On the construction of low-parametric families of min-stable multivariate exponential distributions in large dimensions

Abstract Min-stable multivariate exponential (MSMVE) distributions constitute an important family of distributions, among others due to their relation to extreme-value distributions. Being true multivariate exponential models, they also represent a natural choicewhen modeling default times in credit portfolios. Despite being well-studied on an abstract level, the number of known parametric families is small. Furthermore, for most families only implicit stochastic representations are known. The present paper develops new parametric families of MSMVE distributions in arbitrary dimensions. Furthermore, a convenient stochastic representation is stated for such models, which is helpful with regard to sampling strategies.

motivation of modeling lifetimes. Such "true" multivariate exponential distributions have been de ned in the statistical literature, see, e.g., [12]. In [5], one can nd further support for the application of these dependence concepts. In the past, some of the criticism surrounding copula modeling in the context of portfolio credit risk can be attributed to the lack of such a natural link between the copula model and the intuition of lifetime modeling. Though being a very useful class of distributions and being very well studied on an abstract level, (see, e.g., [20,Chapter 6]; [24,35]), the number of known parametric MSMVE families, in particular in large dimensions, is rather small. Furthermore, the number of parametric models for which concrete stochastic representations are known is even smaller. For most families, only the spectral representation of [7], or an implicit stochastic representation as a limit distribution can be stated. However, concrete stochastic representations are helpful for practical applications as they allow for a more intuitive understanding and can serve as a starting point for developing e cient simulation algorithms. Motivated by the above considerations, there is some recent work aiming at the construction of new and exible parametric MSMVE families in large dimensions, see, e.g., [1,10,13,41]. In the same spirit, the aim of the present paper is to develop new parametric models that have a convenient stochastic representation. We will develop two similar classes of MSMVE distributions, giving rise to a huge quantity of parametric MSMVE models in arbitrary dimensions (see Theorems 1 and 2). The underlying stochastic model is given by a frailty construction, i.e. the components can be de ned as rst-passage times of a stochastic process across independent trigger levels. Similar to [19], the aim is to introduce new classes of models, while an application of these in practice requires a further, very detailed investigation of particular parametric families, which lies outside the scope of this work. Our ndings have at least three important implications: (i) They give rise to a huge class of parametric stable tail dependence functions in arbitrary dimensions. (ii) The underlying stochastic model can be used for e cient simulations, in particular in large dimensions, to construct non-exchangeable extensions (see Lemma 1), and to investigate statistical properties of the associated MSMVE distribution. (iii) Based on the speci c stochastic model, which can be interpreted as a so-called frailty construction, one further advantage is observed: When considering large homogeneous credit portfolios, Glivenko-Cantelli type approximations for the portfolio loss process as presented in [28] are available. Since the default times are conditionally independent and identically distributed in such models, for a large portfolio, the portfolio loss can be approximated by the conditional default probability. The famous Vasicek formula is derived using the same idea, see [42]. These approximations allow, e.g., for semi-analytic evaluation of collateralized debt obligations or similar non-linear derivatives depending on the portfolio loss.
The remainder of the paper is organized as follows. In Section 2, the necessary mathematical concepts are introduced, including a brief introduction to MSMVE distributions and their connection to so-called IDT subordinators. In Section 3, a speci c family of IDT subordinators and the related family of MSMVE distributions is investigated. A similar example is sketched in Section 4. A note on simulation can be found in Section 5. Section 6 concludes.

