A ruin model with a resampled environment

TL;DR

This paper extends classical ruin theory to a model where environment-driven parameters are resampled at Poisson epochs, providing new asymptotic approximations, bounds, and an efficient importance-sampling method for risk assessment.

Contribution

It introduces a resampled environment model in ruin theory, extending classical results and deriving explicit bounds and an importance-sampling algorithm for complex, environment-dependent risk processes.

Findings

Extended Cramér-Lundberg approximation for resampled environments

Derived explicit bounds on ruin probability without spectral restrictions

Developed an importance-sampling algorithm with bounded relative error

Abstract

This paper considers a Cram\'er-Lundberg risk setting, where the components of the underlying model change over time. These components could be thought of as the claim arrival rate, the claim-size distribution, and the premium rate, but we allow the more general setting of the cumulative claim process being modelled as a spectrally positive L\'evy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs we resample the model components from a finite number of $d$ settings. It results in a setup that is particularly suited to describe situations in which the risk reserve dynamics are affected by external processes (such as the state of the economy, political developments, weather or climate conditions, and policy regulations). We extend the classical Cram\'er-Lundberg approximation (asymptotically characterizing the all-time ruin probability in a light-tailed setting) to this more general setup. In addition, for the situation that the driving L\'evy processes are sums of Brownian motions and compound Poisson processes, we find an explicit uniform bound on the ruin probability, which can be viewed as an extension of Lundberg's inequality; importantly, here it is not required that the L\'evy processes be spectrally one-sided. In passing we propose an importance-sampling algorithm facilitating efficient estimation, and prove it has bounded relative error. In a series of numerical experiments we assess the accuracy of the asymptotics and bounds, and illustrate that neglecting the resampling can lead to substantial underestimation of the risk.

Tables

Table 1. Table 1 . . Numerical results varying the speed q 𝑞 q of the background process.

$u=175$	Exact	Thm. 1	Thm. 2	No modulation
$q=3\cdot 4^{-2}=0.1875$	$9.21\cdot {10}^{-3}$	$9.21\cdot {10}^{-3}$	$1.12\cdot {10}^{-2}$	$6.26\cdot {10}^{-6}$
$q=3\cdot 4^{-1}=0.75$	$1.90\cdot {10}^{-4}$	$1.89\cdot {10}^{-4}$	$2.11\cdot {10}^{-4}$	$6.26\cdot {10}^{-6}$
$q=3\cdot 4^0=3$	$1.86\cdot {10}^{-5}$	$1.86\cdot {10}^{-5}$	$1.98\cdot {10}^{-5}$	$6.26\cdot {10}^{-6}$
$q=3\cdot 4^1=12$	$8.36\cdot {10}^{-6}$	$8.36\cdot {10}^{-6}$	$8.80\cdot {10}^{-6}$	$6.26\cdot {10}^{-6}$
$q=3\cdot 4^2=48$	$6.72\cdot {10}^{-6}$	$6.72\cdot {10}^{-6}$	$7.05\cdot {10}^{-6}$	$6.26\cdot {10}^{-6}$

Table 2. Table 2 . . Numerical results varying the initial reserve u 𝑢 u .

$q=\frac{3}{4}$	Exact	Thm. 1	Thm. 2	No modulation
$u=175$	$1.90\cdot {10}^{-4}$	$1.89\cdot {10}^{-4}$	$2.11\cdot {10}^{-4}$	$6.26\cdot {10}^{-6}$
$u=162.5$	$3.48\cdot {10}^{-4}$	$3.47\cdot {10}^{-4}$	$3.87\cdot {10}^{-4}$	$1.47\cdot {10}^{-5}$
$u=150$	$6.38\cdot {10}^{-4}$	$6.37\cdot {10}^{-4}$	$7.10\cdot {10}^{-4}$	$3.44\cdot {10}^{-5}$
$u=137.5$	$1.17\cdot {10}^{-3}$	$1.17\cdot {10}^{-3}$	$1.30\cdot {10}^{-3}$	$8.07\cdot {10}^{-5}$
$u=125$	$2.15\cdot {10}^{-3}$	$2.14\cdot {10}^{-3}$	$2.39\cdot {10}^{-3}$	$1.89\cdot {10}^{-4}$

