Closed-form formulas for the distribution of the jumps of doubly-stochastic Poisson processes

We study the obtainment of closed-form formulas for the distribution of the jumps of a doubly-stochastic Poisson process. The problem is approached in two ways. On the one hand, we translate the problem to the computation of multiple derivatives of the Hazard process cumulant generating function; this leads to a closed-form formula written in terms of Bell polynomials. On the other hand, for Hazard processes driven by L\'evy processes, we use Malliavin calculus in order to express the aforementioned distributions in an appealing recursive manner. We outline the potential application of these results in credit risk.


Introduction
Consider an ordered series of random times τ 1 ≤ ... ≤ τ m accounting for the sequenced occurence of certain events. In the context of credit risk, these random times can be seen as credit events such as the firm's value sudden deterioration, credit rate downgrade, the firm's default, etcetera. The valuation of defaultable claims (see [5,18]) is closely related to computation of the quantities P(τ n > T |F t ), t ≥ 0, n = 1, ..., m, where the reference filtration F = (F t ) t≥0 accounts for the information generated by all state variables. * Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, E-08007 Barcelona, Spain. E-mail: arturo@valdivia.xyz An interesting possibility to model these random times consists in considering τ 1 , ..., τ m as the succesive jumps of a doubly-stochastic Poisson process (DSPP). That is, a timechanged Poisson process (P Λt ) t≥0 , where the time change (Λ t ) t≥0 is a non-decreasing càdlàg F-adapted process starting at zero; and the Poisson process (P t ) t≥0 has intensity rate equal to 1, and it is independent of F. We refer to (Λ t ) t≥0 as the Hazard process.
The purpose of this note is to study the obtainment of closed-form formulas for the distributions of the n-th jump of a doubly-stochastic Poisson process. We address the problem from two different approaches. First, we relate this problem to the computation of the first n derivatives of the Hazard process cumulant generating function. As shown below, the result is written in closed-form in terms of Bell polynomials on the aforementioned derivatives -see [6,12,16] for details on these polynomials. If Λ T has a finite conditional n-th moment ( i.e., E [ Λ n T | F t ] < ∞), then the following equation holds true where B k is the k-th Bell polynomial.
In light of this result, two considerations are in order. On the one hand, it is desirable to consider a model for (Λ t ) t≥0 having a cumulant generating function Ψ being analytic (around i), so that arbitrary jumps of the doubly-stochastic Poisson process can be handled.
On the other hand, it is straightforward to compute (1.1) in closed-form given a tratactable expression for the cumulant generating function Ψ. See examples in Section 2.
As a second approach, we compute the aforementioned distributions direcly, by means of the Malliavian calculus. For this approach we consider a strictly positive pure-jump Lévy process (L t ) t≥0 with Lévy measure ν, and having moments of all orders -see [1,17] and [7] for a general exposition about Lévy processes and Malliavin calculus. We then assume that the Hazard process is of the form where N is the compensated Poisson random measure associated (L t ) t≥0 , and σ is a deterministic function, integrable with respect to N . Assume further that F is given by the natural filtration generated by the driving Lévy process (L t ) t≥0 . In this setting, we have the following result.
Theorem 1.2. The conditional distribution of the n-th jump of doubly-stochastic Poisson process with Hazard process satisfying (1.2) is given by where the quantities m 0 , m 1 , ..., m n are given recursively according to

3)
and for r ≥ 1 The rest of the paper is organized as follows. In Section 2 we present relevant examples appearing the literature. Finally in Section 3 we provide the proofs of our results.
Let us remark that eventhough our study is motivated by the valuation of defaultable claims, our results can potentially be also used in other areas; see for instance [2,14,21] and references therein.