Mathematical prerequisites and notation
Relying on the min-stability as a characterizing property re ects the philosophy of [12] to lift the concepts and properties of the one-dimensional exponential law to higher dimensions. Furthermore, these distributions can be seen as possible distributions of asymptotic component-wise minima, as stated in [32], or as limiting extreme-value distributions with (negative) exponential marginals, as stated in [8].
It is well-known, see, e.g., [20,Theorem 6.2,p. 174], that the survival functionF of an MSMVE distribution can be written asF Additionally assuming all marginal laws to be unit exponentials, such a function is called a stable tail dependence function and (e i ) = is ful lled for all unit vectors e i , ≤ i ≤ d. The previously mentioned link between MSMVE distributions and extreme-value distributions can also be stated using the concept of copulas (for an introduction, see, e.g., the textbooks [20] and [31]): C is the copula of an extreme-value distribution, a so-called extreme-value copula (see [14]), if and only if with a stable tail dependence function. Furthermore, the survival copula of an MSMVE vector (X , . . . , X d ) is exactly of this kind. One of the major ndings in multivariate extreme-value theory, at least known since [9] and many times rediscovered and re-formulated since then, is a one-to-one relationship between d-dimensional MSMVEs and certain measures on a subspace of R d + , which is somehow comparable with the one-to-one relationship between in nitely divisible distributions and their associated Lévy measures. A quite recent, purely analytical derivation of this result can be retrieved from [36]. Additionally, the latter reference shows that a d-variate functionF : [ , ∞) d → [ , ] is an MSMVE survival function with unit exponential marginals if and only if the function is homogeneous of order , fully d-max-decreasing, and (e i ) = , ≤ i ≤ d, so these are necessary and sufcient conditions for : R d + → R+ to be a stable tail dependence function. This re nes a result of [16]. The homogeneity property is a reformulation of the extreme-value property of the underlying extreme-value copula in terms of the function . The fully d-max-decreasingness property is essentially a reformulation of the d-increasingness property of the associated MSMVE's distribution function after transformation to survival functions and an application of the log-transform. However, it is not easy to investigate this property analytically. Another characterization of stable tail dependence functions in terms of so-called "max-zonoids" is given in [30]. In the well-studied bivariate case, is characterized by the so-called Pickands dependence function A : [ , ] → [ / , ], which is de ned by A(t) := (t, − t). Su cient conditions for a function to be a bivariate Pickands dependence function are as follows, see [14,Theorem 2.3]: A is a bivariate Pickands dependence function if and only if A is convex and max{t, − t} ≤ A(t) ≤ , for all t ∈ [ , ]. Concerning measures of dependence, like concordance measures and tail dependence coe cients, it is well-known for a bivariate MSMVE vector (X , X ) that these measures can be computed easily from A (see, e.g., [14,17]), e.g. Note that ρ ≥ for all MSMVE distributions. Assuming the dependence structure, respectively function A, to be of a given parametric form, maximum likelihood, maximum pseudo-likelihood or rank-based moment estimators are available, see the literature on estimation of (bivariate extreme-value) copulas, e.g., [21,Chapter 5].
Our approach is dimension-free in the sense that instead of random vectors (X , . . . , X d ) we consider MSMVE sequences {X k } k∈N . These are sequences of random variables such that the min-stability property holds for all nite subsets of N, i.e. min i∈I {c i X i } is exponentially distributed for all nite I ⊂ N and c i > , for all i ∈ I.