Equations

$$\log \mathbb{E}\exp (-\alpha X_i(1))={\phi }_i(\alpha );$$

$$X(t):=X(T_{n})+\big{(}X_{J_{n}}(t)-X_{J_{n}}(T_{n})\big{)}.$$

$$\bar{X}:=\underset{t⩾0}{\sup }X(t),{\bar{X}}_{\mathrm{d}}:=\underset{n\in {\mathbb{N}}_0}{\sup }X(T_n);$$

$$\pi (u):=$$

$${\pi }_{\mathrm{d}}(u):=$$

$$\kappa :=\sum _{i=1}^d{p}_i{\phi }_i^{\prime }(0)>0.$$

$${\phi }_i(\alpha )=r_i\alpha -{\lambda }_i+{\lambda }_i{b}_i(\alpha ),$$

$$\underset{u\to \infty }{\lim }\frac{1}{u}\log {\pi }_{\mathrm{d}}(u)=-{\omega }^{\star },$$

$$\sum _{i=1}^d{p}_i{\int }_0^{\infty }q{\mathrm{e}}^{-qt}{\mathrm{e}}^{{\phi }_i(-\omega )t}\mathrm{d}t=\sum _{i=1}^d{p}_i\frac{q}{q-{\phi }_i(-\omega )}=1$$

$$\mathbb{E}{\mathrm{e}}^{-\alpha {\tau }_i}=\sum _{k=1}^{\infty }{p}_i^{k-1}\left(1-p_i\right){\left(\frac{q}{q+\alpha }\right)}^k=\frac{(1-p_i)q}{\alpha +(1-p_i)q},$$

$$Q=q\mathbf{e}{\mathbf{p}}^{\top }-q{I}_d,$$

$$M(\alpha )=Q+\Phi (\alpha )=q\mathbf{e}{\mathbf{p}}^{\top }-q{I}_d+\Phi (\alpha )=q\mathbf{e}{\mathbf{p}}^{\top }+\bar{\Phi }(\alpha ),$$

$${\mathbb{E}}\big{(}{\rm e}^{-\alpha X(t)}1_{\{J(t)=j\}}\,|\,J(0)=i\big{)}=({\rm e}^{M(\alpha)t})_{i,j}.$$

$$\underset{t\to \infty }{\lim }\frac{1}{t}\log \mathbb{E}{\mathrm{e}}^{\alpha X(t)}=\mu (M(\alpha )).$$

$$\mathbb{E}{\mathrm{e}}^{-\alpha \bar{X}}=\alpha {\mathbf{\ell }}^{\top }(M(\alpha ){)}^{-1}\mathbf{e},$$

$$\eta (v):=\inf \{t⩾0:-X(t)⩾v\},$$

$$\det (M(\alpha )-\theta I_d)=\det (\bar{\Phi }(\alpha )-\theta I_d)\det (I_d+(\bar{\Phi }(\alpha )-\theta I_d{)}^{-1}q\mathbf{e}{\mathbf{p}}^{\top }).$$

$$\det (M(\alpha )-\theta I_d)=$$

$$=$$

$$\frac{1}{q}=\sum _{i=1}^d{p}_i\frac{1}{q-{\phi }_i(\alpha )+\theta }=:{\Psi }_{\alpha }(\theta ).$$

$$\mathbb{E}{\mathrm{e}}^{-\alpha \bar{X}}=\alpha {\mathbf{\ell }}^{\top }S(\alpha )(\Theta (\alpha ){)}^{-1}{S}^{-1}(\alpha )\mathbf{e},$$

$$q{\mathbf{p}}^{\top }\mathbf{x}-q{x}_j+{\phi }_j(\alpha )x_j=\theta x_j.$$

$$\frac{x_j}{x_i}=\frac{q-{\phi }_i(\alpha )+\theta }{q-{\phi }_j(\alpha )+\theta },$$

$${\bar{\phi }}_i(\alpha ):={\phi }_{b(i,\alpha )}(\alpha )-q,$$

$$\underset{\theta \uparrow {\bar{\phi }}_i(\alpha )}{\lim }{\Psi }_{\alpha }(\theta )=-\infty ,\underset{\theta \downarrow {\bar{\phi }}_i(\alpha )}{\lim }{\Psi }_{\alpha }(\theta )=\infty ,$$

