A note on a Poissonian functional and a $q$-deformed Dufresne identity

In this note, we compute the Mellin transform of a Poissonian exponential functional, the underlying process being a simple continuous time random walk. It shows that the Poissonian functional can be expressed in term of the inverse of a $q$-gamma random variable. The result interpolates between two known results. When the random walk has only positive increments, we retrieve a theorem due to Bertoin, Biane and Yor. In the Brownian limit ($q \rightarrow 1^-$), one recovers Dufresne's identity involving an inverse gamma random variable. Hence, one can see it as a $q$-deformed Dufresne identity.


Introduction
Let P + and P − be two standard Poisson processes with respective intensities 1 and z = q µ ∈ [0, 1). Then ζ t := P + t − P − t is a compound Poisson process with jumps +1 or −1, and drifting to +∞. We investigate the law of the "perpetuity": where 0 < q < 1. We can also rewrite I (q,z) as: q Bn ε n where (ε n ; n ≥ 1) are i.i.d exponential random variables and B n is a standard random walk drifting to +∞. More precisely P (B n+1 − B n = 1) = 1 − P (B n+1 − B n = −1) = 1 1 + z > 1 2 In insurance and mathematical finance, the perpetuity can be understood as the total discounted cash-flow from a perpetual bond, with random interest rates. Corollary 2.2 allows to simulate it with little effort.
In the denomination of [BY05], the random variable we are interested in is an exponential functional associated to the Lévy process: log(q)ζ t We have: E Ä e s log(q)ζt ä = exp (tψ(s)) where the Lévy-Khintchine exponent is: In theorem 2.1, we give an explicit formula for the Mellin transform of I (q,z) . This allows to recognize that I (q,z) has the same law as the product of two independent random variables, one of which is the inverse of a q-gamma random variable -to be defined later in the text.
On the one hand, notice that if we specialize z to zero, we obtain: q n ε n which is exactly the object studied by Bertoin, Biane and Yor in [BBY04]. In that respect, our result is a refinement of their theorem 1.
On the other hand, consider the compound Poisson process ζ t properly rescaled in time and space thanks to the parameter q. Thanks to Donsker's invariance principle, it will converge to a Brownian motion as q → 1, in the uniform Skorohod topology. The exponential functionals will naturally converge too, although the claim requires some work because the exponential functionals are not continuous in the Skorohod topology. This way, one recovers Dufresne's identity involving the inverse of the usual gamma random variable.

Acknowledgments
The author would like to thank Dr. Alexander Watson and Prof. Juan Carlos Pardo for encouraging him to write this note.

Notations
If E is a topological space, we will denote by Bor(E) the σ-algebra of its Borel sets. And if X is a random variable valued in a measurable space (E, E), L (X) = P X is the probability measure on (E, E) known as the distribution of X. Equality in law is denoted by L =.
We will also make use of the Vinogradov symbol:

Main theorem and consequences
Recall the definition of the n-th q-Pochhammer symbol, with n ∈ N ⊔ {∞}: ∀a ∈ C, (a; q) n = n j=0 Ä 1 − aq j ä and the q-gamma function: The main result, whose proof is postponed to section 3, is the following: Theorem 2.1. Recall that z = q µ . The Mellin transform of I (q,z) for ℜ(s) < µ is given by: We denote by G a a geometric random variable with parameter 0 ≤ a < 1. Following [BBY04], if (G a , G aq , G aq 2 , . . . ) are independent geometric random variables with the specified parameters, we define the random variable R (q) a as: Remarkably, its Mellin transform for ℜ(s) ≥ 0 is expressed thanks to q-exponentials. If a = q κ , κ > 0: By taking q → 1 − , it is easy to see the convergence in law of R (q) q κ to a gamma random variable with parameter κ. In fact, R (q) a is the q-analogue of the gamma variable as constructed by Pakes ([Pak96] (5.5) ).
Let us comment on the proof of [BBY04] when z = 0. It goes as follows. Consider two independent random variables R (q) q and I (q) : after computing the entire moments of I (q) , one can notice that those of (1 − q)R (q) q I (q) match the moments of a standard exponential random variable ε. Since the exponential distribution is characterised by its moments, we can deduce (1 − q)R (q) q I (q) L = ε. One concludes by taking the Mellin transform on both sides of the previous equality in law. In the proof of theorem 2.1 though, there is a small catch. If the random variable I (q) has entire moments, I (q,z) does not for z > 0. Therefore, we will have to take a different path -through complex analysis.