. The IDT-frailty model
The starting point of our considerations is a result by [27] which shows that for an exchangeable MSMVE sequence {X k } k∈N there exists a stochastic representation as a frailty model where the processes H ( ) , . . . , H (n) are iid copies of H. Property (2) corresponds to the de nition of IDT processes as used in, e.g., [29] or [11], restricted to non-decreasing processes with some additional technical constraints, which guarantee that {X k } k∈N is well de ned. It is well-known for strong IDT processes, see, e.g., [29], that H t is in nitely divisible for every t > and thus, de ning Ψ H (x) := − log E exp(−x H ) yields a so-called Bernstein function. A function Ψ : ( , ∞) → R is a Bernstein function if and only if it admits a (unique) representation via with c ≥ the so-called killing term, a ≥ the drift term, and ν a measure on ( , ∞) satisfying ( ,∞) ( ∧ u) ν(du) < ∞, the Lévy measure. In the following, we will always ignore the so-called killing term as we only consider distributions on [ , ∞), i.e. we set it to zero. Bernstein functions are often extended to the domain [ , ∞) setting Ψ( ) := . The set of all Lévy measures is denoted M. For further information on Bernstein functions, see [39]. For the Laplace transform of the one-dimensional margins of a strong IDT subordinator, one has Thus, for every strong IDT subordinator, there exists a Lévy subordinator with the same marginal distributions. Actually, Lévy subordinators are the best studied example of strong IDT subordinators. The marginal distribution of H t is of relevance for the exchangeable sequence constructed in Equation (1), since Theoretically, construction (1) allows to de ne new parametric families of MSMVE distributions (and thus stable tail dependence functions) by de ning parametric families of strong IDT subordinators. Furthermore, using Lemma 1, it is possible to construct not only exchangeable sequences, but also non-exchangeable vectors, e.g. based on factor-model motivations.

Lemma 1 (Multi-factor MSMVE distributions).
Consider a probability space (Ω, F, P) supporting n + ∈ N independent, non-decreasing strong IDT subordinatorsH (i) = {H (i) t } t≥ , i = , . . . , n, and an independent iid sequence E , . . . , E d of exponential random variables with unit mean. Moreover, let A = (a i,j ) ∈ R d×(n+ ) be an arbitrary matrix with non-negative entries, having at least one positive entry per row. We de ne the vector-valued stochastic process , whose component processes are all strong IDT subordinators. Then, the random vector (X , . . . , X d ) de ned via has an MSMVE law.
Proof. See [26,Lemma 4.4]. Note that by de ning the entries of the matrix A appropriately and interpreting the processesH (i) as stochastic drivers, dedicated factor models can be constructed.
Besides Lévy subordinators there exist only few parametric families of (non-trivial) strong IDT subordinators and those are not very well investigated, yet. The aim of the present paper is to de ne new classes of MSMVE distributions by de ning suitable parametric families of strong IDT subordinators in a rst step.

. A class of strong IDT subordinators
Consider an arbitrary Lévy subordinator {Λ t } t≥ , i.e. a stochastically continuous process with independent and stationary increments, which is almost surely càdlàg and non-decreasing, see, e.g., [37, De nition 1.6], where we assume that for every ω ∈ Ω, Λ t (ω) is right-continuous in t, non-decreasing, and Λ (ω) = (see, e.g. [37, p. 197]). We consider instances of the general example H t := ∞ f (s/t) dΛs, with f a function ful lling certain conditions, see Lemma 2. This general example can be found, using slightly di ering notation, e.g., in [29], and in [2] as an example of an in nitely temporally selfdecomposable process. The integral could be de ned using the de nition of integrals with respect to independently scattered random measures by [34] or [38]. However, as we restrict ourselves to pathwise non-decreasing processes, it is possible to use pathwise the usual Lebesgue-Stieltjes integral for expressions of the form ∞ . . . dΛs, see, e.g., [23,Example 1.56]. This coincides a.s. with the more complex integral de nition in [34,38], so we can apply their results on properties of the integral.

Lemma 2 (A class of strong IDT subordinators). De ning pathwise
with H := for f a measurable, non-negative, non-increasing, left-continuous function, f ≢ , ful lling where a Λ and ν Λ are the drift and the Lévy measure of the subordinator Λ, yields a strong IDT subordinator H = {H t } t≥ in the sense of [27].
Proof. It is possible to de ne the integral (3) pathwise as a Lebesgue-Stieltjes integral. However, one has to admit the value +∞. The two integral conditions stated above are necessary and su cient conditions for the existence of the integral (for t = ) with respect to independently scattered random measures as stated in [34,Theorem 2.7]. One can show that the existence of that integral ensures the existence of the pathwise integral, and furthermore, the two de nitions coincide a.s..
The next two sections are devoted to a detailed investigation of two speci c families, namely families based on f (s) = ( − s)+ and f (s) = log + ( /s). We derive the related strong IDT subordinators and the resulting MSMVE distributions.

