2.1. The main decomposition result

Let $Y\equiv \{Y(t), \, t \geqslant 0\} $ be a general real-valued Lévy process. In addition, let $\bar Y$ be the associated running maximum process, $\overline Y(t) \,:\!=\, \sup_{s \in [0,t]} Y(s)$ . We consider the setting in which, for some intensity $\omega>0$ , the Lévy process is inspected at Poisson( $\omega$ ) moments $I_1,I_2,\dots$ , so that the number of inspections up to time t has a Poisson distribution with mean $\omega t$ . We consider the resulting inspected process until ‘killing’, which happens at an exponentially distributed time $T_\beta$ with parameter $\beta\geqslant 0$ , sampled independently from Y.

To analyze the process Y at inspection moments, we denote by $Z_m$ the increment of the Lévy process Y(t) between the two consecutive inspection times $I_{m-1}$ and $I_m$ (with $I_0 \,:\!=\,0$ ), conditioned on the process not having been killed. We also define $S_0\,:\!=\,0$ , $S_n\,:\!=\,\sum_{m=1}^n Z_m$ for $n\in{\mathbb N}$ , and the corresponding running maximum process $\overline S_n\,:\!=\,\max\{S_0,S_1,\ldots,S_n\}$ . We wish to analyze the running maximum of the inspected process until killing. The number of inspections $N_{\beta,\omega}$ before killing is shifted-geometric, with the ‘killing probability’ given by $\beta/(\beta+\omega)$ :

\begin{equation*}{\mathbb P}(N_{\beta,\omega} = n) = \bigg(\frac{\omega}{\beta+\omega}\bigg)^n \frac{\beta}{\beta+\omega}.\end{equation*}

The random variable of interest is

(2.1) \begin{equation} Y_{\beta,\omega} \,:\!=\, \overline S_{N_{\beta,\omega}} = \sup_{n=0,1,\ldots,N_{\beta,\omega}} S_n.\end{equation}

Observe that we have the identity

(2.2) \begin{equation} {\mathbb E}\,{\rm e}^{-\alpha Y_{\beta,\omega}} = \sum_{n=0}^\infty \bigg(\frac{\omega}{\beta+\omega}\bigg)^n \frac{\beta}{\beta+\omega} {\mathbb E}\,{\rm e}^{-\alpha \overline S_n}.\end{equation}

Theorem 2.1. For any $\alpha>0$ ,

(2.3) \begin{equation} {\mathbb E}\,{\rm e}^{-\alpha Y_{\beta,\omega}} = \exp\bigg({-}\int_0^\infty\int_{(0,\infty)} \frac{1}{t} {\rm e}^{-\beta t}(1-{\rm e}^{-\omega t}) (1-{\rm e}^{-\alpha x})\, \,{\mathbb P}(Y(t)\in {\rm d}x)\,{\rm d}t\bigg).\end{equation}

Proof. As pointed out in [Reference Kyprianou18], applying the Wiener–Hopf theory for random walks,

\begin{equation*} \sum_{n=0}^\infty (1-p)^n p \,{\mathbb E}\,{\rm e}^{-\alpha \overline S_n} = \exp\Bigg({-}\int_{(0,\infty)} \sum_{n=1}^\infty \frac{1}{n} (1-{\rm e}^{-\alpha x})(1-p)^n\,{\mathbb P}(S_n\in {\rm d}x)\Bigg);\end{equation*}

see also, e.g., [Reference Debicki and Mandjes13, Section 3.3]. As a consequence,

\begin{equation*} {\mathbb E}\,{\rm e}^{-\alpha Y_{\beta,\omega}} = \exp\Bigg({-}\int_{(0,\infty)} \sum_{n=1}^\infty \frac{1}{n} (1-{\rm e}^{-\alpha x})\bigg(\frac{\omega}{\beta+\omega}\bigg)^n\,{\mathbb P}(S_n\in {\rm d}x)\Bigg).\end{equation*}

Now realize that, with ${\rm E}(n,a)$ an Erlang random variable with shape parameter $n\in{\mathbb N}$ and scale parameter $a>0$ , conditional on the process not having been killed, $S_n\stackrel{\rm d} = Y({\rm E}(n, \beta+\omega))$ , so that

\begin{equation*} {\mathbb P}(S_n\in {\rm d}x) = \int_0^\infty (\beta+\omega)^n t^{n-1} \frac{{\mathrm{e}}^{-(\beta+\omega)t}}{(n-1)!}\,{\mathbb P}(Y(t)\in {\rm d}x)\,{\rm d}t.\end{equation*}