Factorisation into independent random variables
The random variable I (q,z) can be factored into the product of two independent random variables, one of which is an inverse q-gamma: Corollary 2.2. The following equality in law holds: where I (q) and R (q) z are independent random variables. In particular, as q → 1, (1 − q) 2 I (q,q µ ) converges in law to 1 γµ with γ µ a gamma random variable with parameter µ.
Proof. For the equality in law, Mellin transforms of both sides are equal.
Since we already know that R (q) q µ converges in law to γ µ it is sufficient to prove that (1 − q)I (q) converges to 1 in probability, which is implied by: Remark 2.3. This last identity in law is, to the author, quite puzzling. Indeed, the perpetuity on the left-hand side involves adding both positive and negative powers of q randomly. On the right-hand side, we see that this tantamounts in law to paying the price of certain negative power of q that is a sum of geometric random variables, then multiplying by a random variable that has the same law as the one obtained when the random walk B only goes upward. A better understanding of such an equality would certainly be desirable.

Scaling limit
Now recall that the acronym càdlàg stands for "continuà droite, limitéà gauche" in French and means right-continuous with left-limits. Let D ∞ be the Skorohod space of functions on R: Let d be the local uniform distance between functions: The induced topology is called the uniform Skorohod topology and is finer than the usual Skorohod topology. However it is not separable. For a definition of the usual Skorohod topology, the reader is referred to section 12 in [Bil99]. D is the ball σ-field for the local uniform distance -or equivalently the Borel σ-field for the usual Skorohod topology ( (15.2) in [Bil99]).
We are concerned with the weak convergence of the following sequence of compound Poisson processes indexed by q: (1−q) 2 ; t ≥ 0 å along with their exponential functionals I Ä W (q,µ) ä where: The following theorem deals with the sequence of probability measures . We choose to work in the uniform Skorohod topology as the functional I is better controlled under variations of its argument's uniform norm. In the usual Skorohod topology, one would have to deal with a troublesome time rescaling as well.
(1 − q) 2 I (q,z) = I Ä W (q,µ) ä and we have the weak convergence on (D ∞ × R + , D ⊗ Bor (R + )): Therefore, we see that putting together proposition 2.4 and corollary 2. where the Lévy-Khintchine exponent is: It satisfies (8) and (9) in [BY05], thereby confirming the existence of negative moments for I (q,z) (theorem 3 in [BY05]). Its Mellin transform is therefore holomorphic on the left half of the complex plane. Let s = −r, with ℜ(r) > 0. One has (see equation (11) in [BY05]): Inspecting this formula, we see that E Ä I (q,z) ä −r admits a holomorphic extension to ℜ(r) > −µ and a meromorphic extension to all of C.
Also, if we define the meromorphic function ϕ on the right half plane by: then, we find using the previous recurrence that, outside of poles: Now, by examining its constituents, the function ϕ is holomorphic on the strip −µ < ℜ(r) < 1. Moreover, this strip is of length larger than one. Because it is 1-periodic, it is an entire function that vanishes on the integers. In order to see it vanishes identically, we invoke Carlson's theorem ( 9.2.1 in [Boa54]); which requires from ϕ not to grow too fast on vertical strips. In order to exclude maps such as z → sin(πz) πz , we need to check that ϕ is a function of exponential type at most π.
(2) There is a q 0 > 0 and a set S 0 ∈ D such that the probability measures (3) The exponential functional is continuous on S 0 .
The rest of the proof is broken down into these 3 steps.
Step 1: Weak convergence in the uniform Skorohod topology Using Wald's identity: Thanks to Donsker's invariance principle, we have weak convergence to W (µ) in the usual Skorohod topology. Because the statement is usually phrased for the rescaled discrete random walk, one can adapt theorem 14.1 in [Bil99] to the case of a compound Poisson process: because of the independence of increments, the convergence of finite dimensional marginals is obtained from the convergence of Lévy exponents. The two previous estimates on mean and variance give tightness. At this point, we need to upgrade to upgrade weak convergence to the finer (locally uniform) topology. This is achieved by applying theorem 6.6 in [Bil99]. Indeed, the Wiener measure is supported on the space of continuous functions; which is separable for the locally uniform topology. Moreover convergence in the usual Skorohod topology to a continuous function implies local uniform convergence.
Step 2: Definition of a set S 0 of full measure Consider 0 < ν < µ and q 0 close enough to 1 so that: Then, for q 0 < q < 1, t → W (q,µ) t −νt drifts to ∞ almost surely. Therefore, the probability measures Ä L Ä W (q,µ) ä ; q ≥ q 0 ä are supported on: tep 3: Continuity of the exponential functional on S 0 Let x ∈ S 0 and consider a sequence (x n ) n∈N in S 0 such that d(x, x n ) → 0. If we set: Then, using the inequality: ∀(x, y) ∈ R 2 , |e −2x − e −2y | ≤ e −2x∧y 1 ∧ |x − y|, The above integral is clearly dominated by 2 ∞ 0 e −2νs−2m < ∞ and the integrand goes to zero because of the uniform convergence x n → x on compact sets. Thanks to the dominated convergence theorem, we can conclude that I(x n ) → I(x).