Examples
In many traditional models (e.g., [8]) the Hazard processes (Λ t ) t≥0 is assumed to be absolutely continuous with respect to the Lebesgue measure, that is, where the process (λ t ) t≥0 is usually refer to as the hazard rate, and it is seen as the instantaneous rate of default in the credit risk context. The following two examples show how to use Theorem using two prominent particular cases for the hazard rate -and consequently for the Hazard process.
Example 2.1. The integrated square-root process (Λ intSR t ) t≥0 (see [9]) defined by means of (2.1) where the hazard rate is given by the solution of where (W t ) t≥0 is a Brownian motion, and we assume σ > 0 and ϑκ ≥ σ 2 in order to ensure that (λ SR t ) t≥0 remains positive. Take now F as the natural filtration generated by (W t ) t≥0 . It is well-known that the correspondent Hazard process has an analytic cumulant generating function given by where the functions A and B are given by The simplicity of Ψ intSR allows to compute its partial derivatives involved in (1.1). And finally we can use the n-th Bell polynomial B n characterization given by where in each column the remaining entries below the −1 are equal to zero. For instance, one can easily see that the first three Bell polynomials are Example 2.2. The integrated non-Gaussian Ornstein-Uhlenbeck processes (see [3]) defined by means of where ϑ, λ 0 > 0 are free parameters, λ 0 being random, and (L t ) t≥0 a non-decreasing purejump positive Lévy process. Equivalently, we can consider again the model in (2.1) where this time (λ t ) t≥0 is given by the solution of An interesting property of this Hazard rate process is that it has continuous sample paths.
It can be shown that where we take F 0 = σ(λ 0 ), that is, the σ-algebra generated by λ 0 . Particular cases of interest are the following. On the one hand, we have the so-called Gamma(a, b)-OU process which is obtained by taking (L t ) t≥0 as a Compound Poisson process where (Z t ) t≥0 is a Poisson process with intensity aϑ, and (x n ) n≥1 is a sequence of independent identically distributed Exp(b) variables. In this case, the correspondent Hazard process in (2.2) has a finite number of jumps in every compact time interval. Moreover, On the other hand, we have the so-called Inverse-Gausssian(a, b)-OU process (see [13] and Tompkins and Hubalek (2000)) which is obtained by taking (L t ) t≥0 as the sum of two independent processes, where (Z t ) t≥0 is a Poisson process with intensity 1 2 ab, and (x n ) n≥1 is a sequence of independent identically distributed Normal(0, 1) variables. In this case, the correspondent Hazard process in (2.2) jumps infinitely often in every interval. Moreover, the equation where, using c := −2b −2 iuϑ −1 , the function A is defined by In both of the cases above, we can see that the simplicity of Ψ allows to compute (1.1) in a straightforward way.
This traditional approach reduces the analytical tractability of the model, along with its parameters calibration. Indeed, suffices to say the Laplace transform of a Hazard process as in (2.1) is known in closed-form only for a reduced number of Hazard rates models.
That is one the reasons why in more recent contributions the modelling focus is set on the Hazard process itself, without requiring to make a reference to the Hazard rate -see for intance [4,13]. In this line, consider a Hazard process (Λ t ) t≥0 as given in (1.2). The following example provides an explicit computation the quantities involved in Theorem The Gamma process and the Inverse Gaussian process can be seen as particular cases by taking Y = 0 and Y = 1 2 , respectively, see [18]. Consider now a Hazard process of the form This is equivalent to take, in (1.
Finally, let us remark that we when considering a model like (1.2), the quantities appearing in Theorem 1.1 and Theorem 1.2 can be related according to the following.
Example 2.4. Let the Hazard process (Λ t ) t≥0 be given as in (1.2). It can be seen that in this case (Lemma 3.2 below) the cumulant generating function is given by Consequently, if the function σ has finite momentŝ T 0ˆR 0 σ k (s, z)dsν(dz) < ∞, k = 1, ..., n, (2.4) then the n-th derivative of Ψ is given by Indeed, these equations can be obtained by succesive differentiation under the integral sign due to the assumption (2.4).

Proofs
Let us start by the construction of the doubly-stochastic Poisson process that we shall consider in what follows.
Let F = (F t ) t≥0 denote our reference filtration; we shall assume that it satisfies the usual conditions of P-completeness and right-continuity. Let the i.i.d. random variables η 1 , ..., η m be exponentially distributed with parameter 1, all being independent of F ∞ . Then the n-th jump of the doubly-stochastic Poisson process with Hazard rate (Λ t ) t≥0 can be characterized as This construction leads to the following expression for the conditional distribution of the DSPP n-th jump Indeed, by construction, since conditioned to F ∞ the random variable η 1 + ... + η n has an Gamma distribution. The result then follows by preconditioning to F t -recall that (Λ t ) t≥0 is F-adapted.
Notice first that by conditioning (3.2) to F t we get Then the purpose of Theorem 1.1 and Theorem 1.2 is to provide a way to compute the conditional expectations in the equation above.

