SOME PROPERTIES OF EXPONENTIAL INTEGRALS OF L´EVY PROCESSES AND EXAMPLES

The improper stochastic integral Z = R ∞− 0 exp( − X s − ) dY s is studied, where { ( X t , Y t ) , t ≥ 0 } is a L´evy process on R 1+ d with { X t } and { Y t } being R -valued and R d -valued, respectively. The condition for existence and ﬁniteness of Z is given and then the law L ( Z ) of Z is considered. Some suﬃcient conditions for L ( Z ) to be selfdecomposable and some suﬃcient conditions for L ( Z ) to be non-selfdecomposable but semi-selfdecomposable are given. Attention is paid to the case where d = 1, { X t } is a Poisson process, and { X t } and { Y t } are independent. An example of Z of type G with selfdecomposable mixing distribution is given.


Introduction
Let {(ξ t , η t ), t ≥ 0} be a Lévy process on R 2 . The generalized Ornstein-Uhlenbeck process {V t , t ≥ 0} on R based on {(ξ t , η t ), t ≥ 0} with initial condition V 0 is defined as where V 0 is a random variable independent of {(ξ t , η t ), t ≥ 0}. This process has recently been well-studied by Carmona, Petit, and Yor [3], [4], Erickson and Maller [7], and Lindner and Maller [10]. Lindner and Maller [10] find that the generalized Ornstein-Uhlenbeck process {V t , t ≥ 0} based on {(ξ t , η t ), t ≥ 0} turns out to be a stationary process with a suitable choice of V 0 if and only if Here L stands for "the distribution of". If {ξ t , t ≥ 0} and {η t , t ≥ 0} are independent, then P (L t = η t for all t) = 1. Keeping in mind the results in the preceding paragraph, we study in this paper the exponential integral ∞− 0 e −Xs− dY s , where {(X t , Y t ), t ≥ 0} is a Lévy process on R 1+d with {X t } and {Y t } being R-valued and R d -valued, respectively. In Section 2 the existence conditions for this integral are given. They complement a theorem for d = 1 of Erickson and Maller [7]. Then, in Section 3, some properties of are studied. A sufficient condition for µ to be a selfdecomposable distribution on R d is given as in Bertoin, Lindner, and Maller [2]. Further we give a sufficient condition for µ not to be selfdecomposable. Recall that, in the case where X t = t, t ≥ 0, and {Y t } is a Lévy process e −s dY s is always selfdecomposable if the integral exists and is finite (see e. g. [16], Section 17). In particular, we are interested in the case where {X t } and {Y t } are independent and {X t } is a Poisson process; we will give a sufficient condition for µ to be semi-selfdecomposable and not selfdecomposable and also a sufficient condition for µ to be selfdecomposable. In Section 4, we are concerned with µ of (1.6) when {X t } is a Brownian motion with positive drift on R, {Y t } is a symmetric α-stable Lévy process on R with 0 < α ≤ 2, and {X t } and {Y t } are independent. We will show that in this case µ gives a type G distribution with selfdecomposable mixing distribution, which is related to results in Maejima and Niiyama [12] and Aoyama, Maejima, and Rosiński [1].

Existence of exponential integrals of Lévy processes
We keep this set-up throughout this section. Let (a X , ν X , γ X ) be the Lévy-Khintchine triplet of the process {X t } in the sense that The following result is a d-dimensional extension of Theorem 2 of Erickson and Maller [7].
if and only if lim t→∞ X t = +∞ a. s. and where |y| is the Euclidean norm of y ∈ R d .
Proof. First, for d = 1, this theorem is established in [7]. Second, for j = 1, . . . , d, the jth coordinate process {Y (j) t , t ≥ 0} is a Lévy process on R with Lévy measure ν Y (j) (B) = R d 1 B (y j )ν Y (dy) for any Borel set B in R satisfying 0 ∈ B, where y = (y 1 , . . . , y d ). Third, the property (2.2) is equivalent to s exists and is finite = 1 for j = 1, . . . , d. (2.4) Next, we claim that the following (2.5) and (2.6) are equivalent: Put f (u) = log u/h X (log u) for u ≥ e c . This f (u) is not necessarily increasing for all u ≥ e c . We use the words increasing and decreasing in the wide sense allowing flatness. But f (u) is increasing for sufficiently large u ( > M 0 , say). Indeed, consider for x > c with n(y) = ν X ( (y, ∞) ). If n(c) = 0, then h X (x)/x = h X (c)/x, which is decreasing. If n(c) > 0, then we have, with d/dx meaning the right derivative, for sufficiently large x (note that n(x) → 0 and that x c (n(x) − n(y))dy = − x c ν X ((y, x])dy is nonpositive, decreasing, and eventually negative since ν X ((c, ∞)) = n(c) > 0). Thus f (u) is increasing for large u. Using this, we see that (2.5) implies (2.6), because, letting M 1 = M ∨M 0 , we have In order to show that (2.6) implies (2.5), let g(x) = h X (x) for x ≥ c and g(x) = h X (c) for −∞ < x < c. Then g(x) is positive and increasing on R. Assume (2.6). Let M 1 = M ∨ M 0 . Then, using the concavity of log(u + 1) for u ≥ 0, we have The first integral in each summand is finite due to (2.6) and the second integral is also finite because the integrand is bounded. This finishes the proof of equivalence of (2.5) and ( t } is not the zero process for some j. Hence, by the theorem for d = 1, lim t→∞ X t = +∞ a. s. and I j < ∞ for such j. For j such that {Y (j) t } is the zero process, ν Y (j) = 0 and I j = 0. Hence we have (2.6) and thus (2.3) holds due to the equivalence of (2.5) and (2.6).
