3.1. Exponentially distributed inter-inspection times

In this subsection we consider the case of inter-inspection times that are exponentially distributed with parameter $\omega>0$ . Note that the time until the first event, which can be either an inspection or killing, is $T_{\beta+\omega}$ (i.e. exponentially distributed with parameter $\beta+\omega$ ). We define the two nonnegative random quantities

\[Z^+ (\beta,\omega) \equiv Z^+ \,:\!=\, \bar{Y}(T_{\beta+\omega}), \qquad Z^- (\beta,\omega) \equiv Z^- \,:\!=\, -\underline{Y}(T_{\beta+\omega}).\]

The random variable $Z^-$ is exponentially distributed with parameter $\theta\,:\!=\, \psi(\beta+\omega)$ , and by virtue of the Wiener–Hopf decomposition $Z^-$ and $Z^+$ are independent. Below we use the compact notation $Z\,:\!=\, Z^+-Z^-$ . Note that, due to (1),

(2) \begin{equation}\mathrm{E} \,\mathrm{e}^{-\alpha Z^{+}} =\xi(\alpha,\beta+\omega)=\frac{\alpha -\psi(\beta+\omega)}{\varphi(\alpha)-\beta-\omega}\frac{\beta+\omega}{\psi(\beta+\omega)}.\end{equation}

3.1.1. Equation for transform.

We shall evaluate the quantity $\pi(\alpha,\beta\,|\,T_\omega)$ by conditioning on the first event. If $Y(\cdot)$ has to exceed level u at an inspection time, then this first event time should be an inspection (i.e. not killing), which happens with probability $\omega/(\beta+\omega)$ . Then there are two options: at this inspection time we are either already above u and, hence, ruin occurs or we are at some level v below u and we still need to bridge the remaining $u-v$ . By following this line of reasoning, we obtain an expression in terms of our target quantity ${\pi}(\alpha,\beta\,|\,T_\omega)$ , which we can solve. To this end, we first we write

\[{{\mathfrak p}_\beta(u\,|\,T_\omega)}=\frac{\omega}{\beta+\omega}\Big(\mathrm{P}(Z > u)+\int_{-\infty}^u\mathrm{P}(Z\in\mathop{}\!\mathrm{d} v)\,{{\mathfrak p}_\beta(u-v\,|\,T_\omega)}\Big).\]

In order to evaluate the associated transform, we multiply the entire equation by $\mathrm{e}^{-\alpha u}$ and integrate over positive u:

(3) \begin{align}{\pi}(\alpha,\beta\,|\,T_\omega)=\frac{\omega}{\beta+\omega}\Bigg(&\underbrace{\int_0^\infty \mathrm{e}^{-\alpha u}\,\mathrm{P}(Z>u)\mathop{}\!\mathrm{d} u}_{\text{(i)}}\nonumber\\&+\underbrace{\int_{u=0}^\infty \mathrm{e}^{-\alpha u}\int_{v=-\infty}^u\mathrm{P}(Z\in\mathop{}\!\mathrm{d} v)\,{{\mathfrak p}_\beta(u-v\,|\,T_\omega)}\mathop{}\!\mathrm{d}u}_{\text{(ii)}}\Bigg).\end{align}

We proceed by separately analyzing the integrals (i) and (ii) appearing in (3).

1. (i) As a first step, we condition on the value of $Z^-$ and change the order of integration. With $\theta\,:\!=\, \psi(\beta+\omega)$ , recalling that $Z^-$ is exponentially distributed with parameter $\theta$ , we readily find that

   \begin{align*}\int_0^\infty \mathrm{e}^{-\alpha u}\ \mathrm{P}(Z > u)\mathop{}\!\mathrm{d} u&=\theta\int_{u=0}^\infty \mathrm{e}^{-\alpha u}\int_{v=0}^\infty \mathrm{e}^{-\theta v}\mathrm{P}(Z^+> u+v)\mathop{}\!\mathrm{d} v\mathop{}\!\mathrm{d} u.\end{align*}
   Performing the change of variable $x\,:\!=\, u+v$ , we have that this equals
   \begin{align*}\theta\int_{x=0}^\infty \mathrm{e}^{-\theta x}\mathrm{P}(Z^+> x)\Big(\int_{u=0}^x\mathrm{e}^{u(\theta-\alpha)}\mathop{}\!\mathrm{d} u\Big)\mathop{}\!\mathrm{d} x.\end{align*}
   Upon evaluating the inner integral, this reduces to
   \begin{align*}\frac{\theta}{\theta-\alpha}\Big(\int_{0}^\infty \mathrm{e}^{-\alpha x}\mathrm{P}(Z^+> x)\mathop{}\!\mathrm{d} x -\int_{0}^\infty \mathrm{e}^{-\theta x}\mathrm{P}(Z^+> x)\mathop{}\!\mathrm{d} x\Big).\end{align*}
   Recalling that $\xi(\alpha,\beta+\omega)= \mathrm{E} \,\mathrm{e}^{-\alpha Z^{+}}$ , observe that
   \begin{align*}\int_0^\infty \mathrm{e}^{-\alpha x} \mathrm{P}(Z^+ > x)\mathop{}\!\mathrm{d} x &=\int_{x=0}^\infty \mathrm{e}^{-\alpha x} \int_{y=x}^\infty\mathrm{P}(Z^+\in\mathop{} \!\mathrm{d} y)\mathop{}\!\mathrm{d} x\\&=\int_{y=0}^\infty \Big(\int_{x=0}^y \mathrm{e}^{-\alpha x} \mathop{}\!\mathrm{d}x\Big) \mathrm{P}(Z^+ \in \mathop{}\!\mathrm{d} y)\\&=\frac{1}{\alpha}\Big(\int_{0}^\infty \mathrm{P}(Z^+\in\mathop{}\!\mathrm{d}y) -\int_{0}^\infty \mathrm{e}^{-\alpha y}\mathrm{P}(Z^+\in\mathop{}\!\mathrm{d}y)\Big)\\ &= \frac{1}{\alpha}\big(1-\xi(\alpha,\beta+\omega)\big)\end{align*}
   A similar reasoning applies when replacing $\alpha$ by $\theta$ . Combining the above, we thus conclude that
   \begin{align*}\int_0^\infty \mathrm{e}^{-\alpha u}\, \mathrm{P}(Z> u)\mathop{}\!\mathrm{d} u &=\frac{\theta}{\theta-\alpha}\Big(\frac{1}{\alpha}\big(1-\xi(\alpha,\beta+\omega)\big)-\frac{1}{\theta}\big(1-\xi(\theta,\beta+\omega)\big)\Big)\\&=\frac{\alpha\xi(\theta,\beta+\omega)-\theta\xi(\alpha,\beta+\omega)}{\alpha(\theta-\alpha)}+\frac{1}{\alpha}\,=\!:\, F(\alpha,\beta, \omega).\end{align*}

2. (ii) We continue by evaluating the second integral, starting by splitting it into two parts:

   (4) \begin{align}\int_{u=0}^\infty \mathrm{e}^{-\alpha u}\int_{v=-\infty}^u&\mathrm{P}(Z\in\mathop{}\!\mathrm{d} v)\,{{\mathfrak p}_\beta(u-v\,|\,T_\omega)}\,\mathop{}\!\mathrm{d} u={\pi}_1(\alpha,\beta\,|\,T_\omega)+{\pi}_2(\alpha,\beta\,|\,T_\omega),\end{align}
   where the first term ${\pi}_1(\alpha,\beta\,|\,T_\omega)$ corresponds to the triangular integration area $\{0\leqslant v\leqslant u\}$ and the second term ${\pi}_2(\alpha,\beta\,|\,T_\omega)$ to the rectangular integration area $\{v\leqslant 0\leqslant u\}$ .