Proof of Theorem 1.1
Let µ t stand for the conditional (to F t ) law of Λ T , so that the assumption on the n-th conditional moment readsˆR As in the unconditional case (cf. [11, Theorem 13.2]), the condition above ensures that the conditional characteristic function has continuous partial derivatives up to order n, and furthermore the following equation holds true Now, let us recall that the n-th Bell polynomial B n can also be written as B n (x 1 , ..., x n ) = n k=0 B n,k (x 1 , ..., x n−k+1 ), (3.4) where B n,k stands for th partial (n, k)-th Bell polynomial, i.e., B 0,0 := 1 and where the sum runs over all sequences of non-negative indices such that j 1 + j 2 + · · · = k and j 1 + 2j 2 + 3j 3 + · · · = n. Using the Bell polynomials we have an expression for the chain rule for higher derivatives: where the superscript denotes the correspondent derivative, i.e., f (k) := d k dx k f and g (k) := d k dx k g, which are assumed to exist. This expression is known as the Riordan's formula -for these results on Bell polynomials we refer to [6,12,16].

Proof of Theorem 1.2
From this moment on, we shall work with a strictly positive pure-jump Lévy process (L t ) t≥0 having a Lévy measure ν satisfyinĝ for every ε > 0 and certain p > 0. This condition implies in particular that (L t ) t≥0 have moments of all orders, and the polynomials are dense in L 2 (dt × ν). Notice that this condition is always satisfied if the Lévy measure has compact support.
In other to prove the corollary we need the following.

Preliminaries on Malliavin calculus via chaos expansions
Let us now introduce basic notions of Malliavin calculus for Lévy processes which we shall use as a framework. Here we mainly follow [7].
For constant values f 0 ∈ R we set I 0 (f 0 ) := f 0 . In these terms, the Wiener-Itô chaos expansion for Poisson random measures, due to [10], states that every F T -measurable random variable F ∈ L 2 (P) admits a representation via a unique sequence of elements f n ∈ L 2 T ((dt × ν) n ). In virtue of this result, each random field (X t,z ) (t,z)∈[0,T ]×R 0 has an expassion T ]×R 0 . Now we are in position to define the Skorohod integral and the Malliavin derivative.
and has Skorohod integral with respect to N δ(X) =ˆT I n+1 (f n ).
Definition. Let D 1,2 be the stochastic Sobolev space consisting of all F T -measureble random variables F ∈ L 2 (P) with chaos expansion F = ∞ n=0 I n (f n ) satisfying For every F ∈ D 1,2 its Malliavin derivative is defined as nI n−1 (f n (·, t, z)).
We have the following theorems are central for the results below; for their proof and more details we refer to [7] and references therein.

A recursive formula
Lemma 3.1. For every deterministic Skorohod integrable function f and non-negative integer n, define If Y ∈ D 1,2 is bounded, then the Malliavin derivative of Y X n is given (dt × ν−a.e.) by Proof. By the product rule we have Moreover, since D s,z F = f (s, z), then an application of the chain rule tells us that Combining these expressions we get Thus, rewritting the last equivalence in terms of X 1 , ..., X n , we get the result. Proof. Since the integrands µ and σ are deterministic, then the increment Λ T − Λ t is independent of F t and Notice that for every deterministic function f the process (E t (f )) t≥0 defined by is a Doléans-Dade exponental martingale. Thus E[E T (f )] = 1, and so In our case this reads as . where the second line follows from the duality formula, and the last one from Lemma 3.1 by setting Y = 1. The result then follows by the linearity of the expectation.

Proof of Theorem 1.2
Notice that (3.3) can be rewritten as Indeed, it suffices to expand the factor and use that Λ t is F t -measurable. Now, since the integrand in (1.2) is deterministic, we have that the increment Λ T − Λ t is independent of F t and thus Applying