via a -transform and the martingale

where is a random positive measure on which is a slight modification of the exploration process defined in [10, 20] and (since does not drift to ).

Again, in the recurrent case (i.e. if ), we observe that the previous quantity equals and we recover the -transform of [8] in the spectrally positive case. Indeed, for general Lévy processes, the authors consider the law of the Lévy process conditioned to stay positive which is defined via the -transform

where is the past infimum process and is a local time at 0 for . In the particular spectrally positive case, and is the identity. Then, in the recurrent case, our conditioned process is the same as the process defined in [6, 7, 8].

However, when drifts to , i.e., when , the probability measure is different from those defined in the previously mentioned papers.

Under , the height process can be compared to the left height process studied in [9]. In that paper, T. Duquesne gives a genealogical interpretation of a continuous-state branching process with immigration by defining two continuous contour processes and that code the left and right parts of the infinite line of descent. We construct two similar processes for conditioned splitting trees in Section 3.

We also obtain a path decomposition of at its minimum: under , the pre-minimum and post-minimum are independent and the law of the latter is which is, roughly speaking, the excursion measure of conditioned to reach ”infinite height”. Since under , starts at 0, this probability can be viewed as the law of the Lévy process conditioned to reach high heights and starting from 0. For similar results, see [6, 7, 8] and references therein. As in [8], the decomposition of under implies the convergence of as to . Recently, in [25], L. Nguyen-Ngoc studied the penalization of some spectrally negative Lévy processes and get similar results as ours about path decomposition.

The paper is organized as follows. In Section 2, we treat the finite variation case and investigate the limiting process after stating some properties about splitting trees. Section 3 is devoted to studying the conditioned splitting tree and Section 4 to considering Lévy processes with infinite variation and to giving properties of the conditioned process.

2. Finite variation case

2.1. Definitions and statement of result

Let be a positive measure on such that and

and let be a spectrally positive Lévy process with Lévy measure and such that

where is the law of conditioned to and

We denote by the natural filtration of . We will suppose that that is, is recurrent () or drifts to (). Then the hitting time is finite almost surely. Observe that since is spectrally positive, the first hitting time of is .

Definition 2.1.

The height process associated with is defined by

We set

and we denote the jumps of by for and (see Figure 1).

The assumption implies that the paths of have finite variation and then for all positive , is finite a.s. (Lemma 3.1 in [20]).

Remark 2.2.

The process can be seen as a LIFO (last in-first out) queue [20, 27]. Indeed, a jump of at time corresponds to the entrance in the system of a new customer who requires a service . This customer is served in priority at rate 1 until a new customer enters the system. Then, the ’s are the remaining service times of the customers present in the system at time .

The sequence can be seen as a random positive measure on non-negative integers which puts weight on . Its total mass is and its support is . We denote by the set of measures on with compact support. For in , set and

Then, according to [27, p.200], the process is a -valued Markov process. Its infinitesimal generator is defined by

(3)
Figure 1. A trajectory of the process started at and killed when it reaches 0 and the remaining service times at time for .

In the following proposition, we condition the Lévy process to reach arbitrarily large heights before .

Theorem 2.3.
  1. Assume that is finite and

    Then, for and ,

    where

    In particular, if , then . Moreover, the process is a -martingale under .

  2. If , the conditioning with respect to is trivial in the sense that for all ,

Observe that if , the process is simply a compensated compound Poisson process whose jumps occur at rate an are distributed as .

The proof of this result will be made in Section 2.3. It uses the fact that can be viewed as the contour process of a splitting tree visited at speed 1. The integrability hypotheses about are made in order to use classical properties of (sub)critical BGW processes that appear in splitting trees.

Notice that the case where is a Lévy process with Laplace exponent and can be treated in a same way if . Indeed, in that case, is still the contour process of a splitting tree but it is visited at speed . Theorem 2.3 is still valid but the martingale becomes

Before the proof, we define the splitting trees and recall some of their properties.

2.2. Splitting Trees

Most of what follows is taken from [19]. We denote the set of finite sequences of positive integers by

where .

Definition 2.4.

A discrete tree is a subset of such that

  1. (root of the tree)

  2. if for , then (if an individual is in the tree, so is its mother)

  3. , , ( is the offspring number of ).

If , then its generation is , its ancestor at generation is denoted by and if we denote by the concatenation of and

In chronological trees, each individual has a birth level and a death level . Let , be the two canonical projections of on and . We will denote by the projection of on

Definition 2.5.

A subset of is a chronological tree if

  1. (the root)

  2. is a discrete tree

  3. , such that if and only if . (resp. ) is the birth (resp. death) level of

  4. if , then (an individual has only children during its life)

  5. if then implies (no simultaneous births).

For , we denote by its lifetime duration. For two chronological trees and such that (not a death point) and for any (not a birth point), we denote by the graft of on at

Recall that is a -finite measure on such that . A splitting tree [11, 13] is a random chronological tree defined as follows. For , we denote by the law of a splitting tree starting from an ancestor individual with lifetime . We define recursively the family of probabilities . Let be the atoms of a Poisson measure on with intensity measure where is the Lebesgue measure. Then is the unique family of probabilities on chronological trees such that

where, conditional on the Poisson measure, the are independent splitting trees and for , conditional on , has law .
The measure is called the lifespan measure of the splitting tree and when it has a finite mass , there is an equivalent definition of a splitting tree:

  • individuals behave independently from one another and have i.i.d. lifetime durations distributed as ,

  • conditional on her birthdate and her lifespan , each individual reproduces according to a Poisson point process on with intensity ,

  • births arrive singly.