As before, to evaluate the first double integral in (4), we interchange the order of the integrals. We obtain

\begin{align*}{\pi}_1(\alpha,\beta\,|\,T_\omega)={\pi}(\alpha,\beta\,|\,T_\omega)\int_0^\infty\mathrm{e}^{-\alpha v}\mathrm{P}(Z\in\mathop{}\!\mathrm{d} v),\end{align*}

which, by conditioning on the value of $Z^-$ , equals

\begin{align*}\theta\ {\pi}(\alpha,\beta\,|\,T_\omega)\int_{v=0}^\infty \mathrm{e}^{-\alpha v}\Big(\int_{x=0}^\infty \mathrm{e}^{-\theta x}\mathrm{P}\big(Z^+-v\in\mathop{}\!\mathrm{d}x\big)\Big)\mathop{}\!\mathrm{d} v .\end{align*}

By evaluating the inner integral with a change of variable $y\,:\!=\, v+x$ , this expression can be further rewritten as

\begin{align*}&\theta\ {\pi}(\alpha,\beta\,|\,T_\omega)\int_{v=0}^\infty\int_{y=v}^\infty \mathrm{e}^{v(\theta-\alpha)}\mathrm{e}^{-\theta y}\mathrm{P}(Z^+\in\mathop{}\!\mathrm{d} y)\mathop{}\!\mathrm{d} v\\& \qquad=\theta\ {\pi}(\alpha,\beta\,|\,T_\omega)\int_{y=0}^\infty\Big(\int_{v=0}^y\mathrm{e}^{v(\theta-\alpha)}\mathop{}\!\mathrm{d} v\Big)\mathrm{e}^{-\theta y}\mathrm{P}(Z^+\in\mathop{}\!\mathrm{d} y),\end{align*}

where we again changed the order of the integrals. Upon evaluating the obtained inner integral, this reduces to

\begin{align*}\theta\ {\pi}(\alpha,\beta\,|\,T_\omega)\int_{0}^\infty \mathrm{e}^{-\theta y}\Big(\frac{\mathrm{e}^{\theta y-\alpha y}}{\theta-\alpha}-\frac{1}{\theta-\alpha}\Big)\,\mathrm{P}(Z^+\in\mathop{}\!\mathrm{d}y).\end{align*}

Some elementary manipulations lead us to the conclusion that

\begin{align*}{\pi}_1(\alpha,\beta\,|\,T_\omega)& =\theta\ {\pi}(\alpha,\beta\,|\,T_\omega)\frac{\xi(\alpha,\beta+\omega)}{\theta-\alpha}-H(\alpha,\beta,\omega),\end{align*}

where

\[H(\alpha,\beta,\omega) \,:\!=\, \theta\ {\pi}(\alpha,\beta\,|\,T_\omega)\frac{\xi(\theta,\beta+\omega)}{\theta-\alpha}.\]

In order to evaluate (4), we are left with considering the second integral, i.e. ${\pi}_2(\alpha,\beta\,|\,T_\omega)$ . Performing the change of variable $x\,:\!=\, u-v$ , and swapping the order of the integrals, we have

\begin{align*}{\pi}_2(\alpha,\beta\,|\,T_\omega)&=\int_{v=-\infty}^0 \int_{x=-v}^\infty\mathrm{e}^{-\alpha v}\mathrm{e}^{-\alpha x}\ {{\mathfrak p}_\beta(x\,|\,T_\omega)}\,\mathop{}\!\mathrm{d}x\,\mathrm{P}(Z\in\mathop{}\!\mathrm{d} v)\\&=\int_{x=0}^\infty\int_{v=-x}^0 \mathrm{e}^{-\alpha v}\mathrm{P}(Z\in\mathop{}\!\mathrm{d}v)\,\mathrm{e}^{-\alpha x}\ {{\mathfrak p}_\beta(x\,|\,T_\omega)}\mathop{}\!\mathrm{d} x .\end{align*}

Now note that the inner integral v is only evaluated in nonpositive values, so $-\infty\lt v\leqslant 0$ . Since both $Z^-,Z^+\geqslant 0$ , we have $Z^+ -Z^-\leqslant 0$ if and only if $Z^-\geqslant Z^+$ . The probability of $Z^+-Z^-$ being less than v can hence be evaluated as follows: for $v\leqslant 0$ , realizing that $\mathrm{P}(Z^-\geqslant Z^+)=\xi(\theta,\beta+\omega)$ ,

\begin{align*}\mathrm{P}(Z < v) &=\mathrm{P}(Z < v\ |\ Z^-\geqslant Z^+)\,\mathrm{P}(Z^-\geqslant Z^+) =\mathrm{e}^{\theta v}\ \xi(\theta,\beta+\omega),\end{align*}

using the memoryless property of the exponential distribution. We thus find that

\begin{align*}{\pi}_2(\alpha,\beta\,|\,T_\omega)=\theta\ \xi(\theta,\beta+\omega)\int_{x=0}^\infty\Big(\int_{v=-x}^0\mathrm{e}^{v(\theta-\alpha)}\mathop{}\!\mathrm{d} v\Big) \,\mathrm{e}^{-\alpha x}\,{{\mathfrak p}_\beta(x\,|\,T_\omega)}\,\mathop{}\!\mathrm{d} x.\end{align*}

Evaluating the (standard) inner integral and performing similar manipulations as before, we conclude that

\begin{align*}{\pi}_2(\alpha,\beta\,|\,T_\omega)&=H(\alpha,\beta,\omega)-G(\alpha,\beta,\omega),\end{align*}

where

\[G(\alpha,\beta,\omega)\,:\!=\, \theta\ \xi(\theta,\beta+\omega)\frac{{\pi}(\theta,\beta\,|\,T_\omega)}{\theta-\alpha}.\]

Upon combining the above, we obtain

\[{\pi}_1(\alpha,\beta\,|\,T_\omega)+{\pi}_2(\alpha,\beta\,|\,T_\omega)=\theta\ {\pi}(\alpha,\beta\,|\,T_\omega)\frac{\xi(\alpha,\beta+\omega)}{\theta-\alpha}-G(\alpha,\beta,\omega).\]

We now use the results obtained above in order to derive an expression for the transform $ {\pi}(\alpha,\beta\,|\,T_\omega)$ . Upon adding up the expressions that we found in (i) and (ii), we obtain

(5) \begin{align}{\pi}(\alpha,\beta\,|\,T_\omega)=\frac{\omega}{\beta+\omega}\Bigg(F(\alpha,\beta,\omega)+\frac{\theta\,\xi(\alpha,\beta+\omega)}{\theta-\alpha}{\pi}(\alpha,\beta\,|\,T_\omega)-G(\alpha,\beta,\omega)\Bigg).\end{align}

Substituting the known expression for $\xi(\alpha,\beta+\omega)$ , as was given in (2), into the relation (5), we obtain the following result characterizing ${\pi}(\alpha,\beta\,|\,T_\omega)$ .

Proposition 1. For any $\alpha\geqslant 0$ and $\beta>0$ ,

(6) \begin{align}{\pi}(\alpha,\beta\,|\,T_\omega)\Big(1-\frac{\omega}{\beta+\omega-\varphi(\alpha)}\Big)=\frac{\omega}{\beta+\omega}\Big(F(\alpha,\beta,\omega)-G(\alpha,\beta,\omega)\Big).\end{align}

3.1.2. Determination of the unknown constant.

Whereas $F(\alpha,\beta,\omega)$ is an explicit expression in terms of the model primitives, $G(\alpha,\beta,\omega)$ contains the unknown quantity ${\pi}(\theta,\beta\,|\,T_\omega)$ . This unknown can be identified by observing that any root (in the right-half of the complex $\alpha$ plane, that is) of the left-hand side of (6) is necessarily also a root of the right-hand side, using the fact that ${\pi}(\alpha,\beta\,|\,T_\omega)$ is finite for all $\alpha,\beta>0$ . Solving

\begin{align*}1=\frac{\omega}{\beta+\omega-\varphi(\alpha)},\end{align*}

leads us to $\varphi(\alpha)=\beta$ or $\alpha=\psi(\beta)$ since $\psi(\cdot)$ is the right inverse of the Laplace exponent $\varphi(\cdot)$ . It requires an elementary calculation to verify that we thus end up with

\begin{align*}{\pi}(\theta,\beta\,|\,T_\omega)=\frac{\omega\ \psi(\beta)\,\xi(\theta,\beta+\omega)-\beta\ \theta+\beta\ \psi(\beta)}{\theta\ \omega\ \psi(\beta)\,\xi(\theta,\beta+\omega)}.\end{align*}

Plugging this expression into ${\pi}(\alpha,\beta\,|\,T_\omega)$ , that we found in Proposition 1, after considerable calculus one arrives at the following result.