$$\underset{\theta \to -\infty }{\lim }{\Psi }_{\alpha }(\theta )=\underset{\theta \to \infty }{\lim }{\Psi }_{\alpha }(\theta )=0.$$

$$\frac{1}{q}=\sum _{i=1}^d{p}_i\frac{1}{q+ϵ{\phi }_i^{\prime }(0)+\delta ϵ+O({ϵ}^2)}=\frac{1}{q}\sum _{i=1}^d{p}_i\left(1-\frac{{\phi }_i^{\prime }(0)+\delta }{q}ϵ\right)+O({ϵ}^2).$$

$$\delta =-\sum _{i=1}^d{p}_i{\phi }^{\prime }(0),$$

$${\omega }^{\star }:=-\sup \{\alpha <0:\bar{\theta }(\alpha )=0\},$$

$$\zeta(\alpha):=\int_{0}^{\infty}e^{-\alpha u}\pi(u){\rm d}u=\frac{1}{\alpha}\big{(}1-{\mathbb{E}}\,{\rm e}^{-\alpha\bar{X}}\big{)}=\frac{1}{\alpha}-{\boldsymbol{\ell}}^{\top}S(\alpha)(\Theta(\alpha))^{-1}S^{-1}(\alpha)\,{\boldsymbol{e}}.$$

$$i^{\star }:=\underset{i=1,\dots ,d}{\mathrm{argmax}}{\theta }_i(-{\omega }^{\star }),$$

Full text

We extend the classical Cramér-Lundberg approximation (asymptotically characterizing the all-time ruin probability in a light-tailed setting) to this more general setup. In addition, for the situation that the driving Lévy processes are sums of Brownian motions and compound Poisson processes, we find an explicit uniform bound on the ruin probability, which can be viewed as an extension of Lundberg’s inequality; importantly, here it is not required that the Lévy processes be spectrally one-sided. In passing we propose an importance-sampling algorithm facilitating efficient estimation, and prove it has bounded relative error. In a series of numerical experiments we assess the accuracy of the asymptotics and bounds, and illustrate that neglecting the resampling can lead to substantial underestimation of the risk.

Keywords. Lévy risk processes $\circ$ parameter uncertainty $\circ$ ruin probabilities $\circ$ Cramér-Lundberg asymptotics $\circ$ Lundberg’s inequality

Affiliations. Corina Constantinescu and Leonardo Rojas Nandayapa are with the Institute for Financial and Actuarial Mathematics, Department of Mathematical Sciences, University of Liverpool, L69 3 BX Liverpool, United Kingdom.

Email: {C.Constantinescu|leorojas}@liverpool.ac.uk.

Guusje Delsing is with Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands, and with Rabobank, Croeselaan 18, 3521 CB Utrecht, the Netherlands. Email: g.a.delsing@uva.nl.

Michel Mandjes is with Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands. He is also with Eurandom, Eindhoven University of Technology, Eindhoven, the Netherlands, and Amsterdam Business School, Faculty of Economics and Business, University of Amsterdam, Amsterdam, the Netherlands. His research is partly funded by NWO Gravitation project Networks, grant number 024.002.003. Email: m.r.h.mandjes@uva.nl.

1. Introduction

Risk theory focuses on analyzing models that describe an insurer’s vulnerability to ruin. Starting from the seminal works by Cramér [13] and Lundberg [29, 30] a substantial research effort has been spent on determining the ruin probability in a broad range of risk models. In the basic model, independent and identically distributed claims are assumed to arrive according to a Poisson process, whereas premiums arrive at a constant rate. The ruin probability is the probability that the capital surplus drops below [math].

After the above mentioned pioneering papers, various extensions and generalizations have been considered to make the model more realistic. In this respect, multiple directions can be distinguished. Without pursuing to provide a complete overview, we include a brief account of a few important branches. In the first place, the classical model has been extended to include time-dependent ruin, i.e. ruin before a specified point in time; see e.g. [7, Ch. V]. Secondly, the assumption of the cumulative claim process being of compound Poisson type has been generalized to that of compound Poisson perturbed by diffusion [19, 21], and later to that of (spectrally one-sided) Lévy input; see e.g. [7, Ch. X and XI] and [27]. Thirdly, returns on investment have been included, and also level-dependent risk models have been considered; see e.g. [7, Ch. VIII] and [4]. A major other branch in the literature focuses on computing or approximating ruin probabilities for specific claim-size distributions; see for instance [12] for the case of Gamma claims and [31] for the case of heavy-tailed claims. Finally, we mention the direction of research in which the effect of specific dependence structures is assessed; see e.g. [11] and, for an overview, [7, Ch. XIII]. We also refer to [20, 28, 32] for further background on risk theory in general.