Family F1
We examine the construction of Lemma 2 using the function f (s) = ( − s)+ := max{ − s, }, which obviously ful lls the conditions stated in Lemma 2: As we can estimate f (s) ≤ 1 [ , ] (s), it follows that the rst integral expression is bounded above by a Λ + ∞ ( ∧ x) ν Λ (dx), and the second integral expression is bounded above with H := . The corresponding family of distributions resulting from construction (1) is called F1. The process H has an alternative representation using integration by parts, namely which can be seen as some kind of moving average of the increasing process Λ. From Equation (4) it can be seen that pathwise, H t equals a Williamson -transform evaluated at /t, see, e.g., [43] for the de nition of Williamson d-transforms. Consequently, H t = ψ( /t) with ψ a (random) convex and non-increasing function. It can be seen from the representation in Equation (5) that H t equals the product of a di erentiable function and a function that is a.e. di erentiable, a.s.. Consequently, the paths of H are a.e. di erentiable, a.s.. Furthermore, whereΛ is an independent copy of Λ andH the corresponding independent copy of H. Consequently, the increments, given the path of Λ up to time x, can be decomposed into a stochastic component independent of the previous evolution and a component measurable with respect to Fx := σ(Λs , ≤ s ≤ x). However, as the value of Λx can not be recovered from Hx, H is not Markovian.

. Attainable marginal distributions
In a rst step, we analyze possible marginal distributions of H that can arise from this construction, as we have seen above that these are of relevance for the corresponding distribution. Let Ψ Λ be the Laplace exponent of {Λ t } t≥ , ν Λ the corresponding Lévy measure, and a Λ its drift term. For the resulting strong IDT subordinator H, we denote its associated Bernstein function by Ψ H with Lévy measure ν H and drift a H . Let Φ denote the considered integral transform, i.e. Φ : L(Λ ) → L ( − s) dΛs , where L(.) denotes the law of some random variable, which we will use simultaneously on the level of considered Lévy measures Φ : ν Λ → ν H . Furthermore, de ne The corresponding class of distributions is called "Jurek class" or class of "s-selfdecomposable distributions", see [22], restricted to distributions on R+. It is shown in [22] that Φ : M → U is one-to-one, i.e. in particular every Bernstein function Ψ H possessing a non-increasing density can be attained by the given construction.

Lemma 3 (Lévy measures associated with H).
(i) Φ : M → U and the mapping is one-to-one. (ii) The Lévy density g H of the measure ν H is given by x , y > .

(iii) For any non-negative, measurable function h
Thus, for any non-negative, measurable function h, applying a substitution in the last step. This proves the claim.
Furthermore, it can be seen from Equation (6) that Φ is a so-called "Upsilon transform" in the sense of [4], with dilation measure γ(dx) = 1 [ , ] dx, from which more results can be derived, e.g., on continuity properties of the transform Φ. Actually, it is even possible to compute the pre-image Φ − (g(x)dx) explicitly, given a non-increasing Lévy density g. Using the previous results, we can conclude that for every Lévy subordinator with marginal distributions in the Jurek class, we can nd another non-decreasing process with the same marginal distributions and a.e. di erentiable paths, a.s..