Theorem 1. For any $\alpha\geqslant 0$ and $\beta>0$ ,

\begin{align*}{\pi}(\alpha,\beta\,|\,T_\omega)=\frac{1}{\alpha}\frac{\omega\ \varphi(\alpha)(\theta-\alpha)\,\psi(\beta) +\alpha\ \beta\big(\varphi(\alpha)-\beta-\omega\big)(\theta-\psi(\beta))}{(\varphi(\alpha)-\beta)(\beta+\omega)(\theta-\alpha)\psi(\beta)}.\end{align*}

Define $\varrho(\alpha,\beta\,|\,\Omega)\,:\!=\, {\mathrm{E}}\,\mathrm{e}^{-\alpha\bar Y_\Omega(T_\beta)}$ , where

\[\bar Y_\Omega(t)\,:\!=\, \sup\{Y(\bar\Omega_n): \bar\Omega_n\leqslant t\}.\]

We note that ${\pi}(\alpha,\beta\,|\,T_\omega)$ and $\varrho(\alpha,\beta\,|\,T_\omega)$ can be expressed in one another, using the well-known ‘translation formula’ $\varrho(\alpha,\beta\,|\,T_\omega)=1-\alpha\,{\pi}(\alpha,\beta\,|\,T_\omega)$ . We eventually find the following result, which is a true generalization of the time-dependent version of the Pollaczek–Khinchine formula, i.e.

\begin{align*}\varrho(\alpha,\beta\,|\,T_\omega)=\frac{\alpha-\psi(\beta)}{\varphi(\alpha)-\beta}\frac{\beta}{\psi(\beta)}\frac{\varphi(\alpha)-\beta-\omega}{\alpha-\psi(\beta+\omega)}\frac{\psi(\beta+\omega)}{\beta+\omega};\end{align*}

cf. [Reference Boxma and Mandjes14, Proposition 1]. Indeed, as $\omega\to\infty$ , which corresponds to the inspection process taking place at an increasingly high frequency, thus approaching permanent inspection of the process Y(t), we find that the transform corresponds to that appearing in the time-dependent Pollaczek–Khinchine formula

\[\lim_{\omega\to\infty}\varrho(\alpha,\beta\,|\,T_\omega)= \frac{\alpha-\psi(\beta)}{\varphi(\alpha)-\beta}\frac{\beta}{\psi(\beta)};\]

see e.g. [Reference Dębicki and Mandjes18, Thm.4.1].

3.2. Hyperexponentially distributed inter-inspection times

We now consider the bankruptcy model with hyperexponential-d inter-inspection times, by mimicking the line of reasoning that has been used in the previous subsection for the case of exponentially distributed inter-inspection times.

With probability $p_i> 0$ the inter-inspection time is exponentially distributed with rate $\omega_i>0$ for $i\in \{1,\ldots,d\}$ and $\sum_{i=1}^d p_i=1$ . This means that, with ${\boldsymbol\omega}=(\omega_1,\ldots,\omega_d)$ and ${\boldsymbol p}=(p_1,\ldots,p_d)$ , a generic inter-inspection time $T_{{\boldsymbol\omega,\boldsymbol p}}$ is characterized by the density,

\[{\mathrm{P}}(T_{{\boldsymbol \omega},{\boldsymbol p}}\in\mathrm{d}t) =\sum_{i=1}^d p_i\,\omega_i\mathrm{e}^{-\omega_i t}\mathrm{d}t.\]

Recalling the definition of the two independent, nonnegative random variables $Z^+ (\beta,\omega_i) \,:\!=\, \bar{Y}(T_{\beta+\omega_i})$ and $Z^- (\beta,\omega_i) \,:\!=\, -\underline{Y}(T_{\beta+\omega_i})$ , we find that

\[\xi_i(\alpha,\beta)\,:\!=\, \mathrm{E} \,\mathrm{e}^{-\alpha Z^+(\beta,\omega_i)}=\xi(\alpha,\beta+\omega_i)=\frac{\alpha-\psi(\beta+\omega_i)}{\varphi(\alpha)-\beta-\omega_i}\frac{\beta+\omega_i}{\psi(\beta+\omega_i)},\]

and that $Z^- (\beta,\omega_i)$ is exponentially distributed with parameter $\theta_i\,:\!=\, \psi(\beta+\omega_i)$ .

3.2.1. Equation for transform.

The main idea is that at time 0 we sample which of the d exponentially distributed inter-inspection times will be used. Given this is the ith inter-inspection time (with parameter $\omega_i$ , that is), we can reuse the argument developed for the exponential case, but with $Z^+(\beta,\omega)$ replaced by $Z^+(\beta,\omega_i)$ and with $Z^-(\beta,\omega)$ replaced by $Z^-(\beta,\omega_i)$ . Hence, as the counterpart of the relation for the transform ${\pi}(\alpha,\beta\,|\,T_{\omega})$ that we derived in the case of exponentially distributed inter-inspection times, we now obtain

\begin{align*}{\pi}(\alpha,\beta\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})= \sum_{i=1}^d p_i\ \frac{\omega_i}{\beta+\omega_i}\Bigg(&\frac{\alpha\ \xi_i(\theta_i,\beta)-\theta_i\ \xi_i(\alpha,\beta)}{\alpha(\theta_i-\alpha)}+\frac{1}{\alpha}\\&+\theta_i{\pi}(\alpha,\beta\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})\frac{\xi_i(\alpha,\beta)-\xi_i(\theta_i,\beta)}{\theta_i-\alpha}\\&+\theta_i\ \xi_i(\theta_i,\beta)\frac{{\pi}(\alpha,\beta\,|\,T_{{\boldsymbol\omega},{\boldsymbol p}})-{\pi}(\theta_i,\beta\,|\,T_{{\boldsymbol\omega},{\boldsymbol p}})}{\theta_i-\alpha}\Bigg);\end{align*}

cf. the decomposition featuring in (5). Following the same procedure as in the case of exponentially distributed inter-inspection times, this can be rewritten as

\[{\pi}(\alpha,\beta\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})= \sum_{i=1}^d p_i \frac{\omega_i}{\beta+\omega_i}\Big(F(\alpha,\beta,\omega_i)+\frac{\theta_i\,\xi_i(\alpha,\beta)}{\theta_i-\alpha}{\pi}(\alpha,\beta\,|\,T_{{\boldsymbol\omega},{\boldsymbol p}})-G_i(\alpha,\beta,{\boldsymbol \omega})\Big).\]

Here the function $F(\alpha,\beta,\omega)$ is as defined in the subsection on exponentially distributed inter-inspection times, so that, for $i=1,\ldots,d$ ,

\[F(\alpha,\beta, \omega_i)= \frac{\alpha \xi(\theta_i,\beta+\omega_i)-\theta_i\xi(\alpha,\beta+\omega_i)}{\alpha(\theta_i-\alpha)}+\frac{1}{\alpha};\]

in addition, for $i=1,\ldots,d$ ,

\[G_i(\alpha,\beta,{\boldsymbol \omega})\,:\!=\, \bar G(\alpha,\beta,\omega_i) \,z_i,\]

where

\[\bar G(\alpha,\beta,\omega_i)\,:\!=\, \frac{\theta_i \xi_i(\theta_i,\beta)}{\theta_i-\alpha},\:\:\:\: z_i\equiv z_i(\beta,{\boldsymbol\omega}) \,:\!=\,{\pi}(\theta_i,\beta\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}}).\]

We proceed by isolating our target object ${\pi}(\alpha,\beta\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})$ . By inserting the expression for $\xi_i(\alpha,\beta)$ , we thus end up with the following characterization.