We thus obtain

\begin{align*} {\mathbb E}\,{\rm e}^{-\alpha Y_{\beta,\omega}} & = \exp\Bigg({-}\int_0^\infty\int_{(0,\infty)} \frac{1}{t}\sum_{n=1}^\infty \frac{1}{n!} (1-{\rm e}^{-\alpha x})({\omega}t)^n\,{\rm e}^{-(\beta+\omega)t}\,{\mathbb P}(Y(t)\in {\rm d}x)\, {\rm d}t\Bigg) \\ & = \exp\bigg({-}\int_0^\infty\int_{(0,\infty)} \frac{1}{t}({\rm e}^{\omega t}-1) (1-{\rm e}^{-\alpha x})\,{\rm e}^{-(\beta+\omega)t} \,{\mathbb P}(Y(t)\in {\rm d}x)\,{\rm d}t\bigg) \\ & = \exp\bigg({-}\int_0^\infty\int_{(0,\infty)} \frac{1}{t} {\rm e}^{-\beta t}(1-{\rm e}^{-\omega t}) (1-{\rm e}^{-\alpha x})\, \,{\mathbb P}(Y(t)\in {\rm d}x)\,{\rm d}t\bigg).\end{align*}

This proves the claim.

We can let $\omega \rightarrow \infty$ in the above theorem to recover a classical Wiener–Hopf factorization result for general Lévy process Y.

We now state and prove our main result, a decomposition theorem for a Lévy process with Poisson inspection epochs.

Theorem 2.2. The following distributional equality applies:

(2.4) \begin{equation} \overline Y({T_\beta}) \stackrel{\rm d} = Y_{\beta,\omega}+ \overline Y({T_{\beta+\omega}}), \end{equation}

with the two terms on the right-hand side being independent.

Proof. By applying the Wiener–Hopf theory for Lévy processes, as presented in, e.g., [Reference Kyprianou17, Theorem 6.15] or [Reference Debicki and Mandjes13, Section 3.3], for any $\zeta>0$ ,

\begin{equation*} {\mathbb E} \,{\rm e}^{-\alpha \overline Y({T_\zeta})}= \exp\bigg({-}\int_0^\infty\int_0^\infty \frac{1}{t} {\rm e}^{-\zeta t} (1-{\rm e}^{-\alpha x})\, \,{\mathbb P}(Y(t)\in {\rm d}x)\,{\rm d}t\bigg).\end{equation*}

Taking $\zeta=\beta$ and $\zeta = \beta + \omega$ , and using Theorem 2.1, we obtain ${\mathbb E} \,{\rm e}^{-\alpha \overline Y({T_\beta})} = {\mathbb E}\,{\rm e}^{-\alpha Y_{\beta,\omega}}\cdot {\mathbb E} \,{\rm e}^{-\alpha \overline Y({T_{\beta+\omega}})}$ , which implies the stated result.

Remark 2.2. As is to be expected, $S_{N_{\beta,\omega}}$ can also be decomposed as the sum of $\overline S_{N_{\beta,\omega}}$ and $\underline S_{N_{\beta,\omega}} \,:\!=\, \inf_{n=0,1,\ldots,N_{\beta,\omega}} S_n$ , with the latter two quantities being independent. To verify this, first observe that ${\mathbb E} \, {\rm e}^{\alpha {\rm i} Z_m} = (\beta + \omega)/\big(\beta + \omega - {\rm log} \, {\mathbb E} \, {\rm e}^{\alpha{\rm i} Y(1)}\big)$ . Hence,

\begin{align*} {\mathbb E}\,e^{\alpha{\rm i} S_{N_{\beta,\omega}}} & = \frac{\beta}{\beta+\omega -\omega\, {\mathbb E}\,{\mathrm{e}}^{\alpha{\rm i}Z}} = \frac{1 + \omega/(\beta - {\rm log} \, {\mathbb E} \, {\rm e}^{\alpha{\rm i} Y(1)})}{1 + \omega/\beta} \\ & = \exp\bigg({-}\int_0^\infty\int_{({-}\infty,\infty)} \frac{1}{t} {\rm e}^{-\beta t}(1-{\rm e}^{-\omega t}) (1-{\rm e}^{\alpha{\rm i} x})\, \,{\mathbb P}(Y(t)\in {\rm d}x)\,{\rm d}t\bigg). \end{align*}

In the last step we have used the Frullani integral [Reference Kyprianou17, Lemma 1.7],

\begin{equation*} 1 + \frac{\omega}{a} \, = \, {\rm exp} \bigg(\int_0^{\infty} \frac{1}{t} {\rm e}^{-at} (1 - {\rm e}^{-\omega t}) \, {\rm d}t \bigg), \end{equation*}

with both $a = \beta$ and $a = \beta - \log {\mathbb E}\,{\mathrm{e}}^{\alpha{\rm i} Y(1)}$ . Finally, observe that, by symmetry, the running minimum can be dealt with in precisely the same manner as the running maximum, in that the transform of $\underline S_{N_{\beta,\omega}}$ is as given in (2.3), but with the integration interval $(0,\infty)$ replaced by $({-}\infty,0)$ .

