Transition densities of spectrally positive Lévy processes

We deduce the asymptotic behavior of transition densities for a large class of spectrally one-sided Lévy processes of unbounded variation satisfying mild condition imposed on the second derivative of the Laplace exponent or, equivalently, on the real part of the characteristic exponent. We also provide sharp two-sided estimates on the density when restricted additionally to processes without Gaussian component.


Introduction
The aim of this paper is to discuss the behavior of the transition densities of spectrally one-sided Lévy processes of unbounded variation or, in other words, spectrally one-sided processes that are not subordinators with drift. Such processes, due to their specific structure, find natural applications in financial models, in particular, insurance risk modeling and queue theory, and therefore they have been intensively analyzed from that point of view. The prominent example here and, at the same time, one of the first questions we would like to ask in the financial setting is the so-called exit problem, the identification of the distribution of the pair ( I ), where is an open interval, which has been intensively discussed over last decades. We refer to prominent works of Zolotarev [26], Takács [25], Emery [5], and Rogers [21]. Vast majority of results is expressed in terms of so-called scale functions, which have been of independent interest later on; see, for example, [10] or [16]. Also, specific structure of spectrally one-sided processes considerably simplifies the fluctuation theory, which for general Lévy processes is rather implicit. For details, we refer to books of Bertoin [1, Sects. VI, VII], Kyprianou [17], or Sato [23]. This short list is far from being complete, and for further discussion, we refer to the works mentioned and the references therein.
The abundant number of papers related to financial applications stays in stark contrast with the fact that surprisingly little is known about transition densities of general spectrally one-sided Lévy processes, although it seems that such knowledge can potentially be an important and useful tool. We may find, for example, asymptotic series expansion for the particular case of stable processes in the book of Zolotarev [27,Thm. 2.5.2] or the asymtotics with a fixed time variable under rather implicit assumptions in a recent article by Patie and 44 Ł. Leżaj Vaidyanathan [18], but to the author's best knowledge, general results are not obtained. Therefore the purpose of this paper is filling this gap in the theory and analysis of the behavior of the transition densities of spectrally one-sided Lévy processes in a feasibly wide generality.
Let us briefly describe our results. From theoretical point of view, the absence of negative (positive) jumps allows us to exploit techniques involving the Laplace transform, which can be easily proved to exist (see e.g., the book of Bertoin [1,Sect. VII]). We exploit that property in the first part, where we concentrate on the asymptotic behavior of the transition density, which is covered by Theorem 1. Note that the result is very general, as the only assumption is the lower scaling property with index 0. In particular, we assume neither the upper scaling property nor the absolute continuity of the Lévy measure (d ). Note that the independent sum of Brownian motion and a subordinator satisfying the scaling condition is admissible. The case = 2 is also included. Recall that without Gaussian component, the condition of having unbounded variation is tantamount to satisfying the integral condition (see Preliminaries for elaboration) In fact, if this is the case, then the assumption 0 may seem superfluous at first sight, as the integrability condition (1.1), roughly speaking, requires enough singularity of order at least 1. We may, however, construct a bit pathological example of a Lévy process of unbounded variation but with lower scaling index strictly smaller than 1. We refer the reader for details to Remark 1 and Example 2, but we highlight here that such processes are also included. Let us note in passing that, in some cases, it is easier to impose scaling condition on the real part of the characteristic exponent instead of on the second derivative of the Laplace exponent. These two are in fact equivalent, and we state that result in Corollary 6. If any of them is true, then by (4.5) and (4.6) we have, for some 0 0, Here is the Laplace exponent, and is the characteristic exponent of Lévy process. Admittedly, one of the strengths of Theorem 1 lies in the fact that the expression in the exponent is given explicitly and there is no hidden unknown constant. Nonetheless, in view of the equation above, we can substitute the Laplace exponent with characteristic exponent, if necessary, at the cost of losing an exact formula and implicit constant, which will appear instead.
Next, we restrict ourselves to processes of unbounded variation without Gaussian component and focus on upper and lower estimates on the transition density. Whereas the former, covered by Theorem 2, requires no additional assumptions and is independent of the previous results, the latter, consisting of Lemmas 3 and 4, requires apparently stronger conditions, 1 and 1, respectively. They provide local and tail lower estimates on the transition density, and the proof of the latter relies strongly on Theorem 1. As above, we point out that the condition 1 is not very restrictive in the class of processes of unbounded variation. Finally, we merge all previous results in Theorem 4 to obtain sharp two-sided estimates. They require both lower and upper scaling condition with indices strictly separated from 1 and 2, that is, 1 2. Let us note here that in contrast to symmetric processes, where a lot is already known, the general nonsymmetric case is still under development. It usually requires either implying familiar structure or imposing complex assumptions on the process; see, for example, [8,12,13,14,15,19] and the references therein. Recently, estimates for subordinators in a general setting were obtained (see, e.g., [3,4,7]). Still we are not aware of any paper that comprehensively treats the transition densities of general spectrally one-sided Lévy processes of unbounded variation. By inspecting the proofs of Lemmas 3 and 4, Theorem 3, and Proposition 9 we see that covering the limit cases using our methods is not possible, and it is not very surprising, as in general, they usually require more sophisticated methods or sometimes even a completely different approach. Nonetheless, the asymptotic behavior displayed by Theorem 1 covers both = 1 and = 2.
The paper is organized as follows. In Section 2, we introduce our setting and prove some basic properties of the Laplace exponent . Section 3 is devoted to the proof of the asymptotic behavior of the transition density. Upper and lower estimates are derived in Section 4, whereas in Section 5, we combine all previous results to obtain sharp two-sided estimates on the transition density.