Proposition 2. For any $\alpha\geqslant 0$ and $\beta>0$ ,

(7) \begin{align}{\pi}(\alpha,\beta\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})\Big(1-\sum_{i=1}^d p_i\frac{\omega_i}{\beta+\omega_i-\varphi(\alpha)}\Big)=\sum_{i=1}^dp_i\frac{\omega_i}{\beta+\omega_i}\Big(F(\alpha,\beta,\omega_i)-G_i(\alpha,\beta,{\boldsymbol\omega})\Big).\end{align}

3.2.2. Determination of the unknown constants.

Observe that, on the right-hand side of (6) in Proposition 2, the object $F(\alpha,\beta,\omega_i)$ is explicitly expressed in terms of the model primitives, and so is $\bar G(\alpha,\beta,\omega_i)$ . The constants $z_1,\ldots,z_d$ however are yet to be determined. In the remainder of this subsection we point out how these constants can be identified.

Lemma 1. For any $\beta$ and ${\boldsymbol\omega}$ , the equation

(8) \begin{equation}\sum_{i=1}^d p_i \frac{\omega_i}{\beta+\omega_i-\varphi(\alpha)}=1\end{equation}

has d roots in the right half-plane, say, $\alpha_1\equiv\alpha_1(\beta,{\boldsymbol\omega}),\ldots,\alpha_d\equiv\alpha_d(\beta,{\boldsymbol\omega}).$

Proof. The main idea behind the proof (as well as other proofs later in the paper) is the following. Consider a d-dimensional Markov additive process (MAP), say $(W(\cdot),J(\cdot))\equiv(W(t),J(t))_{t\geqslant 0}$ ; we refer to, e.g. [Reference Asmussen8, Section XI.2] for a definition. With $q_{ij}$ , for $i,j=1,\ldots,d$ , denoting the transition rates corresponding to the modulating continuous-time Markov chain $J(\cdot)$ on $\{1,\ldots,d\}$ , and with $\varphi_1(\cdot),\ldots,\varphi_d(\cdot)$ Laplace exponents of d Lévy processes, define the (i, j) th entry of the $(d\times d)$ -matrix $M(\alpha,{\boldsymbol \eta})$ by

\[m_{ij}(\alpha,{\boldsymbol \eta}) \,:\!=\, (\varphi_i(\alpha)-\eta_i)\,1\{i=j\}+q_{ij}\]

for a given componentwise positive vector ${\boldsymbol\eta}$ . From [Reference Ivanovs, Boxma and Mandjes23, Theorem 1 and Remark 2.1] we know that, if the d Lévy processes featuring in $(W(\cdot),J(\cdot))$ are no subordinators, then the equation $\textrm{det}\, M(\alpha,{\boldsymbol\eta})=0$ has, for any componentwise positive vector ${\boldsymbol\eta}$ , exactly d roots in the right-half of the complex plane. To prove the lemma, the objective is now to construct a MAP $(W(\cdot),J(\cdot))$ such that the corresponding equation $\textrm{det}\, M(\alpha,{\boldsymbol \eta})=0$ is, for some choice of the vector ${\boldsymbol\eta}$ , equivalent to (8).

Consider the MAP defined by a transition rate matrix Q that is given by

\begin{equation*}Q=\boldsymbol{p}\boldsymbol{\omega}^{\top}-\mathrm{diag}(\boldsymbol{\omega}),\end{equation*}

and Laplace exponents $\varphi_1(\alpha)=\cdots=\varphi_d(\alpha)=\varphi(\alpha)$ . The entries of the componentwise positive vector ${\boldsymbol\eta}$ are defined to be $\eta_i=\beta$ , for $i=1,\ldots,d$ . We now observe that

\begin{equation*}M(\alpha,{\boldsymbol \eta})=Q+ \textrm{diag}({\boldsymbol\varphi(\alpha)-\boldsymbol\eta})=\boldsymbol{p}\boldsymbol{\omega}^\top+\textrm{diag}(\varphi(\alpha){\boldsymbol1}-{\boldsymbol\omega}-{\beta}{\boldsymbol 1}),\end{equation*}

with ${\boldsymbol 1}$ a d-dimensional all-ones vector. As a consequence, $M(\alpha,\eta)$ is the sum of a diagonal matrix and a rank-one matrix.

The last step is to evaluate $\textrm{det}\,M(\alpha,{\boldsymbol \eta})$ , to show that the equation $\textrm{det}\,M(\alpha,{\boldsymbol \eta})=0$ is equivalent to (8). To this end, define $\Delta\,:\!=\,\textrm{diag}(\varphi(\alpha){\boldsymbol1}-{\boldsymbol\omega}-{\beta}{\boldsymbol 1})$ , and let $\delta_i$ be the (i,i)th entry of this diagonal matrix. Observe that $M(\alpha,{\boldsymbol \eta})=\Delta+\boldsymbol{p}\boldsymbol{\omega}^\top$ . As a consequence, we can write

\begin{equation*}\text{det}\,M(\alpha,{\boldsymbol\eta})=\text{det}\,\Delta\cdot\text{det}\,(I_d+\boldsymbol{p}\boldsymbol{\omega}^\top \Delta^{-1}).\end{equation*}

For A of dimension $m\times n$ , and B of dimension $n\times m$ , we have $\textrm{det}\,(I_m+AB)=\textrm{det}\,(I_n+BA)$ . We thus conclude that

\begin{align*}\textrm{det}\,M(\alpha,{\boldsymbol\eta})&=\textrm{det}\,\Delta\cdot\textrm{det}\,(I_1 +\boldsymbol\omega^\top \Delta^{-1}\boldsymbol{p})=\left(\prod_{i=1}^d\delta_i\right)\Bigg(\sum_{i=1}^dp_i\frac{\omega_i}{\varphi(\alpha)-\beta-\omega_i}+1\Bigg),\end{align*}

so that indeed $\textrm{det}\,M(\alpha,{\boldsymbol \eta})=0$ is equivalent to (8).

To identify the unknown quantities $z_i$ , we use the property that any root of the left-hand side of (7) in Proposition 2 (in the right-half of the complex $\alpha$ plane) is necessarily a root of the right-hand side as well, noting that ${\pi}(\alpha,\beta\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})$ is finite for all $\alpha\geqslant 0$ and $\beta>0$ . Let $\alpha_1,\ldots,\alpha_d$ be the roots of the left-hand side in the right-half of the complex $\alpha$ plane; there are precisely d of these, by Lemma 1. Consequently, we are to solve a system of linear equations: we have, for any $j=1,\ldots,d$ ,

(9) \begin{equation}\sum_{i=1}^d p_i\ \frac{\omega_i}{\beta+\omega_i}\Big(F(\alpha_j,\beta,\omega_i)-\bar G(\alpha_j,\beta,\omega_i)\,z_i\Big)=0.\end{equation}

It can be argued that the roots $\alpha_j$ are distinct; see Remark 2. Hence, the unknown vector ${\boldsymbol z}$ can be identified by solving the j linear equations (9). We thus arrive at the following result.