More often than not, in the models that have been considered the corresponding model primitives (in terms of parameters and distributions) are fixed. For instance in the classical Cramér-Lundberg model a specific claim arrival rate, premium rate, and claim-size distribution are held constant, in the sense that they cannot change over time. In reality, however, such a setup is typically not valid: as a consequence of various ‘external circumstances’ the model primitives may fluctuate. In this context one could think of exogenous factors affecting the claim arrival process, such as the state of the economy, the political situation, weather conditions, and policy regulations. Neglecting the parameter uncertainty (by using the conventional Cramér-Lundberg model with time-averaged parameters) could evidently lead to a substantial underestimation of the risk.

An intuitively appealing mechanism to introduce parameter uncertainty is to periodically resample them. A very basic example of such a model would be an adaptation of the classical Cramér-Lundberg framework, in which (say every day, week or month) the arrival rate is resampled from a given distribution. Evidently, in principle also the other model primitives (i.e., premium rate and claim-size distribution) can be periodically resampled. In [22], for a different class of models, a similar mechanism to introduce parameter fluctuations has been proposed.

In this paper we consider the setup in which the claim arrival process is a spectrally one-sided Lévy process, thus covering the frequently used compound Poisson case. The special feature concerns the resampling mechanism described above: after exponentially distributed times, the Laplace exponent of this driving Lévy process is resampled from a set of $d\in{\mathbb{N}}$ possible settings. There is a connection between this model and the one in which the claim arrival process is a so-called Markov additive process (MAP) [7, Ch. VII]. Importantly, due to our specific resampling mechanism that we impose in this paper, the results we obtain are relatively explicit (compared to their counterparts under a MAP claim arrival process). Throughout this paper we assume the claim-size distributions are light-tailed (in line with what is assumed in the classical Cramér-Lundberg framework).

The main contributions of our paper are the following. (i) In the first place, for an initial capital reserve level $u$, we identify the exact asymptotics of the ruin probability in the regime that $u$ grows large. This result can be seen as the counterpart of the Cramér-Lundberg asymptotics for our resampling model. (ii) In the second place, restricting ourselves to the situation that the driving Lévy processes are sums of Brownian motions and compound Poisson processes, we find an explicit upper bound on the ruin probability that is uniform in $u\geqslant 0.$ This bound can be seen as an extension of Lundberg’s inequality. In this context it is important to note that it is not required that the Lévy processes be spectrally one-sided. (iii) In passing we propose an importance-sampling algorithm that facilitates the efficient estimation of small ruin probabilities. We prove that this procedure has bounded relative error, which effectively means that the number of runs needed to obtain an estimate of a given precision is hardly affected by the value of $u$. (iv) We conclude this paper by a series of numerical experiments, in which we systematically assess the accuracy of the asymptotics and bounds. An important observation is that neglecting the resampling (by using the Cramér-Lundberg model with time-averaged parameters) typically leads to a significant underestimation of the risk.

This paper is organized as follows. Section 2 provides a formal model description and some preliminaries. Then in Section 3 the exact asymptotics are established. Section 4 presents the counterpart of Lundberg’s inequality, together with the importance-sampling algorithm. Numerical examples are provided in Section 5; this section also provides explicit expressions for the asymptotics and bounds in case the number of environmental states $d$ equals $2$.

2. Model and preliminaries

In this section we introduce our resampling model, and provide preliminaries. In our model, the risk process is expressed in terms of a spectrally-positive Lévy process, whose characteristics are resampled at Poisson epochs.

2.1. Model

We start by constructing the net cumulative claim process $X(\cdot)$. To this end, we first introduce spectrally-positive scalar-valued Lévy processes $X_{i}(\cdot)$ for $i=1,\ldots,d$, where we assume that $X_{i}(0)=0$. These processes are characterized by their respective Laplace exponents $\varphi_{1}(\cdot)$ up to $\varphi_{d}(\cdot)$, meaning that, for $\alpha\geqslant 0$,

[TABLE]

see e.g. [27].