2.2. Relation to a result in [Reference Albrecher and Ivanovs4]

In this subsection we outline the relation between Theorem 2.2 and some results from [Reference Albrecher and Ivanovs4]. In preparation, we take a closer look at the increments $Z_m$ between inspection epochs, and we mention the powerful concept of Wiener–Hopf factorization for Lévy processes; see, e.g., [Reference Kyprianou17, Chapter 6] or [Reference Debicki and Mandjes13, Section 3.3].

The Wiener–Hopf decomposition entails each $Z_m$ being written as $Z_m = Z_m^+ - Z_m^-$ , with $Z_m^+$ and $Z_m^-$ independent and both non-negative. Here, $Z_m^+$ (resp. $Z_m^-$ ) is distributed as the supremum (resp. minus the infimum) of the Lévy process Y, when started anew at zero at inspection epoch $I_{m-1}$ , over the interval between $I_{m-1}$ and $I_m$ (whose length is exp( ${\beta+\omega}$ )). To get some feeling for this property, it is useful to observe that a time-reversibility argument for Lévy processes implies that, with $\underline Y(t)$ denoting the running minimum process, we have

(2.5) \begin{equation} Y(t) - \underline Y(t) = Y(t) - \inf_{s \in [0,t]}Y(s) = \sup_{s \in [0,t]} (Y(t) - Y(s)) \stackrel{{\rm d}} = \sup_{s \in [0,t]} Y(s) = \overline{Y}(t)\end{equation}

(but, evidently, for a given t, $\underline Y(t)$ and $\overline{Y}(t)$ are not independent). In the following, $Z^+$ (resp. $Z^-$ ) is a generic random variable distributed as $Z_m^+$ (resp. $Z_m^-$ ).

Albrecher and Ivanovs [Reference Albrecher and Ivanovs4] considered a Lévy process $X\equiv \{X(t),\, t\geqslant 0\}$ , starting at u, which is also being inspected at Poisson( $\omega$ ) epochs $I_0=0,I_1,\dots$ If X attains a negative value, then ruin is said to occur, whereas if it is negative at an inspection epoch, then bankruptcy is said to occur. Recall that the all-time ruin probability starting at surplus level u is denoted by p(u), and the (obviously smaller) corresponding bankruptcy probability by $\tilde{p}(u)$ . The starting point in [Reference Albrecher and Ivanovs4] is their elegant Proposition 1, which (in our notation) states that

(2.6) \begin{equation} p(u) = {\mathbb E} \,\tilde{p}(u-Z^+) ,\end{equation}

where the process X relates to our Y through the relation $X(t)=u-Y(t)$ for $t\geqslant 0$ . The focus in [Reference Albrecher and Ivanovs4] lies not on deriving decompositions, but the proof of (2.6) in fact implicitly reveals such a decomposition, which can be used to rederive our decomposition of Theorem 2.2. They introduced (again adapted to our notation) the partial sums, for $i=1,2,\dots$ ,

\begin{alignat*}{2} \hat{\sigma}_0 & \,:\!=\, 0, & \hat{\sigma}_i & \,:\!=\, - \sum_{j=1}^i Z_j, \\ \sigma_0 & \,:\!=\, -Z_1^+, \qquad & \sigma_i & \,:\!=\, - \sum_{j=1}^i \big(Z_{j+1}^+ -Z_j^-\big) , \end{alignat*}

and then concluded that $\{\hat{\sigma}_i -Z^+ \}_{i=0,1,\ldots} \stackrel{{\rm d}} = \{\sigma_i\}_{i=0,1,\ldots}$ , and hence $p(u) = {\mathbb P}({-} \min_{i \geqslant 0} \sigma_i \geqslant u) = {\mathbb P}\big({-} \min_{i \geqslant 0} \hat{\sigma}_i + Z_1^+ \geqslant u\big) = {\mathbb E} \,\tilde{p}\big(u-Z_1^+\big)$ . Observe that this identity implicitly entails the decomposition $- \min_{i=0,1,\ldots} \sigma_i \stackrel{{\rm d}} = - \min_{i =0,1,\ldots} \hat{\sigma}_i + Z_1^+$ .

Let us now turn to the case with ‘killing’, as considered in Theorem 2.2, i.e. the process ends at $T_\beta \sim {\rm exp}(\beta)$ . We note that [Reference Albrecher and Ivanovs4, Remark 3] briefly mentions the option of killing, at an inspection epoch $I_i$ . It is stated there, without proof, that the finite-time ruin and bankruptcy probabilities before inspection time $I_i$ are related via