Preliminaries
NOTATION. Throughout the paper, 1 1 denote positive constants, which may vary from line to line. By = ( ) we mean that the constant depends only on one parameter . For two functions : (0 ) [0 ], we write if the quotient stays between two positive constants. Analogous rule is applied to symbols and . We set = min and = max . The Borel sets on are denoted by ( ).
Let X = ( : 0) be a spectrally positive Lévy process, that is, a Lévy process on with only positive jumps that is not a subordinator with drift. There is a function : such that for all 0 and , There are 0, , and -finite measure on (0 ) such that for all , By we denote the Laplace exponent of X, that is, By holomorphic extension we can see that Let us introduce a symmetric, continuous, and nondecreasing majorant of Re , ( ) = sup Re ( ) 0 and its generalized inverse function 1 ( ) = sup 0: ( ) = . Then we have , then the underlying process has almost surely bounded variation, and we can rewrite (2.1) as follows: If this is the case, then = + 0 + , where B is a Brownian motion, and T = ( : 0) is a subordinator, that is, a one-dimensional Lévy process with nondecreasing paths that starts from 0. Although it is usually clear from the setting, to avoid unnecessary confusions, we assume in the whole paper that X is of unbounded variation. By [23,Thms. 21.9,24.10] this is true either when 0 or when the following condition is satisfied:

Ł. Leżaj
Therefore in this paper, we exclude the case where X is a subordinator with drift. However, the independent sum of Brownian motion and a subordinator is included. For results concerning heat kernel estimates for subordinators, we refer the reader to [7] and the references therein. By differentiating (2.3) twice we can easily deduce that is convex, and since P( 1 1) 0 (in view of [23,Thm. 24.10]), we have ( ) as . However, it is not necessarily positive. Indeed, differentiating (2.2) with respect to , setting = 1, and taking the limit as 0 + , we have We see that E 1 ( + ]. In particular, if E 1 0, then 0 in a neighborhood of the origin. Let 0 be the largest root of . Similarly, let 1 = inf 0: ( ) 0 . We have 1 0 , and the equality may occur only for the case 0 = 1 = 0. Note that there is always a root of at = 0 and, due to convexity of , at most one root for 0, precisely, in the case 1 0. By the Wiener-Hopf factorization we get that is necessarily of the form where is the Laplace exponent of a (possibly killed) subordinator, known as an ascending ladder height process, with the Lévy measure given by See [10,Sect. 4] and the references therein. It is known that is a Bernstein function, that is, a function in such that its derivative is completely monotone (see, e.g., [24] for a thorough analysis). Following Pruitt [20], we define the concentration functions and by setting ( ) = Clearly, ( ) ( ). Note that by the Fubini-Tonelli theorem Furthermore, by [ [0 ) has weak lower scaling property at infinity if there are , (0 1], and 1 0 such that for all 1 and 1 , ( ) ( ). In such case, we write WLSC ( 1 ). Analogously, we say that has a weak upper scaling property at infinity ( WUSC ( 1 )) if there are , 1, and 1 0 such that for all 1 and 1 ,

( ) ( )
A convenient equivalent definition of scaling property is provided in [2]. Namely, Note here that a nonincreasing function with weak lower scaling property and nondecreasing function with weak upper scaling property have a doubling property. In particular, WLSC ( 2 0 ) has a doubling property.