In a standard ruin-theoretic setting the processes $X_{i}(\cdot)$ would correspond to compound Poisson processes (representing the cumulative claim process) from which a deterministic drift is subtracted (the incoming premiums). Observe however that the framework we consider is significantly richer: the processes $X_{i}(\cdot)$ could contain a Brownian component, and also increasing ‘small-jumps processes’ (such as the Gamma process or the Inverse Gaussian process) can be included [27, Sections 1.2.4–1.2.5].

We now construct our resampling model. Let $T_{n}$ be the jump epochs of a Poisson process with rate $q>0$; we set $T_{0}:=0.$ At these epochs with probability $p_{i}\in[0,1]$ the $i$-th of the above-mentioned $d$ Lévy processes is picked, with the $p_{i}$ summing to 1. Let $J_{n}\in\{1,\ldots,d\}$ be the index of the Lévy process that was picked between $T_{n}$ and $T_{n+1}$, and set $J(t)=J_{n}$ when $t\in[T_{n},T_{n+1})$. Then we recursively define the cumulative claim process by, for $t\in[T_{n},T_{n+1})$,

[TABLE]

In a ruin context, we let $u-X(t)$ represent the capital surplus at time $t$, given the initial reserve was $u>0$. This means that the all-time ruin probability can be expressed as the probability that $X(t)\geqslant u$ for some $t\geqslant 0.$ This is the probability that we will study in this paper.

Define the all-time maxima

[TABLE]

in other words, $X_{\rm d}$ is the all-time maximum, but restricted to jump epochs of the background process. We work in the sequel with

[TABLE]

It is clear that $\bar{X}\geqslant\bar{X}_{\rm d}$, so that $\pi(u)\geqslant\pi_{\rm d}(u).$

Throughout this paper we assume a negative drift, so that the events under consideration are increasingly rare as $u$ grows large. This negative drift assumption entails that we require

[TABLE]

In addition, in this work we assume that we are in the light-tailed setting, meaning that for all $i\in\{1,\ldots,d\}$ the Laplace exponent $\varphi_{i}(\alpha)$ is finite for $\alpha$ in an open neighborhood of the origin. In the $d=1$ case, this is in line with what was assumed to obtain the traditional Cramér-Lundberg asymptotics.

The claim arrival processes $X(\cdot)$ covers a resampled compound Poisson process as a special case. Then we can write the Laplace exponent of the $i$-th Lévy process (i.e., $X_{i}(\cdot)$) as

[TABLE]

where $r_{i}$ is the deterministic drift, $\lambda_{i}$ the claim arrival rate, and $b_{i}(\cdot)$ the Laplace transform of the claim sizes.

2.2. Preliminaries

In this paper the focus lies in particular on the above probabilities’ exact asymptotics (and related upper bounds) in the light-tailed domain. It is not hard to guess what the decay rate of the tail is. In the first place, one would expect that the logarithmic asymptotics of $\pi(u)$ and $\pi_{\rm d}(u)$ match (this we later prove). Secondly, observe that $(X(T_{n}))_{n\in{\mathbb{N}}}$ is a random walk; the increments $Y_{n}:=X(T_{n})-X(T_{n-1})$ (for $n\in{\mathbb{N}}$) are independent and identically distributed (say, as a generic random variable $Y$). For this setting it is well-known [25] that

[TABLE]

with $\omega^{\star}$ the unique positive root of ${\mathbb{E}}\,{\rm e}^{\omega Y}=1$. This means that $\omega^{\star}$ solves

[TABLE]

(where it is implicit that $\omega^{\star}$ is such that $q>\varphi_{i}(-\omega^{\star})$ for all $i\in\{1,\ldots,d\}$). The existence of the root $\omega^{\star}$ is assumed; it implies that there are $\alpha<0$ such that $\varphi_{i}(-\alpha)$ is finite (for all $i\in\{1,\ldots,d\}$), which means that we are in the regime that the upward jumps of $X_{i}(\cdot)$ are light-tailed; cf. e.g. [15, Section 8.1].