Theorem 2. For any $\alpha\geqslant 0$ and $\beta>0$ ,

\begin{align*}{\pi}(\alpha,\beta\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})=\frac{\displaystyle \sum_{i=1}^dp_i\frac{\omega_i}{\beta+\omega_i}\Big(F(\alpha,\beta,\omega_i)-\bar G(\alpha,\beta, \omega_i)\,z_i\Big)}{\displaystyle 1-\sum_{i=1}^d p_i\frac{\omega_i}{\beta+\omega_i-\varphi(\alpha)}},\end{align*}

where the vector ${\boldsymbol z}$ follows from the linear equations (9).

Remark 2. The d roots are actually real, as follows from a reasoning similar to that in [Reference Constantinescu16, Section 3.2]. First realize that without loss of generality we can assume the rates $\omega_i$ are distinct (because if they were not, some of the $p_i$ could be ‘lumped’ to make the $\omega_i$ distinct). Putting (without loss of generality) these $\omega_i$ in increasing order, we obtain that there is precisely one real root between $x_{i-1}$ and $x_i$ , with $x_0\,:\!=\, 0$ and $x_i\,:\!=\,\psi(\beta+\omega_{i})$ for $i=1,\ldots,d$ . Observe that the smallest root is $\psi(\beta)$ . This implies that the roots are distinct.

4.1. Erlang distributed inter-inspection times

We consider the bankruptcy model with Erlang-k inspection times, with shape parameter $k\in{\mathbb N}$ and scale parameter $\omega$ . This means that a generic inter-inspection time $E_{k,\omega}$ is characterized via the density

\[\mathrm{P}\big(E_{k,\omega}\in\mathop{}\!\mathrm{d} t\big)=\omega\frac{(\omega t)^{k-1}}{(k-1)!}\mathrm{e}^{-\omega t}\,\mathrm{d}t.\]

In this subsection again a crucial role is played by the two nonnegative random quantities

\[Z^+(\beta,\omega)\equiv Z^+\,:\!=\, \bar{Y}(T_{\beta+\omega}), \qquad Z^-(\beta,\omega)\equiv Z^-\,:\!=\, -\underline{Y}(T_{\beta+\omega}).\]

In what follows, let $Z^+_k\equiv Z^+_k(\beta,\omega)$ denote the sum of k i.i.d. copies of $Z^+$ , and analogously $Z^-_k\equiv Z^-_k(\beta,\omega)$ the sum of k i.i.d. copies of $Z^-$ .

Recalling ${\mathrm{E}}\,\mathrm{e}^{-\alpha Z^+}=\xi(\alpha,\beta+\omega)$ , which can be evaluated by applying (1), we write

\[\xi_k(\alpha,\beta+\omega)\,:\!=\, {\mathrm{E}}\,\mathrm{e}^{-\alpha Z_k^+}=(\xi(\alpha,\beta+\omega))^k.\]

In addition, we recall that $Z^-$ is exponentially distributed with parameter $\theta\,:\!=\, \psi(\beta+\omega)$ . It is also noted that $Z^-$ and $Z^+$ are, by virtue of the Wiener–Hopf decomposition, independent. This entails that $Z_k^-$ is Erlang distributed with shape parameter k and scale parameter $\theta$ , and that $Z^-_k$ and $Z^+_k$ are independent. Below we use the compact notation $Z_k\,:\!=\, Z_k^+-Z_k^-$ .

4.1.1. Equation for transform.

The objective is to evaluate the transform

\[\pi(\alpha,\beta\,|\,E_{k,\omega})\,:\!=\, \int_0^\infty \mathrm{e}^{-\alpha u} \,{{\mathfrak p}_\beta(u\,|\,E_{k,\omega})}\,\textrm{ d}u.\]

To this end, first note that

\begin{equation*}{{\mathfrak p}_\beta(u\,|\,E_{k,\omega})}=\left(\frac{\omega}{\beta+\omega}\right)^k\Big(\,{\mathrm{P}}(Z_k> u)+\int_{-\infty}^u {\mathrm{P}}(Z_k\in\mathrm{d}v)\,{{\mathfrak p}_\beta(u-v\,|\,E_{k,\omega})}\,\Big).\end{equation*}

In order to evaluate the associated transform, we multiply the entire equation by $\mathrm{e}^{-\alpha u}$ and integrate over u, so as to obtain

(10) \begin{align}&{\pi}(\alpha,\beta\,|\,E_{k,\omega})\nonumber\\&\quad=\Big(\frac{\omega}{\beta+\omega}\Big)^k\Bigg(\underbrace{\int_0^\infty \mathrm{e}^{-\alpha u}\,\mathrm{P}(Z_k>u)\mathop{}\!\mathrm{d} u}_{\text{(i)}}+\underbrace{\int_{u=0}^\infty \mathrm{e}^{-\alpha u}\int_{v=-\infty}^u\mathrm{P}(Z_k\in\mathop{}\!\mathrm{d} v)\,{{\mathfrak p}_\beta(u-v\,|\,E_{k,\omega})}\mathop{}\!\mathrm{d} u}_{\text{(ii)}}\Bigg).\end{align}

As before, we deal with the terms (i) and (ii) separately. Lemma 2 below, proven in Appendix A, provides an expression for (i). Define, with $\xi^{(m)}(\alpha,\beta)$ denoting the mth derivative of $\xi(\alpha,\beta)$ with respect to $\alpha$ ,

(11) \begin{align}&I(\alpha,\beta,\omega)\nonumber\\&\qquad\,:\!=\,\:\left(\frac{\theta}{\theta-\alpha}\right)^k\frac{1-\xi_k(\alpha,\beta+\omega)}{\alpha}-\frac{1}{\theta}\sum_{n=0}^{k-1}\left(\frac{\theta}{\theta-\alpha}\right)^{k-n}\left(1-\sum_{m=0}^n\xi_k^{(m)}(\theta,\beta+\omega)\frac{(-\theta)^m}{m!}\right).\end{align}

Lemma 2. For any $\alpha\geqslant 0$ ,

\begin{align*}\int_0^\infty \mathrm{e}^{-\alpha u}\,{\mathrm{P}}(Z_k>u)\,\mathrm{d}u=&\:I(\alpha,\beta, \omega).\end{align*}

The right-hand side of (11) can be further simplified by swapping the order of summation and evaluating the finite geometric series, but we leave this computation out.

We now consider integral (ii). To this end, define

\[\delta_{n,k}\,:\!=\, \frac{(\!-\theta)^n}{n!}\xi_k^{(n)}(\theta,\beta+\omega).\]

Also,

\begin{align*}J(\alpha,\beta,\omega)&\,:\!=\,\sum_{n=0}^{k-1}\delta_{k-1-n,k}\left(\frac{\theta}{\theta-\alpha}\right)^{n+1}\sum_{m=0}^n\frac{(\alpha-\theta)^m}{m!}\pi^{(m)}(\theta,\beta\,|\,E_{k,\omega})=\sum_{i=0}^{k-1} \bar J_i(\alpha,\beta,\omega)\,z_i,\end{align*}

where (by swapping the order of the two sums) $J_i(\alpha,\beta,\omega)$ and $z_i\equiv z_i(\beta,\omega)$ are

(12) \begin{equation} \bar J_i(\alpha,\beta,\omega)\,:\!=\, \sum_{n=i}^{k-1} \delta_{k-1-n,k}\left(\frac{\theta}{\theta-\alpha}\right)^{n+1}\frac{(\alpha-\theta)^i}{i!},\qquad z_i\,:\!=\, \pi^{(i)}(\theta,\beta\,|\,E_{k,\omega}).\end{equation}

Observe that $\bar J_i(\alpha,\beta,\omega)$ is explicitly known (in terms of the model primitives, that is), whereas the k coefficients $z_0,\ldots,z_{k-1}$ are yet to be determined. The following lemma is proven in Appendix A.