Then lim t→∞ X t = +∞ a. s. and h X (x) is positive and bounded for large x. Thus (2.2) holds if and only if (2.7) Here log + u = 0 ∨ log u. For d = 1 this is mentioned in the comments following Theorem 2 of [7].
(ii) As is pointed out in Theorem 5.8 of Sato [17], lim t→∞ X t = +∞ a. s. if and only if one of the following (a) and (b) holds: In other words, lim t→∞ X t = +∞ a. s. if and only if one of the following (a ′ ) and (b ′ ) holds: ∞) )dy = ∞ and (2.8) holds. See also Doney and Maller [5]. (iii) If lim t→∞ X t = +∞ a. s., then h X (x) > 0 for all large x, as is explained in [7] after their Theorem 2.
When {X t } and {Y t } are independent, the result in Remark 2.2 (i) can be extended to more general exponential integrals of Lévy processes. if and only if We use the following result, which is a part of Proposition 4.3 of [19]. Proof of Theorem 2.3. Let E[X 1 ] = b. By assumption, 0 < b < ∞. By the law of large numbers for Lévy processes (Theorem 36.5 of [16]), we have lim t→∞ X t /t = b a. s. Hence Conditioned by the process {X t }, the integral t 0 e −|Xs−| α dY s can be considered as that with X s , s ≥ 0, frozen while Y s , s ≥ 0, maintains the same randomness. This is because the two processes are independent. Hence we can apply Proposition 2.4. Thus, if (2.10) holds, then Conversely, if (2.10) does not hold, then the conditional probability is less than 1 and (2.9) does not hold.
3 Properties of the laws of exponential integrals of Lévy processes.
Let µ be a distribution on R d . Denote by µ(z), z ∈ R d , the characteristic function of µ. We call µ selfdecomposable if, for every b ∈ (0, 1), there is a distribution ρ b on R d such that If µ is selfdecomposable, then µ is infinitely divisible and ρ b is uniquely determined and infinitely divisible. If, for a fixed b ∈ (0, 1), there is an infinitely divisible distribution ρ b on R d satisfying (3.1), then µ is called b-semi-selfdecomposable, or of class L 0 (b, R d ). If µ is b-semi-selfdecomposable, then µ is infinitely divisible and ρ b is uniquely determined. If µ is b-semi-selfdecomposable and ρ b is of class L 0 (b, R d ), then µ is called of class L 1 (b, R d ). These "semi"-concepts were introduced by Maejima and Naito [11]. We start with a sufficient condition for selfdecomposability of the laws of exponential integrals of Lévy processes.
Suppose in addition that {X t } does not have positive jumps and 0 < EX 1 < +∞ and that Then µ is selfdecomposable.
When d = 1 and Y t = t, the assertion is found in [9]. When d = 1, the assertion of this theorem is found in the paper [2] with a key idea of the proof. This fact was informed personally by Alex Since we are assuming that X t does not have positive jumps and that 0 < EX 1 < +∞, we have T c < ∞ and X(T c ) = c a. s. Then we have Since c is arbitrary, it follows that the law of Z is selfdecomposable.
We turn our attention to the case where {X t } is a Poisson process and {X t } and {Y t } are independent. The suggestion of studying this case was personally given by Jan Rosiński to the authors. In this case we will show that the law µ of the exponential integral can be selfdecomposable or non-selfdecomposable, depending on the choice of {Y t }. A measure ν on R d is called discrete if it is concentrated on some countable set C, that is, ν(R d \ C) = 0. Then the following statements are true.
(ii) Suppose that either {Y t } is a strictly α-stable Lévy process on R d , d ≥ 1, with 0 < α ≤ 2 or {Y t } is a Brownian motion with drift with d = 1. Then, µ is selfdecomposable and of class L 1 (e −1 , R d ).
(iii) Suppose that d = 1 and {Y t } is integer-valued, not identically zero. Let Then µ is not selfdecomposable and, furthermore, the Lévy measure ν µ of µ is discrete and the set of points with positive ν µ -measure is dense in D.