Actually, the precise asymptotics of $\pi_{\rm d}(u)$ have been identified already. Recalling that $X(T_{n})$ can be written as the sum of $n$ independent and identically distributed increments $Y_{1}$ up to $Y_{n}$, the exceedance probability $\pi_{\rm d}(u)$ can be interpreted as the probability that a random walk with negative drift (cf. condition (1)) and light-tailed increments ever exceeds level $u$. For this setting in e.g. [25] a positive constant $\gamma$ is found that $\pi_{\rm d}(u)\,{\rm e}^{\omega^{\star}u}\to\gamma$. As these exact asymptotics of $\pi(u)$ have not been identified so far, it is one of the main objectives of this paper to derive these; see Section 3. Another objective concerns a uniform upper bound on $\pi(u)$; see Section 4.

3. Asymptotics

In order to identify the exact asymptotics of $\pi(u)$, we first verify that our model actually corresponds to the maximum value attained by a specifically chosen Markov additive process (in the sequel abbreviated to MAP); see e.g.  [15, Section 11.4]. To this end, recall that a MAP behaves as a Lévy process $X_{i}(\cdot)$ whenever the background process $J(\cdot)$ (whose transition rate matrix we denote by $Q=(q_{ij})_{i,j=1}^{d}$) is in state $i\in\{1,\ldots,d\}$. Let us construct the matrix $Q$, by considering the transition rates from state $i$. Observe that the process $J(\cdot)$ stays for an exponential amount of time (with rate, say, $\bar{q}_{i}$) in $i$; after this time, it jumps to state $j\not=i$ with probability $p_{j}/(1-p_{i})$. The parameter $\bar{q}_{i}$ can be determined by computing the Laplace-Stieltjes transform of the time spent in state $i$, say $\tau_{i}$. We obtain

[TABLE]

from which we conclude that $\tau_{i}$ is exponentially distributed with parameter $\bar{q}_{i}=(1-p_{i})\,q$. We thus observe that $q_{ij}=\bar{q}_{j}\cdot p_{j}/(1-p_{i})=q\,p_{j}$ for $i\not=j$, whereas $q_{ii}=-\bar{q}_{i}$. We thus arrive at

[TABLE]

with ${\boldsymbol{e}}$ an all-ones vector and $I_{d}$ the $d$-dimensional identity matrix. The conclusion is that our process $X(\cdot)$ corresponds to a MAP with the transition rate matrix $Q$ given by (3), and state-dependent Laplace exponents $\varphi_{i}(\cdot)$ for $i\in\{1,\ldots,d\}.$ In the sequel, we use that $X(\cdot)$ has a MAP-representation; in particular we make use of the fact that for this setting the Laplace transform of $\bar{X}$ is known.

3.1. Transform of $\bar{X}$

In this subsection we provide the Laplace transform of $\bar{X}$, and show how this can be simplified, owing to the special structure of the matrix $Q$.

Define $\Phi(\alpha):={\rm diag}\{\varphi_{1}(\alpha),\dots,\varphi_{d}(\alpha)\}$ and $\bar{\Phi}(\alpha)=-qI_{d}+\Phi(\alpha)$. We can now introduce

[TABLE]

which can be seen as a ‘matrix-valued Laplace exponent’ in the sense that, for any $i,j\in\{1,\ldots,d\}$,

[TABLE]

By a Perron-Frobenius based argumentation, one can show that the eigenvalue of $M(\alpha)$ with largest real part, which we denote by $\mu(M(\alpha))$, is actually real (where we note that in our specific setting we argue below that all $d$ eigenvalues are real). It thus follows that

[TABLE]

Due to the fact that this concerns a limiting logarithmic moment generating function, we thus conclude that $\mu(M(\alpha))$ is a convex function of $\alpha$.

The Laplace transform of $\bar{X}$ in $\alpha$ can also be expressed in terms of this matrix $M(\alpha).$ More specifically, as can be found in e.g. [15, Eqn. (11.1)], there is the following ‘matrix counterpart’ of the celebrated Pollaczek-Khinchine formula:

[TABLE]

for a vector ${\boldsymbol{\ell}}$ determined in e.g. [14, 17].

Remark 1**.**

In [14] a compact representation for the vector ${\boldsymbol{\ell}}$ is given. We provide a brief account of this representation here. First, split the $d$ background states as follows. Let for states $i\in\{d-d^{-}+1,\ldots,d\}$ the Lévy process $X_{i}(\cdot)$ correspond to a decreasing subordinator; obviously, in these states the process $X(\cdot)$ cannot cross the level $u$ (if there are no states corresponding to decreasing subordinators, we put $d^{-}:=0$). In the other states, corresponding to $i\in\{1,\ldots,d-d^{-}\}$, the level $u$ can be crossed.