Properties of the Laplace exponent ϕ
First, note that by differentiating (2.3) and using the inequality e 1 e , 0, we get that for all 0, Thus we get the claim for in the case 1 = 0 and (0) = 0. If 1 = 0 but (0) 0, then we clearly have (0) ( ) for all 0 and 1, and we get the claim with 1 = 2. Finally, if 1 0, then it remains to prove that there is 0 such that for all 2 1 0 and 1, Since ( 1 ) = 0, by the monotonicity of and we obtain , and (2.11) follows. Now, with the first part proved, a similar argument applies to the second and therefore is omitted.
Furthermore, for all Proof. We first note that the first inequality of (2.12) follows from (2.9). Let 2 0 and 1 . By the monotonicity of we have ( ) ( ) ( ) ( ). Put = 1 and = 2. By Proposition 1 we have ( ) ( ) where˜is taken from Proposition 1, and the first part follows. For the proof of the second part, it remains to note that by the monotonicity of , for Thus for 2 1 , we get the claim.
The implied constant depends only on 0 .
Proof. In view of Proposition 2, it remains to prove that if 0 0, then there is 0 such that for all . By [11,Lemma 3.9.34], for any + , Hence, for 0 2, , which, together with (2.14), implies (2.13), and the claim follows. Now we deduce some properties of and its derivatives following from scaling properties.
Proof. We proceed as in the proof of Proposition 1. Let 1 and 1 0 . By the weak scaling property of we have and the claim follows for the case 1 0 . If 1 0 , then by differentiating (2.3) we conclude that ( ) as . Since it is also monotone, there is 1 0 such that ( ) 2 ( 0 ) for all 1 , and, consequently, WLSC ( 1˜1) for some˜(0 1]. Finally, using the continuity and positivity of , we can extend the scaling area to ( 0 ) at the expense of worsening the constant. The proof of the weak scaling property of follows by an analogous argument.
Proof. Assume first that WLSC ( 1 1 ). We claim that (2.15) holds. In view of Proposition 2, it suffices to prove the first inequality. First, let 2 1 1 . By the weak scaling property of we have There is a constant 0 such that for all 0 , Proof. First, assume that = 0; the extension to any is immediate. Let : (0 ) be the function defined as Note that by the Fubini-Tonelli theorem, for 0, we have Since is nondecreasing, for any 0, Hence for any 2, Therefore setting = 2 0 , by the weak scaling property of we have At this stage, we choose 2 such that 2 2 2e 2 2 2 . Then again, by the weak scaling property of , for 0 , which ends the proof.
Since 1 e 2 ( ) by Lemma 1 we immediately obtain the following corollary.
Remark 1. Suppose = 0. It is known (see, e.g., [8,Lemma 2.9]) that the scaling property with the scaling index 1 implies unbounded variation. However, Theorem 1 holds in greater generality. Namely, there are processes of unbounded variation that satisfy the scaling condition with strictly smaller than 1, and we may construct a Lévy measure that satisfies (2.4) and whose corresponding second derivative of the Laplace exponent has lower and upper Matuszewska indices of orders 3 2 and 1 2, respectively. Thus the lower scaling condition for holds only for 1. An example of such a process is constructed in Example 2. We also note that the Gaussian component is not excluded.
Proof of Theorem 1. Let = ( ) and 0. We first show that provided that 0 and 2 ( ) , where for 0, we have set To this end, we recall that, by the Mellin inversion formula, if the limit then the probability distribution of has a density ( ), and Using change of variables twice, we obtain where is a number satisfying ( ). We also have  . For the case E 1 0, it is not, however, optimal, because we do not treat positive that may be in the area from Corollary 4. We also note that = 2 is included. Proof. We verify that for ( ) belonging to the set (3.10), we have 0 and ( ) 0 , where = ( ) 1 ( ). Let 1 2 , where 1 and 2 are taken from Propositions 2 and 1, respectively. By Propositions 2 and 1 we have In view of (3.11), we get that 2 ( )