Lemma 3. For any $\alpha\geqslant 0$ ,

\begin{align}&\int_{u=0}^\infty \mathrm{e}^{-\alpha u}\int_{v=-\infty}^u\mathrm{P}(Z_k\in\mathop{}\!\mathrm{d} v)\,{{\mathfrak p}_\beta(u-v\,|\,E_{k,\omega})}\mathop{}\!\mathrm{d} u\nonumber\\&\qquad=-J(\alpha,\beta,\omega)+\Big(\frac{\theta}{\theta-\alpha}\Big)^k\xi_k(\alpha,\beta+\omega)\,\pi(\alpha,\beta\,|\,E_{k,\omega}).\nonumber\end{align}

We now use the results obtained above to derive an expression for the transform $\pi(\alpha,\beta\,|\,E_{k,\omega})$ . Adding up the expressions that we have found in Lemmas 2 and 3,

(13) \begin{align}&\pi(\alpha,\beta\,|\,E_{k,\omega})\nonumber\\&\qquad=\Big(\frac{\omega}{\beta+\omega}\Big)^k\bigg(&I(\alpha,\beta,\omega)-J(\alpha,\beta,\omega)+\Big(\frac{\theta}{\theta-\alpha}\Big)^k\xi_k(\alpha,\beta+\omega)\,\pi(\alpha,\beta\,|\,E_{k,\omega})\bigg).\end{align}

Noting that

\[\left(\frac{\theta}{\theta-\alpha}\right)^k \xi_k(\alpha,\beta+\omega)=\left(\frac{\beta+\omega}{\beta+\omega-\varphi (\alpha)}\right)^k,\]

we arrive at the following result.

Proposition 3. For any $\alpha\geqslant 0$ and $\beta>0$ ,

(14) \begin{equation}\pi(\alpha,\beta\,|\,E_{k,\omega})\left(1-\left(\frac{\omega}{\beta+\omega-\varphi(\alpha)}\right)^k\right) =\Big(\frac{\omega}{\beta+\omega}\Big)^k\big(I(\alpha,\beta,\omega)-J(\alpha,\beta,\omega)\big).\end{equation}

4.1.2. Determination of the unknown constants.

We observe that in Proposition 3 the right-hand side of (14) contains the object $J(\alpha,\beta,\omega)$ , which contains the unknown quantities $z_0,\ldots,z_{k-1}$ . The aim of the remainder of this subsection is to identify these unknown quantities, leading to an expression for our object of study, i.e. ${\pi}(\alpha,\beta\,|\,E_{k,\omega})$ . The following lemma is proven in Appendix A.

Lemma 4. For any $\beta,\,\omega$ and k, the equation

(15) \begin{equation}\left(\frac{\omega}{\beta+\omega-\varphi(\alpha)}\right)^k=1\end{equation}

has k roots in the right half of the complex plane, say, $\alpha_1\equiv\alpha_1(\beta,{\omega}),\ldots,\alpha_k\equiv \alpha_k(\beta,{\omega}).$

The unknowns $z_i$ can be found relying on Lemma 4, once more using the fact that any root of the left-hand side (in the right-half of the complex $\alpha$ -plane) of (14) in Proposition 3 is also a root of the right-hand side. It entails that we have to solve the linear system

(16) \begin{equation}I(\alpha_j,\beta,\omega) = \sum_{i=0}^{k-1} \bar J_i(\alpha_j,\beta,\omega)\,z_i\end{equation}

for $j=1,\ldots,k.$ Below we write $J(\alpha,\beta,\omega)\,:\!=\,\sum_{i=0}^{k-1} \bar J_i(\alpha,\beta,\omega)\,z_i.$

Theorem 3. For any $\alpha\geqslant 0$ and $\beta>0$ ,

\begin{align*}{\pi}(\alpha,\beta\,|\,E_{k,\omega})= \frac{({\omega}/({\beta+\omega}))^k\big(I(\alpha,\beta,\omega)-J(\alpha,\beta,\omega)\big)}{1-({{\omega}/({\beta+\omega-\varphi(\alpha)})})^k},\end{align*}

where the vector ${\boldsymbol z}$ follows from the linear equations (16).

Remark 3. The entries of ${\boldsymbol z}$ are real, as can be seen as follows.

1. (i) First observe that the roots $\alpha_j$ solve the equation, with $\textrm{i}$ denoting the imaginary unit,

   (17) \begin{equation}\varphi(\alpha) =\omega -\omega \,\mathrm{e}^{2\pi \textrm{i}j/k} +\beta,\end{equation}
   with $j=0,\ldots,k-1.$ This means that any $\varphi(\alpha_j)$ is either real or the complex conjugate of $\varphi(\alpha_{j'})$ for $j\not=j'$ .

2. (ii) Noting that, for $v,w\in{\mathbb R}$ ,

   \[\varphi(v+w\textrm{i})=\log {\mathrm{E}}\mathrm{e}^{-(v+w\textrm{i})Y(1)} = \log{\mathrm{E}} \left[\mathrm{e}^{-vY(1)}(\cos(\!-wY(1))+\textrm{ i}\sin(\!-wY(1))\right],\]
   it requires a straightforward verification to conclude that if $\alpha_j=v+w\textrm{i}$ solves 17 then so does its complex conjugate $\alpha_{j'}=v-w\textrm{i}$ . Hence, the roots of 15 are either real or belong to a pair of complex conjugates.

3. (iii) Consider row j and j’ in the linear system (16), with $\alpha_j$ and $\alpha_{j'}$ being complex conjugates. From their definitions it can be verified that $I(\alpha_j,\beta,\omega)$ is the complex conjugate of $I(\alpha_{j'},\beta,\omega)$ and that $\bar J_i(\alpha_j,\beta,\omega)$ is the complex conjugate of $\bar J_i(\alpha_{j'},\beta,\omega)$ . Now replace rows j and j’ by one row that follows by adding the two rows up, and a second row that follows by subtracting one row from the other, and dividing by $\textrm{i}$ . All coefficients in the new linear system have become real, and hence, the solution ${\boldsymbol z}$ is real.

Remark 4. In the case that $Y(\cdot)$ is of compound Poisson type, an insightful decomposition applies, earlier identified in [Reference Albrecher, Cheung and Thonhauser2, Equation (3.9)]. In the first place, $Z_k^+$ is distributed as the sum of k i.i.d. random variables, each of them characterized by the transform $\xi(\alpha,\beta+\omega)$ . Realizing that this $\xi(\alpha,\beta+\omega)$ can be interpreted in terms of a geometric sum of ladder heights, as can be found in, for instance, [Reference Mandjes and Boxma31, Section 1.4], $(\xi(\alpha,\beta+\omega))^k$ represents a negative binomial number of such ladder heights. In the second place, $Z^-_k$ is distributed as the sum of k independent exponential random variables with parameter $\psi(\beta+\omega)$ , and, consequently, has an Erlang distribution (and is independent of $Z_k^+$ ).

4.2. Hyper-Erlang distributed inter-inspection times

We consider the bankruptcy model with hyper-Erlang-d inter-inspection times, with a common scale parameter. With probability $p_i$ the inter-inspection time is Erlang- $k_i$ distributed (for some $k_i\in{\mathbb N}$ ), with scale parameter $\omega>0$ , for $i\in\{1,\ldots,d\}$ and $\sum_{i=1}^d p_i=1$ . This means that a generic inter-inspection time $E_{\,{\boldsymbol k},{\boldsymbol p},\omega}$ has the density, with ${\boldsymbol k}=(k_1,\ldots, k_d)$ and ${\boldsymbol p}=(p_1,\ldots,p_d)$ ,

\[\mathrm{P}(E_{\,{\boldsymbol k},{\boldsymbol p},\omega}\in\mathop{}\!\mathrm{d} t)=\sum_{i=1}^d p_i \,\omega\frac{(\omega t)^{k_i-1}}{(k_i-1)!}\mathrm{e}^{-\omega t}\,\textrm{ d}t.\]

Without loss of generality, we assume that the $k_i$ are put in increasing order.