(2.7) \begin{align} p(u,I_i) & = {\mathbb P}\Big({-} \min_{j=0,1,\ldots, i-1} \sigma_j \geqslant u\Big) \nonumber \\ & = {\mathbb P}\Big({-} \min_{j=0,1,\ldots, i-1} \hat{\sigma}_j + Z_1^+ \geqslant u\Big) = {\mathbb E} \,\tilde{p}\big(u-Z_1^+,I_{i-1}\big) ;\end{align}

here, p(u, t) (resp. $\tilde p(u,t)$ ) is the probability of ruin (resp. bankruptcy) before time t, given an initial surplus u. To translate this observation to the setting of Theorem 2.2, let us assume that the system is inspected at the Poisson( $\beta+\omega$ ) epochs $I_0=0,I_1,\dots$ The inspection intervals now are exp( $\beta+\omega$ ) distributed and, accordingly, in the distributions of the $Z^+$ and $Z^-$ the parameter $\omega$ should be replaced by $\beta + \omega$ . The ‘ $\beta$ -inspection’ is preceded by $N_{\beta,\omega}$ ‘ $\omega$ -inspections’ at epochs $I_1,\dots,I_{N_{\beta,\omega}}$ . Now consider the three terms in Theorem 2.2, and compare them with the three main random elements featuring in (2.7). Firstly, observe that

(2.8) \begin{equation} Z_1^+ \stackrel{{\rm d}} = \overline{Y}(T_{\beta+\omega}),\end{equation}

as both are distributed as the supremum of the Lévy process Y over an exp( $\beta + \omega$ ) interval. Secondly, noticing that if $I_i = T_\beta$ then $N_{\beta,\omega} = i-1$ , we have

\begin{equation*} - \min_{j=0,1, \ldots, N_{\beta,\omega}} \hat{\sigma}_j = \sup_{j=0,1, \ldots, N_{\beta,\omega}} \sum_{k=0}^j Z_k \stackrel{{\rm d}} = \overline{Y}_{\beta,\omega} .\end{equation*}

Thirdly,

\begin{equation*} - \min_{j=0,1, \ldots, N_{\beta,\omega}} \sigma_j = \sup_{j=0,1, \ldots, N_{\beta,\omega}} \sum_{k=0}^j \big(Z_{k+1}^+ - Z_k^-\big) \stackrel{{\rm d}} = \overline{Y}(T_\beta) ,\end{equation*}

as the latter supremum is the supremum of the Y process until $T_\beta$ . We thus conclude that Theorem 2.2 can be recovered in this way from the middle equality in (2.7).

3.1. The spectrally positive case

Suppose that Y is spectrally positive, i.e. it has no downward jumps. Define its Laplace exponent by $\varphi(\alpha)\,:\!=\, \log {\mathbb E}\exp({-}\alpha Y(1))$ , and $\psi(\beta)$ its right-inverse; cf. [Reference Kyprianou17, Section 3.3]. It is well known [Reference Kyprianou17, Section 6.5.2] that, for $\zeta\geqslant 0$ ,

(3.1) \begin{equation} {\mathbb E} \,{\rm e}^{-\alpha \overline{Y}(T_\zeta)} = \frac{\zeta}{\zeta - \varphi(\alpha)} \frac{\psi(\zeta) - \alpha}{\psi(\zeta)} .\end{equation}

By substituting first $\zeta = \beta$ and then $\zeta = \beta + \omega$ , we obtain from (3.1) and Theorem 2.2 the following expression for the Laplace–Stieltjes transform (LST) of $Y_{\beta,\omega}$ .

Proposition 3.1. If Y is spectrally positive, then, for $\alpha\geqslant 0$ ,

(3.2) \begin{equation} {\mathbb E} \,{\rm e}^{-\alpha Y_{\beta,\omega}} = \frac{\alpha - \psi(\beta)}{\beta - \varphi(\alpha)} \frac{\beta}{\psi(\beta)} \frac{\beta + \omega - \varphi(\alpha)}{\alpha - \psi(\beta+\omega)} \frac{\psi(\beta+\omega)}{\beta+\omega}. \end{equation}

Remark 3.1. We also provide an alternative derivation of Proposition 3.1, using a relation with an associated queueing model. First observe that $Z^-$ is minus the running minimum of Y over an interval between two successive Poisson( $\omega$ ) inspection epochs, given that the latter epoch occurs before the killing epoch $T_\beta$ ; such an inspection interval is exp( $\beta+\omega$ ) distributed. Hence, $Z^-$ is exp( $\psi(\beta+\omega)$ ) distributed, just like $-\underline{Y}(T_{\beta+\omega})$ ; see, e.g., [Reference Kyprianou17, Section 6.5.2]. Furthermore,

\begin{equation*} {\mathbb E}\,{\mathrm{e}}^{-\alpha Z} = {\mathbb E}\,{\mathrm{e}}^{-\alpha Y(T_{\beta+\omega})}=\frac{\beta+\omega}{\beta+\omega - \varphi(\alpha)}. \end{equation*}