Upper and lower estimates on the density
In this section, we assume that WLSC ( 2 0 ) for some (0 1], 0 0, and 0. By Theorem 1 the probability distribution has a density ( ). Let us define ( ) = 2 ( ), 0. Clearly, WLSC ( 0 ). By 1 we denote its right-sided inverse, that is, where ( ) = sup 0 ( ). Clearly, 1 is nondecreasing. Similarly to 1 , we have Note that since for all 0 and 1, Furthermore, for any 0, let be such that 1 ( ) = . Then by (4.2) and (4.1), for any 1, In view of (2.7), we have By Corollary 3 we have ( ) (1 ) for 0 1 0 , which, together with the scaling property, implies That finishes the proof for the case 0 = 0. If 0 0, then we observe that we also have ( ) 1 for all 0 1 0 . Since the second term on the right-hand side of (4.4) is constant, the proof is finished.
In view of (2.8), Proposition 5, and Corollary 3, we have for all 0 . In particular, WLSC ( 0 ) for some (0 1]. Furthermore, for all 0 , Thus, for all 0 , ( ) Re ( ) (4.6) As a corollary, we now present the aforementioned equivalence between the scaling property of the second derivative of the Laplace exponent and the real part of the characteristic exponent. Proof. In view of (4.5) and (4.6), it remains to prove the second implication in the corollary. We first prove that WLSC ( 1 0 ) for some 1 (0 1]. Let 0 and 1. By the scaling property of Re we have and for all ( 0 ), Furthermore, there is 1 such that for all 1 and ( 0 ), Proof. The first inequality of (4.7) follows immediately from (4.5). If 0 = 0, then the second inequality is also a consequence of (4.5). If this is not the case, then note that for 0 , and the first part is proved. Furthermore, it follows that 1 ( 1 ) 1 ( ) 1 ( ) for all ( 0 ). Hence, by Proposition 6, 1 ( ) 1 ( ) for all max ( 0 ) 2 (1 0 ) , and (4.8) follows by standard extension argument. The scaling property of 1 is an easy consequence of (4.8) and Proposition 6.

Since
, by Proposition 7 and (4.5) we immediately obtain the following: and for all Proof. Corollary 2 and Remark 4 yield the first part. The proof of the second is omitted due to similarity to the proof of Proposition 7.

Upper estimates
From this moment on, we additionally assume that = 0. As explained in Preliminaries, that is equivalent to saying that X satisfies the integral condition (1.1). Suppose WLSC ( 2 0 ). Recall that since is positive and continuous on (0 ), if 0 0, then at the cost of worsening the constant , we can extend the area of comparability to any 1 (0 0 ) so that WLSC ( 2 1 ), where depends on 1 . Thus if 1 0 and 0 0, then we may and do assume that 0 is shifted so that 0 1 . With this in mind, let us define : [0 ) [0 ] as Let us comment on the function . As we will see in Theorem 2, it will play a role of majorant on the transition density. In such a setting, it is clear that must be nonnegative. Moreover, in the proof, we will need it to be monotone. Now, if 1 0, then we know that is indeed monotone for (0 1 ) but also negative. Thus a change of sign is required. On the other hand, if 1 = 0, then 0, and there is no need for absolute value and shifting of 0 . In general, however, may be negative in a neighborhood of the origin and change sign at 0 , so we have to be careful in expanding the scaling area to the proper place. Note that by Corollary 1 and Remark 4, , where depends only on 1 . Denote Theorem 2. Let X be a spectrally positive Lévy process of infinite variation with Lévy-Khintchine exponent and Laplace exponent . Suppose that = 0 and WLSC ( 2 0 ) for some (0 1], 0 0, and 0. We also assume that ( ) has an almost monotone density ( ). Then the probability distribution of has a density ( ). Moreover, there is 0 such that for all (0 1 ( 0 )) and , Proof. In the first step, we verify the assumptions of [ and thus, by Corollary 1, ( ) ( ) for all 0. Since is nonincreasing, we conclude that the first assumption is satisfied. Next, we claim that has a doubling property on (0 ). Indeed, since is nonincreasing, by Remark 4, for 0 1 0 , we have with 1 = whenever 0 = 0. Note that if 1 48 ( 0 ), then using the positivity and monotonicity, we can expand the estimate for 1 48 ( 0 ), and the first step is finished. Therefore by [9, Thm. 2.1] there is 0 such that for all (0 1 ( 0 )) and , Now it suffices to prove that 1 (1 ) whenever ( ) ( ) 1 (1 ). First, note that for any (0 1], the condition ( ) ( ) 1 (1 ) implies Indeed, by Corollary 1 and Remark 4 we have ( ) (1 ), which entails (4.12). Furthermore, we have 1 3 1 (1 ) 1, since otherwise by (4.2) we would have 1 contrary to (4.12).