. The corresponding MSMVE family
It is known that for all d ≥ , (X , . . . , X d ) constructed as in Equation (1) is a stable tail dependence function for every d ≥ . A random vector (X , . . . , X d ) with the respective MSMVE distribution can be constructed via

see [39, p. 218], which is a decreasing density and consequently, ν H ∈ U. We can compute the density f of ν Λ as f (x) = ( − α) g(x), which can be checked using Lemma 3(ii). Consequently, the associated Lévy subordinator Λ is an α-stable subordinator. The resulting bivariate Pickands dependence function is
For α ∈ ( , ), this interpolates between complete dependence and independence as can be seen in Figure 1. The lower tail dependence coe cient is given by λ L = − α . Though A appears to exhibit a kink at t = / , we know from previous computations that it is indeed di erentiable.
Another simple class of Bernstein functions is based on the compound Poisson distribution. We present one speci c instance.

which corresponds to a compound Poisson process with intensity ( + a) and Exp(a)-distributed jumps. The related Lévy process is a compound Poisson process with intensity ( + a) and Γ( , a)-distributed jumps, as can be seen from its
, < t ≤ . , and the lower tail dependence coe cient is given by λ L = /( + a), i.e. every value in ( , ] is attainable.  Table 1. In many cases, one can also compute the corresponding Lévy subordinator. If, e.g., H is distributed according to a compound Poisson distribution, the corresponding Lévy subordinator is a compound Poisson process (CP) as well. Figure 1 visualizes di erent attainable shapes of A. Name  (1) where the limit is a.s.. In the context of the construction in Equation (1), this can be interpreted as an intensity model with intensity λs := Λs /s, s > , with λ = a Λ . It is de ned consistently, as lim s λs = a Λ a.s., see [37, p. 351]. This is a peculiar construction, as it follows, assuming the rst and second moment of Λ to exist, that E[λs] = a Λ + ∞ x ν Λ (dx) and Var[λs] = /s ∞ x ν Λ (dx) for s > . Thus, the variance of the intensity is exploding close to and is vanishing for large s.  Table 1. The parameters in the example to the right are chosen such that all models exhibit a Spearman's ρ of . .

. Attainable marginal distributions
From [3,Proposition 2.3] it follows that f ful lls the integrability conditions in Lemma 2. Furthermore, the corresponding integral transform is well-known and thoroughly investigated in arbitrary dimensions, see [3]. We denote the transform restricted to distributions on R+ again by Φ, and de ne The Bernstein functions corresponding to Lévy measures in BO are called complete Bernstein functions. [3] show that Φ(M) = BO and that Φ is one-to-one. We provide a short proof of the rst result as it is helpful for understanding the transform itself. It is based on a characterization of the class BO via complete Bernstein functions given in [39,Remark 6.4]: Ψ is the Laplace exponent of a distribution in BO if and only if it has a representation with σ the so-called Stieltjes measure on ( , ∞), which satis es ∞ ( + t) − σ(dt) < ∞.

Lemma 4 (Attainable marginal distributions using f ). Using f , Ψ H has a representation
with a H = a Λ and σ H (B) : Proof. Using again [34,Proposition 2.6], it follows that with σ H as de ned above. Consequently, Φ(M) ∈ BO follows from the observation that σ H as de ned above is a Stieltjes measure. Φ(M) = BO follows from the observation that σ H as de ned above is a Stieltjes measure if and only if ν Λ is a Lévy measure. This can be easily shown using basic inequalities.
Lemma 4 de nes a direct connection between the characteristics of Ψ H and Ψ Λ as we can write the Stieltjes measure of H in terms of the Lévy measure of Λ. We will make use of this fact below.

. The corresponding multivariate distribution
As we have seen in Lemma 4, for an arbitrary Stieltjes measures σ H one can nd a corresponding Lévy measure. We will use this fact and state the dependence function of the resulting multivariate distribution in terms of the Stieltjes measures, such that arbitrary Stieltjes measures can be plugged in. Notice that Remark 1 also applies to Theorem 2.