Hence, because $Z = Z^+ - Z^-$ , with $Z^+$ and $Z^-$ being independent,

(3.3) \begin{equation} {\mathbb E} \,{\rm e}^{-\alpha Z^+} = \frac{\beta+\omega}{\beta+\omega - \varphi(\alpha)} \frac{\psi(\beta+\omega) - \alpha}{\psi(\beta+\omega)} ; \end{equation}

indeed, cf. (3.1), $Z^+ \stackrel{{\rm d}} = \overline{Y}(T_{\beta+\omega})$ , as we already noticed in (2.8). Now observe, cf. (2.1), that $Y_{\beta,\omega}$ can be interpreted as the waiting time of the $N_{\beta,\omega}$ th customer of an M/G/1 queue with generic interarrival time $Z^-$ and generic service time $Z^+$ , with the first customer arriving in an empty system. Its LST is given by

\begin{equation*} \sum_{n=1}^{\infty} \bigg(\frac{\omega}{\beta+\omega}\bigg)^n \frac{\beta}{\beta+\omega} \,{\mathbb E} \,{\rm e}^{-\alpha W_n}, \end{equation*}

with $W_n$ denoting the waiting time of the nth such customer. The next step is to use the expression from [Reference Cohen12, (II.4.77)] for the generating function of ${\mathbb E} \,{\rm e}^{-\alpha W_n}$ . After some elementary calculations, (3.2) is recovered.

Remark 3.3. In many real-life applications, it may be more natural to have inspection intervals that are constant, or at least have a small coefficient of variation, instead of being exponentially distributed. In this remark we outline how we can use the alternative derivation of Proposition 3.1, as described in Remark 3.1, to determine the LST of the running maximum at Erlang( $k,k \omega$ )-distributed inspection moments, with $k\in{\mathbb N}$ . Note that the mean inspection interval still equals $1/\omega$ , and that its squared coefficient of variation equals $1/k$ , so that the case of a large k emulates constant inspection intervals. We shall again denote by $Y_{\beta,\omega}$ the running maximum at inspection moments until killing epoch $T_\beta \sim {\rm exp}(\beta)$ , and by $N_{\beta,\omega}$ the number of inspections before killing. We now have

\begin{equation*} {\mathbb P}(N_{\beta,\omega} = n) = \bigg(\frac{k \omega}{k \omega + \beta}\bigg)^{kn} \bigg(1 - \bigg(\frac{k \omega}{k \omega + \beta}\bigg)^k \bigg), \qquad n=0,1,\dots \end{equation*}

Indeed, the first factor on the right-hand side denotes the probability that at least kn intervals $ \sim {\rm exp}(k \omega)$ occur before $T_\beta$ , while the second factor denotes the probability that, subsequently, less than k such exp $(k \omega)$ intervals occur, i.e. the ( $n+1$ )th inspection does not occur before $T_\beta$ .

Denote by $Z_1^-,\dots,Z_k^-$ the negatives of the running minima, and by $Z_1^+,\dots,Z_k^+$ the running maxima, over k consecutive exp( $k \omega$ ) intervals which together compose one inspection interval. Since Y is a spectrally positive Lévy process, it follows that $Z_1^-,\dots,Z_k^-$ are independent and identically distributed (i.i.d.) exp $(\psi(\beta + k \omega))$ distributed. We furthermore note that $Z_1^+,\dots,Z_k^+$ are i.i.d., and the reasoning leading to (3.3) shows that their LST is given by

\begin{equation*} {\mathbb E}\, {\rm e}^{-\alpha Z^+} = \frac{\beta+k \omega}{\beta+ k \omega - \varphi(\alpha)} \frac{\psi(\beta+ k \omega) - \alpha}{\psi(\beta+ k \omega)} . \end{equation*}

Moreover, all the $Z_i^-$ and $Z_j^+$ are independent. Now $Y_{\beta,\omega}$ , as a supremum of partial sums $S_n = \sum_{m=1}^n Z_m$ with $Z_m = \sum_{i=1}^k Z_{m,i}^+ - \sum_{i=1}^k Z_{m,i}^-$ , can be interpreted as the waiting time of the $N_{\beta,\omega}$ th customer of an E $_k$ /G/1 queue with generic interarrival time $\sum_{i=1}^k Z_i^-$ and generic service time $\sum_{i=1}^k Z_i^+$ , with the first customer arriving in an empty system. Its LST, and hence the LST of $Y_{\beta,\omega}$ , is given by

\begin{equation*} \sum_{n=1}^{\infty} {\mathbb P}(N_{\beta,\omega}=n) \,{\mathbb E} \,{\rm e}^{-\alpha W_n} = \bigg(1 - \bigg(\frac{k \omega}{\beta+k \omega}\bigg)^k \bigg) \sum_{n=1}^{\infty} \bigg(\frac{k \omega}{\beta+k \omega}\bigg)^{kn} {\mathbb E} \,{\rm e}^{-\alpha W_n}. \end{equation*}

Finally, we can use [Reference Prabhu20, Theorem 25, p. 44] for the generating function of ${\mathbb E} \,{\rm e}^{-\alpha W_n}$ in the E $_k$ /G/1 queue.

We close this remark by once more focusing on the bankruptcy probability in insurance risk. In Section 2.2 we saw that the probability of bankruptcy before $T_\beta$ is given by $\tilde{p}(u,T_\beta) = {\mathbb P}(Y_{\beta, \omega} > u)$ . Hence, the LST of the bankruptcy probability, in the case of a spectrally positive Lévy process and Erlang( $k,k \omega$ ) inspection intervals, immediately follows from the LST of $Y_{\beta,\omega}$ .