We now display a branching process embedded in a splitting tree and which will be useful in the following. According to [19], when , if for , is the number of alive individuals of generation

(4)

then under , is a Bienaymé-Galton-Watson (BGW) process starting at 1 and with offspring distribution defined by

(5)

Notice that under , is still a BGW process with the same offspring distribution but starting at , distributed as a Poisson r.v. with parameter .

Because of the additional hypothesis (resp. ), the splitting trees that we consider are subcritical (resp. critical). Then, in both cases they have finite lengths a.s. and we can consider their associated JCCP (for jumping chronological contour process) as it is done in [19].

It is a càdlàg piecewise linear function with slope which visits once each point of the tree. The visit of the tree begins at the death level of the ancestor. When the visit of an individual of the tree begins, the value of the process is her death level . Then, it visits backwards in time. If she has no child, her visit is interrupted after a time ; otherwise the visit stops when the birth level of her youngest child (call it ) is reached. Then, the contour process jumps from to and starts the visit of in the same way. When the visits of and all her descendance will be completed, the visit of can continue (at this point, the value of the JCCP is ) until another birth event occurs. When the visit of is finished, the visit of her mother can resume (at level ). This procedure then goes on recursively until level 0 is encountered ( = birth level of the root) and after that the value of the process is 0 (see Figure 2). For a more formal definition of this process, read Section 3 in [19].

Figure 2. On the left panel, a splitting tree whose ancestor has lifespan duration (vertical axis is time and horizontal axis shows filiation) and its associated Jumping Chronological Contour Process on the right panel.

Moreover, the splitting tree can be fully recovered from its JCCP and we will use this correspondence to prove Theorem 2.3. It enables us to link the genealogical height (or generation) in the chronological tree and the height process of the JCCP.

Proposition 2.6 (Lambert,[19]).

The process under has the law of the Lévy process under .

Moreover, if, as in Definition 2.1, for , we consider

then is exactly the genealogical height in of the individual visited at time by the contour process.

2.3. Proof of Theorem 2.3

Thanks to Proposition 2.6, the process is the JCCP of a splitting tree with lifespan measure . For , let . Then, . Furthermore, according to the second part of Proposition 2.6, the events { reaches height before } and {the splitting tree is alive at generation } coincide.

We first prove the simpler point (ii) of Theorem 2.3 where . According to [19], if for , denotes the sum of lifespans of individuals of generation in the splitting tree

then under , the process is a Jirina process starting at and with branching mechanism

that is, is a time-homogeneous Markov chain with values in , satisfying the branching property with respect to initial condition, (i.e. if and are two independent copies of respectively starting from and , then has the same distribution as starting from ) and such that

where is the -th iterate of . Hence, since is the genealogical height in the splitting tree ,

by monotone convergence and because the mass of is infinite.

We now make the proof of Theorem 2.3(i) and suppose that is finite.

(6)

We will investigate the asymptotic behaviors of the three probabilities in the last display as .

As previously, is the genealogical height in the splitting tree but in this case, we can use the process defined by (4). As explained above, this process is a BGW process with offspring generating function defined by (5) and such that under , has a Poisson distribution with parameter . With an easy computation, one sees that the mean offspring number equals . Hence the BGW-process is critical or subcritical. We have

and by the branching property,

We first treat the subcritical case. According to Yaglom [28], if is subcritical () and if , then there exists such that

(7)

In the following lemma, we show that this -condition holds with assumptions of Theorem 2.3.

Lemma 2.7.

If , then .

Proof.

According to (5),

by Fubini-Tonelli theorem. Since we have

and

Then if , the -condition of (7) is fulfilled and there exists a constant such that

Then,

Moreover,

and

where is some positive constant. Hence, using the dominated convergence theorem,

(8)

Similarly, if is critical (), since the variance of its reproduction law

is finite, one also knows [3] the asymptotic behavior of . Indeed, we have Kolmogorov’s estimate

(9)

Then

(10)

We are now interested in the behavior of as . In fact, we will show that it goes to 0 faster than (resp. ) if (resp. ). Since , the total number of jumps of before has a Poisson distribution with parameter . Hence, since ,

Thus, using the last equation and equation (8) or (10), the first term of the r.h.s. of (6) vanishes as for .

We finally study the term . For a word , and , we denote by the event { gives birth before age to a daughter which has alive descendance at generation }, that is, if ,

Let be the individual visited at time . Hence, using the Markov property at time and recalling that , we have

and by the branching property,

As previously, since computations for subcritical and critical cases are equivalent, we only detail the first one. We have, with another use of (7),

We want to use the dominated convergence theorem to prove that

and then, using (8),

so that the proof of the subcritical case would be finished. We have almost surely

where is a positive, deterministic constant. Hence, to obtain an integrable upper bound, since , it is sufficient to prove that

(11)

in order to use the dominated convergence theorem. Recall that denotes the time-reversal of at time

and is its associated past supremum process

It is known that the process has the law of under [5, ch.II]. We also have

which is the number of records of the process during and the ’s are the overshoots of the successive records. More precisely, if we denote by the record times of , the overshoots are

We come back to the proof of (11). We have

We denote by the natural filtration of . Thus, using Fubini-Tonelli theorem and the strong Markov property for applied at time

where