Theorem 2 (Constructing parametric MSMVE distributions of family F2). For every complete Bernstein function Ψ H with Stieltjes measure σ H and drift a H such that Ψ H ( ) = , the function
denotes a stable tail dependence function for every d ≥ . A stochastic representation of an MSMVE distributions (X , . . . , X d ) with stable tail dependence function and unit exponential marginals is given by with Λ a Lévy subordinator with drift a Λ = a H and Lévy measure given by ν Λ (B) : , and an iid sequence {E k } k∈N of unit exponential random variables independent of Λ.
Proof. See Appendix B.
At least two approaches are possible when looking for tractable speci cations of family F2. As there exists a direct link between σ H and ν Λ , one can start from both sides. It is, for example, possible to start from σ H corresponding to a desired Ψ H and try to compute the expression in Theorem 2. [39] lists the Stieltjes measures for many of the known complete Bernstein functions. One could also start from a ν Λ such that the Laplace transform of the measure ν Λ (du)/(n u + ) for n ∈ N is known in closed form. This can be seen from the third from last line of the computation in the proof of Theorem 2.

Possible parametrization:
We present one example starting from the Laplace exponent Ψ H . Table 1) is attainable and corresponds to a compound Poisson distribution with intensity ( +a) and jump-size distribution Exp(a). This coincides with Example 2 in the previous section, i.e. it is possible to construct a process H using f which has the same marginal distributions as the process constructed in Example 2 using f . Thus, the minima of subsets of the two di erent resulting MSMVE sequences have the same exponential distributions, though their multivariate distributions di er considerably. The corresponding Stieltjes measure is determined as σ H (ds) = ( +a)δa(s), where δa denotes the Dirac measure at a. It is easy to see that ν Λ = Φ − (σ H ) is given by ν Λ (ds) = ( + a)δ /a (s), so Λ is a Poisson process with xed jump-size /a and intensity ( + a). A closed-form solution for de ned in Equation (9) is given by
The bivariate Pickands dependence function for < t < . can be stated as The dependence functions of this model and the one of Example 2 are compared in Figure 2. It can be observed that both approaches yield considerably di erent dependence functions.

A note on simulation
As mentioned before, the stochastic representation of (X , . . . , X d ) as an IDT-frailty model can be used to develop e cient simulation algorithms. When the involved Lévy subordinators are compound Poisson processes, simulating is straight-forward. Other Lévy subordinators can be approximated by compound Poisson processes or more involved schemes can be developed based on the given representation. We compare Example 2 of family F1 with Example 3 of family F2, which both yield CP1, i.e. Ψ H (x) = ( + a) x/(x + a), a > , as the desired (complete) Bernstein function for H. In Example 2, this corresponds to Λ ( ) being a compound Poisson process with intensity ( + a) and Γ( , a)-distributed jumps, i.e.
has the desired Laplace exponent. For the family F2, as described in Example 3, this corresponds to Λ ( ) being a Poisson process with deterministic jump-size /a and intensity ( + a), i.e.
yields a second construction with the desired marginal distribution. Denoting by τ i , i ∈ N, the jump times of a Poisson process with intensity ( + a), one can rewrite To illustrate the construction, sample paths are shown in Figure  3, where the same jump times are used to emphasize the di erences of the resulting paths. Based on these representations, it is clear how to sample from the construction in Equation (1). Exemplary scatterplots can be found in Figure 4, where we transformed the marginals to uniform distributions on [ , ] so that samples from the related extreme-value survival copulas are obtained for reasons of better comparability. Example 2 yields more samples close to the diagonal, which can be explained by the additional randomness introduced through the random variables G i . High values of G i correspond to a steep increase of H ( ) , which increases the probability of imminent triggering for both components within a short time period.

Concluding remarks
The present paper developed two new families of MSMVE distributions that give rise to many parametric models. The analysis conducted has shown that these new classes are quite exible. Similar to [19], the aim of this paper was to introduce new classes of models, while an application of these in practice requires more detailed investigations of particular parametric models. One clear advantage of the presented models is the availability of concrete stochastic models allowing for e cient simulation even in large dimensions. Furthermore, applying these models in credit-portfolio modeling yields additional advantages.

Acknowledgments
We thank two anonymous referees for helpful remarks on an earlier version of the paper.
The claim follows.