Remark 3.4. The decomposition can be used to determine all moments of $Y_{\beta,\omega}$ from the (known) corresponding moments of $\overline Y(T_\beta)$ and $\overline Y(T_{\beta+\omega})$ . In this remark we demonstrate this by providing such a computation for the mean and variance of $Y_{\beta,\omega}$ . Clearly, it suffices to be able to determine the mean and variance of $\overline{Y}(T_{\zeta})$ for some $\zeta>0$ :

(3.4) \begin{align} {\mathbb E} \,\overline{Y}(T_\zeta) & = \frac{1}{\psi(\zeta)} - \frac{\varphi'(0)}{\zeta} ,\\[3pt] {\mathbb V}{\rm ar}\, \overline{Y}(T_\zeta) & = \frac{\varphi''(0)}{\zeta} + \bigg(\frac{\varphi'(0)}{\zeta}\bigg)^2 - \bigg(\frac{1}{\psi(\zeta)}\bigg)^2. \nonumber\end{align}

Due to the independence of the terms on the right-hand side of the decomposition of Theorem 2.2, the mean and variance of $Y_{\beta,\omega}$ immediately follow by successively plugging in $\zeta = \beta$ and $\zeta = \beta + \omega$ , and subtracting the resulting expressions.

3.2. The spectrally negative case

Suppose that Y is spectrally negative, i.e. it has no upward jumps. Consider an exp( $\zeta$ )-distributed interval. Define the cumulant $\Phi(\alpha) \,:\!=\, \log {\mathbb E} \exp({\alpha Y(1)})$ and its right-inverse $\Psi(\beta)$ . As follows directly from, e.g., [Reference Kyprianou17, Section 6.5.2], the running maximum $\overline Y (T_\zeta)$ is exponentially distributed with rate $\Psi(\zeta)$ . Using this result with $\zeta = \beta$ and $\zeta = \beta + \omega$ , and applying Theorem 2.2, we obtain an expression for the LST of $Y_{\beta,\omega}$ in the spectrally negative case.

Proposition 3.2. If Y is spectrally negative, then, for $\alpha\geqslant 0$ ,

(3.5) \begin{equation} {\mathbb E} \,{\rm e}^{-\alpha Y_{\beta,\omega}} = \frac{\Psi(\beta)}{\Psi(\beta) + \alpha} \frac{\Psi(\beta + \omega) + \alpha}{\Psi(\beta + \omega)} . \end{equation}

Using Proposition 3.2, an elementary computation reveals that $Y_{\beta,\omega}$ has an atom at zero, i.e.

\begin{equation*} {\mathbb P}(Y_{\beta,\omega} = 0) = \frac{\Psi(\beta)}{\Psi(\beta + \omega)},\end{equation*}

and is exp( $\Psi(\beta)$ ) distributed with the complementary probability $1-{\mathbb P}(Y_{\beta,\omega} = 0)$ .

4.1. The bankruptcy probability in the light-tailed case

We assume in this subsection that B is light-tailed in the sense that there is a unique strictly positive solution $\theta^\star$ of the equation $\varphi({-}\theta^\star)=0$ . Our aim is to identify the asymptotic behavior of $\tilde p(u)$ , and to compare it to the classic result for the asymptotic ruin probability p(u) in the same model (which coincides with the asymptotic behavior of the waiting-time tail in the dual M/G/1 queue). The well-known Cramér–Lundberg approximation [Reference Asmussen and Albrecher7, Theorem 5.3] states that

(4.1) \begin{equation} p(u) \sim \gamma {\rm e}^{-\theta^\star u} ,\end{equation}

with $f(u)\sim g(u)$ denoting $f(u)/g(u)\to 1$ as $u\to\infty$ , and

(4.2) \begin{equation} \gamma = - \frac{\varphi'(0)}{\varphi'({-}\theta^\star)} .\end{equation}

To determine the asymptotics of $\tilde{p}(u)$ we return to an observation made in Remark 3.1: $Y_{\beta,\omega}$ can be interpreted as the waiting time of the $N_{\beta,\omega}$ th customer of an M/G/1 queue with generic (exponentially distributed) interarrival time $Z^-$ and generic service time $Z^+$ . For $\beta=0$ , $Y_{0,\omega}$ becomes the steady-state waiting time for that queue. Notice that ${\mathbb E}Z^- = 1/\psi(\omega)$ while, as $Z^+$ is distributed as $\overline{Y}(T_{\omega})$ , it follows from (3.4) that ${\mathbb E}Z^+ = 1/\psi(\omega) - \varphi'(0)/\omega$ . As $\varphi'(0) = r-\lambda {\mathbb E} B > 0$ , we have ${\mathbb E} Z^- > {\mathbb E} Z^+$ , so the steady-state waiting-time distribution indeed exists. As a consequence, we can (again) rely on the Cramér–Lundberg approximation, or (equivalently) the tail asymptotics of the M/G/1 queue with generic interarrival time $Z^-$ and generic service time $Z^+$ , cf. (4.1): $\tilde{p}(u) \sim \tilde{\gamma} {\rm e}^{-\tilde{\theta}^\star u}$ , $u \rightarrow \infty$ . Our next task is to determine the constants $\tilde{\theta}^\star$ and $\tilde{\gamma}$ . The customer arrival rate in the auxiliary M/G/1 queue is $1/{\mathbb E}Z^- = \psi(\omega$ ), the generic service time $Z^+$ has LST (cf. (3.3))