Now consider

[TABLE]

as a process in $v\geqslant 0.$ As argued in e.g. [14], $(\eta(v),J(\eta(v))_{v\geqslant 0}$ is a MAP, with $J(\eta(v))$ attaining values in $\{1,\ldots,d-d^{-}\}$. Let $\bar{\boldsymbol{\pi}}$ be the $(d-d^{-})$-dimensional invariant probability measure pertaining to the Markov process $(J(\eta(v))_{v\geqslant 0}$. Then the vector ${\boldsymbol{\ell}}$ is such that ${\boldsymbol{\ell}}^{\top}=\kappa\,(\bar{\boldsymbol{\pi}}^{\top},{\boldsymbol{0}}^{\top})$, with the scalar $\kappa>0$ as in (1). $\Box$

Interestingly, due to the fact that $M(\alpha)$ is the sum of a diagonal matrix and a rank-one matrix, its eigenvalues can be somehow characterized, applying the following nice (and well-known) idea. To this end, we can write

[TABLE]

For $A$ of dimension $m\times n$, and $B$ of dimension $n\times m$, we have ${\rm det}(I_{m}-AB)={\rm det}(I_{n}-BA)$. We thus conclude that

[TABLE]

We find that the eigenvalues $\theta_{1}(\alpha)$ up to $\theta_{d}(\alpha)$, for a fixed $\alpha$, are the solutions to

[TABLE]

With $\Theta(\alpha):={\rm diag}\{\theta_{1}(\alpha),\ldots,\theta_{d}(\alpha)\}$, using standard machinery from linear algebra we get for a matrix $S(\alpha)$ that

[TABLE]

under the familiar regularity conditions regarding the multiplicities of the eigenvalues. In principle we have now a unique characterization of $\bar{X}$, and hence also, albeit in implicit terms, a way of computing $\pi(u).$ In general this requires numerical inversion, for which there are various algorithms available; see e.g. [2, 23]. In this section we have another objective: we use knowledge of the transform of $\bar{X}$ to identify the corresponding tail asymptotics.

Remark 2**.**

A standard fact from linear algebra is that the columns of $S(\alpha)$ contain the right eigenvectors of $M(\alpha).$ If the eigenvalues $\theta_{1}(\alpha)$ up to $\theta_{d}(\alpha)$ have been found, these can be easily expressed in terms of these eigenvalues. Suppose $\theta$ is such an eigenvalue. Then the eigenvector ${\boldsymbol{x}}$ satisfies $M(\alpha){\boldsymbol{x}}=\theta{\boldsymbol{x}},$ or, equivalently, for $j\in\{1,\ldots,d\}$,

[TABLE]

We conclude that

[TABLE]

so that we can pick $x_{i}=(q-\varphi_{i}(\alpha)+\theta)^{-1}$. $\Box$ **

3.2. Tail asymptotics

The idea is to rely on the Heaviside recipe [15, Recipe 8.1] to find the tail behavior. To this end we first have to identify the rightmost pole on the negative halfline. The poles are the values of $\alpha<0$ for which one of the $\theta_{i}(\alpha)$ equals 0.

To study the behavior of the poles, first observe that $\Psi_{0}(0)=1/q$, so for $\alpha=0$ all $d$ roots equal [math]. Now pick a negative value of $\alpha$, and let the bijection $b(i,\alpha)$ relabel the $\varphi_{i}(\alpha)$ such that, with

[TABLE]

the $\bar{\varphi}_{i}(\alpha)$ are increasing in $i$. Then, using the shape of $\Psi_{\alpha}(\theta)$, it is easily argued that one eigenvalue is larger than $\bar{\varphi}_{d}(\alpha)$, and that for $i=1,\ldots,d-1$ there is one of the eigenvalues $\theta_{j}(\alpha)$ in each of the intervals $(\bar{\varphi}_{i}(\alpha),\bar{\varphi}_{i+1}(\alpha))$, as illustrated in Fig. 3.2; to this end, observe that

[TABLE]

and

[TABLE]

In this argumentation it is tacitly assumed that the $\bar{\varphi}_{i}(\alpha)$ are different, but the reasoning followed extends in an obvious way to the case that some are equal.