4.2.1. Equation for transform.

Based on the decomposition that we developed in (13), corresponding to the transform $\pi(\alpha,\beta\,|\,E_{k,\omega})$ in the case of Erlang distributed inter-inspection times, we now obtain

(18) \begin{align}&\pi(\alpha,\beta\,|\,E_{\,{\boldsymbol k},{\boldsymbol p},\omega})\nonumber\\&\quad=\sum_{i=1}^d p_i\Big(\frac{\omega}{\beta+\omega}\Big)^{k_i}\bigg(I_i(\alpha,\beta,\omega)-J_i(\alpha,\beta,\omega)+\Big(\frac{\theta}{\theta-\alpha}\Big)^{k_i}\xi_{k_i}(\alpha,\beta+\omega)\,\pi(\alpha,\beta\,|\,E_{\,{\boldsymbol k},{\boldsymbol p},\omega})\bigg),\end{align}

with

\begin{gather*}\begin{aligned}I_i(\alpha,\beta,\omega)\,:\!=\,&\left(\frac{\theta}{\theta-\alpha}\right)^{k_i}\frac{1-\xi_{k_i}(\alpha,\beta+\omega)}{\alpha}\\&-\frac{1}{\theta}\sum_{n=0}^{{k_i}-1}\left(\frac{\theta}{\theta-\alpha}\right)^{{k_i}-n}\left(1-\sum_{m=0}^n\xi_{k_i}^{(m)}(\theta,\beta+\omega)\frac{(\!-\theta)^m}{m!}\right),\end{aligned}\\\begin{aligned}J_i(\alpha,\beta,\omega)&\,:\!=\,\sum_{n=0}^{k_{i}-1}\delta_{{k_i}-1-n,k_i}\left(\frac{\theta}{\theta-\alpha}\right)^{n+1} \sum_{m=0}^n\frac{(\alpha-\theta)^m}{m!}\pi^{(m)}(\theta,\beta\,|\,E_{\,{\boldsymbol k},{\boldsymbol p},\omega})\\&=\sum_{j=0}^{k_{i}-1}\bar{J}_{i,j}(\alpha,\beta,\omega)\,z_j,\end{aligned}\end{gather*}

where, again by swapping the order of the two sums,

\begin{align*}\bar{J}_{i,j}(\alpha,\beta,\omega)\,:\!=\,\sum_{n=j}^{k_{i}-1}\delta_{k_{i}-1-n,k_i}\left(\frac{\theta}{\theta-\alpha}\right)^{n+1}\frac{(\alpha-\theta)^j}{j!}\end{align*}

and $z_j\equiv z_j\,(\beta,\omega)\,:\!=\,\pi^{(j)}(\theta,\beta\,|\,E_{\,{\boldsymbol k},{\boldsymbol p},\omega}).$ Observe that $\bar J_{i,j}(\alpha,\beta,\omega)$ is explicitly known, whereas the coefficients $z_0,\ldots,z_{k_{d}-1}$ are yet to be determined. As before, the next step is to isolate the object $\pi(\alpha,\beta\,|\,E_{\,{\boldsymbol k},{\boldsymbol p},\omega})$ in (18). We thus end up with the following characterization of our target quantity.

Proposition 4. For any $\alpha\geqslant 0$ and $\beta>0$ ,

(19) \begin{align}&\pi(\alpha,\beta\,|\,E_{\,{\boldsymbol k},{\boldsymbol p},\omega})\Bigg(1-\sum_{i=1}^dp_i\Big(\frac{\omega}{\beta+\omega-\varphi(\alpha)}\Big)^{k_i}\Bigg)\nonumber\\&\qquad=\sum_{i=1}^dp_i\Big(\frac{\omega}{\beta+\omega}\Big)^{k_i}\big(I_i(\alpha,\beta,\omega)-J_i(\alpha,\beta,\omega)\big).\end{align}

4.2.2. Determination of the unknown constants.

On the right-hand side of (19) in Proposition 4 we have that the object $I_i(\alpha,\beta,\omega)$ is explicitly expressed in terms of the model primitives, and so is $\bar J_i(\alpha,\beta,\omega)$ . The coefficients $z_0,\ldots,z_{k_{d}-1}$ however are still unknown. The aim of the remainder of this subsection is to identify these unknown quantities. The following lemma, proven in Appendix A, plays a crucial role.

Lemma 5. For any $\beta,\,\omega, {\boldsymbol p}$ and ${\boldsymbol k}$ , the equation

(20) \begin{equation}\sum_{i=1}^d p_i\Big(\frac{\omega}{\beta+\omega-\varphi(\alpha)}\Big)^{k_i}=1\end{equation}

has $k_d$ roots in the right half of the complex plane, say, $\alpha_1\equiv\alpha_1(\beta,\boldsymbol{\omega}),\ldots,\alpha_{k_d}\equiv\alpha_{k_d}(\beta,\boldsymbol{\omega}).$

The unknowns $z_1,\ldots,z_{k_d}$ can be found using Lemma 5 in combination with the property that any root (in the right half of the complex $\alpha$ plane) of the left-hand side of (19) in Proposition 4, is necessarily a root of the right-hand side as well. This reasoning leads to the linear system

(21) \begin{equation}\sum_{i=1}^d p_i\Big(\frac{\omega}{\beta+\omega}\Big)^{k_i}I_i(\alpha_\ell,\beta,\omega)=\sum_{i=1}^d p_i\Big(\frac{\omega}{\beta+\omega}\Big)^{k_i}\sum_{j=0}^{k_{i}-1}\bar{J}_{i,j}(\alpha_\ell,\beta,\omega)\,z_j\end{equation}

for $\ell=1,\ldots,k_d$ . As in Remark 3, the entries of ${\boldsymbol z}$ are real. Defining

\[J_i(\alpha,\beta,\omega)\,:\!=\,\sum_{j=0}^{k_{i}-1}\bar{J}_{i,j}(\alpha,\beta,\omega)\,z_j,\]

we have found the following result.

Theorem 4. For any $\alpha\geqslant 0$ and $\beta>0$ ,

\begin{equation*}\pi(\alpha,\beta\,|\,E_{\,{\boldsymbol k},{\boldsymbol p},\omega})=\frac{\displaystyle\sum_{i=1}^d p_i({\omega}/({\beta+\omega}))^{k_i}(I_i(\alpha,\beta,\omega)-{J}_{i}(\alpha,\beta,\omega))}{\displaystyle 1-\sum_{i=1}^d p_i({\omega}/({\beta+\omega-\varphi(\alpha))})^{k_i}},\end{equation*}

where the vector $\boldsymbol z$ follows from the linear equations (21).

5.1. Approximating general distributions, two-moments fit

The inter-inspection time distributions we have dealt with in the previous sections are all contained in the wider class of phase-type distributions. Informally, a phase-type random variable has a distribution that is constructed by taking convolutions and/or mixtures of exponential distributions. Phase-type distributions owe their popularity to the fact that they are capable of approximating any distribution (on the positive half-line, that is) arbitrarily closely [Reference Asmussen8, Theorem III.4.2], while still allowing (semi-)explicit evaluation.

A phase-type random variable is uniquely characterized as the time until absorption of a finite-state continuous-time Markov chain with a single absorbing state. Following the approach presented in [Reference Asmussen8, Section III.4], it is encoded by an initial distribution ${\boldsymbol a}\in{\mathbb R}^m$ and a substochastic $m\times m$ transition rate matrix S, where ‘substochastic’ means that in at least one row the sum of the nondiagonal elements exceeds the diagonal element. For instance, the Erlang distribution with k phases is represented by $a_1=1$ , $S_{ii}=-\omega$ for $i =1,\ldots,k$ and $S_{i,i+1}=\omega$ for $i=1,\ldots,k-1$ (while all other entries of the k-dimensional vector ${\boldsymbol a}$ and of the $k\times k$ matrix S are set equal to 0).