\[{\mathbb E} \,{\rm e}^{-\alpha Z^+} = \frac{\omega}{\omega - \varphi(\alpha)} \frac{\psi(\omega) - \alpha}{\psi(\omega)},\]

and the service speed or premium rate equals one. Hence, $\tilde{\theta}^\star$ is the unique positive solution of $\tilde{\varphi}({-}\tilde{\theta}^\star)=0$ , with $\tilde{\varphi}(\alpha) \,:\!=\, \alpha - \psi(\omega)(1 - {\mathbb E}\,{\rm e}^{-\alpha Z^+})$ :

\begin{equation*} \tilde{\theta}^\star - \psi(\omega) \bigg(1 - \frac{\omega}{\psi(\omega)} \frac{\psi(\omega) + \tilde{\theta}^\star }{\varphi({-}\tilde{\theta}^\star) - \omega}\bigg) = 0.\end{equation*}

It readily follows that $\tilde{\theta}^\star = \theta^\star$ satisfies this equation (as $\varphi({-}\theta^\star)=0$ ), while otherwise there is only the negative solution $\tilde{\theta}^\star = - \psi(\omega)$ . The implication is that the bankruptcy probability $\tilde{p}(u)$ has the same decay rate $\theta^\star$ as the ruin probability p(u).

Let us now determine the prefactor $\tilde{\gamma}$ . Using (4.2) with $\varphi(\alpha)$ replaced by $\tilde{\varphi}(\alpha)$ , we find:

\begin{equation*} \tilde{\gamma} = - \frac{\tilde{\varphi}'(0)}{\tilde{\varphi}'({-}\tilde{\theta}^\star)} = - \frac{1 - \psi(\omega) {\mathbb E}Z^+}{1 +\psi(\omega) \frac{{\rm d}}{{\rm d} \alpha} {\mathbb E} \,{\rm e}^{-\alpha Z^+}\big|_{\alpha = -\tilde{\theta}^\star}} .\end{equation*}

A brief calculation, using (4.2), results in

(4.3) \begin{equation} \tilde{\gamma} = \gamma \frac{\psi(\omega)}{\psi(\omega) + \theta^\star} .\end{equation}

We have established the main result of this subsection.

Proposition 4.1. Assume B is light-tailed. As $u\to\infty$ ,

\begin{equation*} \frac{\tilde{p}(u)}{p(u)}\to \gamma^\star_\omega\,:\!=\, \frac{\psi(\omega)}{\psi(\omega)+\theta^\star}.\end{equation*}

This result shows that $\gamma^\star_\omega\uparrow 1$ as the inspection rate $\omega$ grows large. In addition, using that $\varphi(\alpha)-r\alpha +\lambda\to 0$ as $\alpha\to\infty$ implies that $\psi(\theta)- (\theta+\lambda)/r\to 0$ as $\theta\to\infty$ , it can be verified that $\lim_{\omega\to\infty}\omega(1-\gamma^\star_\omega) = r\theta^\star$ , which means that, for $\omega$ large, $\gamma^\star_\omega$ behaves as $1- r\theta^\star/\omega$ . This relation can be used to determine a ‘rule of thumb’ by which one can determine the minimally required inspection rate $\omega$ such that the information loss due to Poisson inspection is below a given threshold.

4.2. The bankruptcy probability in the heavy-tailed case

In this subsection we study the asymptotic behavior of the bankruptcy probability when B is heavy-tailed. More specifically, we assume that $B \in {\mathscr S}^\star$ , a class introduced in [Reference Klüppelberg16]. A random variable U on ${\mathbb R}$ belongs to ${\mathscr S}^\star$ if and only if its complementary distribution function $\overline F_U(x)\,:\!=\,{\mathbb P}(U>x)$ is positive for all x, and $\int_0^x \overline{F}_U(x-y) \overline{F}_U(y) \,{\rm d}y \sim 2 m_U\, \overline{F}_U(x)$ as $x \rightarrow \infty$ ; here, $m_U$ denotes the mean of U, restricted to the positive half-axis. ${\mathscr S}^\star$ is a class that is contained in, but is also very close to, the well-known class ${\mathscr S}$ of subexponential distributions. The class ${\mathscr S}^\star$ has the convenient property that if $U \in {\mathscr S}^\star$ then both U and $U^{{\rm res}}$ , the latter random variable being characterized through

\begin{equation*} {\mathbb P}(U^{{\rm res}}\leqslant x) = \int_0^x \frac{{\mathbb P}(U>y)}{m_U} \,{\rm d}y,\end{equation*}

are subexponential.