Lower estimates
We begin with an estimate, which, together with Theorem 2, will allow us to localize the supremum of ( ). Note that here we require the scaling condition with 1. Proof. Without loss of generality, we may assume that = 0. By [8,Thm. 5.4], for any 0, there is 0 such that for all (0 1 ( 0 )) and 1 (1 ), Since by Propositions 6 and 7 we have 1 (1 ) 1 1 (1 ) for all (0 1 ( 0 )), it suffices to prove that for some 1 0. To this end, note that for 1 2 0, we have Now put 1 = 1 ( ) and 2 = 1 (1 ). Then using again Propositions 6 and 7, we infer that and the proof is completed.
Now we treat the right tail of the transition density. In general, based on results concerning various kinds of Lévy processes, we expect the decay to be expressed in terms of Lévy measure. For instance, in the case of unimodal Lévy processes satisfying some scaling conditions, it is known [2, Thm. 21 and Corollary 23] that ( ) ( 0) ( ). In the nonsymmetric case, a similar right tail decay is displayed by the transition densities of subordinators (see, e.g., [7]). This is also the case for spectrally one-sided Lévy processes, as the following lemma states. See also Theorem 4 and Remark 6 for comments on two-sided estimate. Let X ( ) be a spectrally positive Lévy processes having the Laplace exponent , 1 2 . First, we note that 2 2 . Thus, for every 0, and, consequently, 1 2 ( ) 2 ( ) ( ) (4.14)

Sharp two-sided estimates
This section is devoted to derivation of sharp two-sided estimates. As mentioned in the Introduction, here we will require the upper scaling condition as well to express the Lévy density in terms of Laplace exponent . However, thanks to strict separation from the limit case = 1, first, we are able to provide a simpler expression for the localization of sup ( ).
Theorem 3. Let X be a spectrally positive Lévy process of infinite variation with Laplace exponent . Suppose that = 0 and WLSC ( 0 ) for some (0 1], 0 0, and 1. We also assume that ( 1 ) = 0. Then for all 1 2 , there is 0 such that for all (0 1 ( 0 2 0 )) and satisfying Proof. First, note that by Proposition 8 there is 1 such that for all (0 1 ( 0 2 0 )), Thus, in view of [8,Thm. 3.1] and Propositions 6 and 7, it suffices to prove the first inequality in (5.1). Next, note that the assumptions of Lemma 3 are satisfied. Let 0 be taken from Lemma 3; for fixed 0 and (0 1 ( 0 2 0 )), we set = ( 1 ( )). By Propositions 2 and 7 and (5.2) there is 1 (0 1] such that Now we are ready to prove our main result of this section. We also assume that the Lévy measure (d ) has an almost monotone density ( ). Then there is 1 (0 ] such that for all (0 1 ( 0 )) and Proof. Indeed, by Proposition 8 there is 1 1 such that for all ( 0 ), , where is taken from (4.9). Then it follows that Thus by Proposition 2 Moreover, also by Proposition 2 with 1 = ( 1) we have that for (0 1 ( 0 )),
Lastly, let us justify Remark 1 by constructing an example of spectrally positive Lévy process of unbounded variation for which the lower scaling property holds only with 1. (2 + 1)! 3 2 [(2 + 1)! (2 + 2)!] By construction, has the lower and upper Matuszewska indices equal to 3 2 and 1 2, respectively, and, consequently, the lower scaling property holds only with 1. Now let us define the Lévy process X by setting = 0, = 0, and as in (5.9). Then we have (2 + 1)! = Therefore X is of unbounded variation.