It is noteworthy that a seemingly pathological Lévy measure appears in a natural way in the assertion (iii). In relation to the infinite divisibility in (i), we recall that where W 0 , W 1 . . . . are independent and identically distributed and W n d = Y (T 1 ) ( d = stands for "has the same law as"). Consequently we have where W 0 and Z ′ are independent and is a Lévy process given by subordination of {Y s } by a gamma process. Here we use our assumption of independence of {N t } and {Y t }. Thus µ is e −1 -semi-selfdecomposable and hence infinitely divisible. An alternative proof of the infinite divisibility of µ is to look at the representation (3.5) and to use that L(Y (T 1 )) is infinitely divisible.
(ii) Use the representation (3.5) with W n d = U 1 , where we obtain a Lévy process {U s } by subordination of {Y s } by a gamma process. Since gamma distributions are selfdecomposable, the results of Sato [18] on inheritance of selfdecomposability in subordination guarantee that L(U 1 ) is selfdecomposable under our assumption on {Y s }. Hence µ is selfdecomposable, as selfdecomposability is preserved under convolution and convergence. Further, since selfdecomposability implies b-semi-selfdecomposability for each b, (3.6) shows that µ is of class L 1 (e −1 , R d ).
(iii) The process {Y t } is a compound Poisson process on R with ν Y concentrated on the integers (see Corollary 24.6 of [16]). Let us consider the Lévy measure ν (0) of Y (T 1 ). Let a > 0 be the parameter of the Poisson process {N t }. As in the proofs of (i) and (ii), Y (T 1 ) where {U s } is given by subordination of {Y s }, by a gamma process which has Lévy measure x −1 e −ax dx. Hence, using Theorem 30.1 of [16], we see that Suppose that {Y t } is not a decreasing process. Then some positive integer has positive ν (0)measure. Denote by p the minimum of such positive integers. Since {Y t } is compound Poisson, P (Y s = kp) > 0 for any s > 0 for k = 1, 2, . . .. Hence ν (0) ({kp}) > 0 for k = 1, 2, . . .. Therefore, for each nonnegative integer n, the Lévy measure ν (n) of e −n Y (T 1 ) satisfies ν (n) ({e −n kp}) > 0 for k = 1, 2, . . .. Clearly, ν (n) is also discrete. The representation (3.5) shows that Hence, ν µ is discrete and The following remarks give information on continuity properties of the law µ. A distribution on R d is called nondegenerate if its support is not contained in any affine subspace of dimension d − 1.
Remark 3.3 (i) Any nondegenerate selfdecomposable distribution on R d for d ≥ 1 is absolutely continuous (with respect to Lebesgue measure on R d ) although, for d ≥ 2, its Lévy measure is not necessarily absolutely continuous. This is proved by Sato [15] (see also Theorem 27.13 of [16]). (ii) Nondegenerate semi-selfdecomposable distributions on R d for d ≥ 1 are absolutely continuous or continuous singular, as Wolfe [20] proves (see also Theorem 27.15 of [16]).

An example of type G random variable
In Maejima and Niiyama [12], an improper integral was studied, in relation to a stationary solution of the stochastic differential equation where {B t , t ≥ 0} is a standard Brownian motion on R, λ > 0, and {S t , t ≥ 0} is a symmetric α-stable Lévy process with 0 < α ≤ 2 on R, independent of {B t }. They showed that Z is of type G in the sense that Z is a variance mixture of a standard normal random variable by some infinitely divisible distribution. Namely, Z is of type G if for some nonnegative infinitely divisible random variable V and a standard normal random variable W independent of each other. Equivalently, Z is of type G if and only if Z d = U 1 , where {U t , t ≥ 0} is given by subordination of a standard Brownian motion. If Z is of type G, then L(V ) is uniquely determined by L(Z) (Lemma 3.1 of [18]). The Z in (4.1) is a special case of those exponential integrals of Lévy processes which we are dealing with. Thus Theorem 3.1 says that the law of Z is selfdecomposable. But the class of type G distributions (the laws of type G random variables) is neither larger nor smaller than the class of symmetric selfdecomposable distributions. Although the proof that Z is of type G is found in [12], the research report is not well distributed. Hence we give their proof below for readers. We will show that the law of Z belongs to a special subclass of selfdecomposable distributions.
Theorem 4.1 Under the assumptions on {B t } and {S t } stated above, Z in (4.1) is of type G and furthermore the mixing distribution for variance, L(V ), is not only infinitely divisible but also selfdecomposable.
Since Γ −1 is selfdecomposable, V is also selfdecomposable due to the inheritance of selfdecomposability in subordination of strictly stable Lévy processes (see [18]). Therefore Z is of type G with L(V ) being selfdecomposable. Also, the selfdecomposability of Z again follows.
In their recent paper [1], Aoyama, Maejima, and Rosiński have introduced a new strict subclass (called M (R d )) of the intersection of the class of type G distributions and the class of selfdecomposable distributions on R d (see Maejima and Rosiński [13] for the definition of type G distributions on R d for general d). If we write the polar decomposition of the Lévy measure ν by where K is the unit sphere {ξ ∈ R d : |ξ| = 1} and λ is a probability measure on K, then the element of M (R d ) is characterized as a symmetric infinitely divisible distribution such that ν ξ (dr) = g ξ (r 2 )r −1 dr