In our analysis of the asymptotics of $\tilde p(u)$ , we shall use a well-known result from [Reference Embrechts, Klüppelberg and Mikosch14] concerning the supremum $M_{\sigma}$ of a random walk $\{S_n\}_{n\in{\mathbb N}}$ over an interval $[0,\sigma]$ , with $\sigma$ some random variable (see also [Reference Foss and Zachary15, Theorem 1] for the more general case of $\sigma$ being a stopping time, and [Reference Asmussen and Albrecher7, p. 309] for the special case of a constant $\sigma$ ). If the increments of the random walk are in ${\mathscr S}^\star$ , with distribution function $F({\cdot})$ and complementary distribution function $\overline F({\cdot})$ , then

(4.4) \begin{equation} \lim_{u \rightarrow \infty} \frac{{\mathbb P}(M_{\sigma} >u)}{\overline{F}(u)} = {\mathbb E}\, \sigma .\end{equation}

Notice that the increments attain values in ${\mathbb R}$ , i.e. not necessarily in ${\mathbb R}^+$ . If we take $F({\cdot})$ to be the distribution function of a claim size minus a claim interarrival time, then we can apply (4.4) by considering the random number of claim arrivals in an exp( $\omega$ ) interval. This yields:

(4.5) \begin{equation} {\mathbb P} \big(\overline{Y}(T_\omega) >u\big) \sim \frac{\lambda}{\omega} \bar{F}(u) \sim \frac{\lambda}{\omega}\, {\mathbb P}(B>u) .\end{equation}

In order to use this result to determine the asymptotic behavior of the bankruptcy probability, we observe the following:

1. (i) $\overline{Y}(T_\omega)$ has the same distribution as $Z^+$ , cf. (2.8) with $\beta = 0$ . Hence, $Z^+ \in {\mathscr S}^\star$ , and $Z^{+,{\rm res}}$ is subexponential.

2. (ii) $Y_{0,\omega}$ can be viewed as the steady-state waiting time in an M/G/1 queue with generic service time $Z^+$ and arrival rate $\psi(\omega)$ , so with load $\rho \,:\!=\, {\mathbb E}Z^+/{\mathbb E}Z^-$ . Hence, we can use a standard result for the waiting-time tail in the M/G/1 queue, in which the residual service time $Z^{+,{\rm res}}$ is subexponential:

   \begin{equation*} {\mathbb P}(Y_{0,\omega} >u) \sim \frac{\rho}{1-\rho} {\mathbb P}(Z^{+,{\rm res}}>u). \end{equation*}
   This result holds equivalently for the ruin probability in the dual Cramér–Lundberg model, see, e.g., [Reference Asmussen and Albrecher7, Theorem X.2.1], and hence we conclude that
   \begin{align*} \tilde{p}(u) = {\mathbb P}(Y_{0,\omega} >u) & \sim \frac{{\mathbb E} Z^+}{{\mathbb E}Z^- - {\mathbb E} Z^+} {\mathbb P}(Z^{+,{\rm res}} >u) \\ & = \frac{1}{{\mathbb E}Z^- - {\mathbb E}Z^+} \int_u^{\infty} {\mathbb P}(Z^+ >y)\, {\rm d} y \\ & = \frac{\omega}{r - \lambda {\mathbb E}B} \int_u^{\infty} \frac{\lambda}{\omega}\, {\mathbb P}(B>y)\, {\rm d} y . \end{align*}
   Here, the last equality follows using ${\mathbb E}Z^- = 1/\psi(\omega)$ and (cf. (3.4)) ${\mathbb E}Z^+ = 1/\psi(\omega) - \varphi'(0)/\omega$ with $\varphi'(0) = r-\lambda {\mathbb E}B >0$ , and applying (4.5).

Combining (i) and (ii) yields the main result of this subsection.

Proposition 4.2. Assume $B\in {\mathscr S}^\star$ . As $u\to\infty$ ,

\begin{equation*} \tilde{p}(u) \sim \frac{\lambda {\mathbb E}B}{r - \lambda {\mathbb E}B} {\mathbb P}(B^{{\rm res}} >u) . \end{equation*}

Notice, looking at the statement in (ii) above regarding the tail behavior in an M/G/1 queue, that $\tilde p(u)$ has the exact same tail asymptotics as p(u). In particular, the asymptotics of $\tilde p(u)$ do not depend on the inspection rate $\omega$ . This may look surprising at first sight, but realize that for $B\in {\mathscr S}^\star$ there is the intuition that ‘a single big jump’ is responsible (with overwhelming probability as $u\to\infty$ ) for exceeding a high level. This suggests that when ruin occurs, it is highly likely that the capital is still below zero at the next inspection moment.