It is noted that the proof of [Reference Asmussen8, Theorem III.4.2] reveals that actually any distribution on the positive half-line can be approximated arbitrarily closely by elements from a specific subclass of phase-type distributions. This smaller class are the mixtures of Erlang distributions, each of them having the same scale parameter, i.e. the class of distributions dealt with in Section 4.2. The proof also indicates that the drawback of working with this specific class of phase-type distributions is that one may need a large dimension m to get an accurate fit. That is, the reason why a two-moment fit has been proposed in which a relatively low value of m suffices. In the one presented in [Reference Tijms33], for distributions with a coefficient of variation less than 1, a mixture of Erlang random variables (with the same scale parameter) is used and, for distributions with a coefficient of variation larger than 1, a hyperexponential random variable is used, all of them of relatively low dimension; this fitting method is compactly summarized in, e.g. [Reference Kuiper27, Section 3.1]. This two-moment fit renders the inter-inspection time distributions covered in Sections 3 and 4 particularly useful.

5.2. Limitations

In the previous sections we succeeded in identifying the transform $\pi(\alpha,\beta\,|\,\Omega)$ for (hyper-)exponential and (hyper-)Erlang inter-inspection times, both belonging to the family of phase-type distributions. This suggests that the analysis may extend to the full class of phase-type distributions. Careful inspection however reveals that such an extension is challenging.

To explain what complications arise, let us analyze the distribution of $Z=Y(\Omega)$ if the inter-inspection times $\Omega$ are of phase type. Let $K_j$ for $j=1,\ldots,m$ be the number of visits of the associated underlying Markov chain to state j before absorption. Then $Z=Z^+-Z^-$ , where, in self-evident notation,

\[Z^+\,:\!=\, \sum_{j=1}^m \sum_{i=1}^{K_j} Z_{ji}^+,\qquad Z^-\,:\!=\,\sum_{j=1}^m \sum_{i=1}^{K_j}Z_{ji}^{-}.\]

While all pair of random variables $Z_{ji}^+$ and $Z_{ji}^{-}$ are independent, for obvious reasons,

\[\sum_{i=1}^{K_j}Z_{ji}^+\quad\text{and}\quad\sum_{i=1}^{K_j}Z_{ji}^-\]

are not. As a consequence, for general phase-type $\Omega$ , it is not clear how to evaluate objects like

\[\int_0^\infty \mathrm{e}^{-\alpha u}\ \mathrm{P}(Z > u)\mathop{}\!\mathrm{d} u\quad\text{and}\quad{\mathrm{P}}(Z < v\,|\,Z < 0),\]

which were the key quantities in the computations of the previous sections.

As is readily verified, the reason why the computations in Sections 3 and 4 worked out is that, for all of the inter-inspection time distributions considered, the random variables $K_j$ are either 0 or 1, so that $Z^+$ and $Z^-$ are independent. A sufficient condition for this property to hold is that the substochastic transition rate matrix S does not allow cycles (so that every state is visited at most once), a condition that is met for (hyper-)exponential and (hyper-)Erlang inter-inspection times; see also Appendix B.

6.1. Change of measure

We start by introducing some useful notation. As before, we consider the net cumulative claim process Y(t) at the inspection epochs $\bar\Omega_1,\bar\Omega_2, \ldots$ . We introduce the increments of Y(t) between two consecutive inspections, i.e. $Z_n \,:\!=\, Y(\bar\Omega_n) -Y(\bar\Omega_{n-1})$ . Note that $(Z_n)_n$ is a sequence with i.i.d. random variables, say, distributed as the generic random variable Z. As a consequence, we can rewrite our target probability as

\[{{\mathfrak p}(u\,|\,\Omega)}={\mathrm{P}}(T(u)<\infty) = {\mathrm{P}}(\tau(u)<\infty),\]

where

\begin{align*}T(u) &\,:\!=\, \inf \left\{n\in{\mathbb N}: \sum_{i = 1}^n Z_i >u\right\}, \qquad \tau(u) \,:\!=\, \inf_{n\in{\mathbb N}:Y(\bar\Omega_n)> u } \bar\Omega_n,\end{align*}

so that $Y(\tau(u)) = \sum_{i=1}^{T(u)}Z_i.$ In this section we assume that the net profit condition is in place, i.e. ${\mathrm{E}}\,Z<0$ or equivalently $\varphi'(0)>0$ , so that bankruptcy becomes increasingly rare as $u\to\infty.$

To find a convenient expression for ${{\mathfrak p}(u\,|\,\Omega)}$ , we work with an alternative measure $\mathbb{Q}$ . We assume that we are in the light-tailed regime, in that there is a $\theta^\star>0$ such that ${\mathrm{E} yy}\,\mathrm{e}^{\theta^\star Z}=1$ . Then the probability measure $\mathbb{Q}$ is defined through the relation

(22) \begin{equation}\mathbb{Q}(Z\in \mathrm{d}x) = \mathrm{P}(Z\in \mathrm{d}x) \,\mathrm{e}^{\theta^\star x},\end{equation}

which integrates to 1 by the definition of $\theta^\star$ . Note that

\begin{align*}{\mathrm{E} y}\,\mathrm{e}^{\theta Z}&= \int_{0}^\infty \mathrm{P}(\Omega\in\mathrm{d}t) \,{\mathrm{E} y}\,\mathrm{e}^{\theta Y(t)}= \int_{0}^\infty\mathrm{P}(\Omega\in \mathrm{d}t) \,\mathrm{e}^{\varphi(-\theta)\,t}= {\mathrm{E}}\,\mathrm{e}^{\varphi(-\theta)\,\Omega}.\end{align*}

As $\Omega\geqslant 0$ , to obtain ${\mathrm{E} yy}\,\mathrm{e}^{\theta^\star Z}=1$ , one should necessarily have $\varphi(\!-\theta^\star) = 0$ , implying that $\mathbb{Q}$ is not affected by the distribution of the inter-inspection times $\Omega$ . Note that due to the convexity of $\varphi(\cdot)$ and $\varphi'(0)>0$ , we have $\varphi'(\!-\theta^\star)<0$ ; throughout we assume that $|\varphi'(\!-\theta^\star)|<\infty.$ Under the measure ${\mathbb Q}$ , the Laplace exponent of the driving Lévy process becomes $\varphi_{\mathbb Q}(\alpha)\,:\!=\,\varphi(\alpha-\theta^\star)$ ; in the following $\psi_{\mathbb Q}(\cdot)$ denotes the right inverse of $\varphi_{\mathbb Q}(\cdot)$ .

Let $\ell(x)$ denote the likelihood ratio $\mathrm{d}\mathrm{P}/\mathrm{d}{\mathbb Q}$ evaluated in the argument x. Then we have the fundamental identity [Reference Asmussen8, Section XIII.3], describing the ‘translation’ from sampling under the alternative measure $\mathbb{Q}$ to sampling under the actual measure $\mathrm{P}$ :

\[{\mathrm{P}}(T(u)<\infty) = {\mathrm{E} y}\,1\{T(u)<\infty\} = {\mathrm{E}}_{\mathbb{Q}}\big[1\{T(u)<\infty\}\,\ell(Z_1)\cdots\ell(Z_{T(u)})\big],\]

where we also use the independence of the random variables $Z_i$ .

It is readily verified that

\[\frac{\mathrm{d}}{\mathrm{d}\theta}{\mathrm{E} y}_{\mathbb{Q}}| \mathrm{e}^{\theta Z} =-\varphi'(\!-\theta-\theta^\star)\,{\mathrm{E} y}\,\big[\Omega\,\mathrm{e}^{\varphi(-\theta-\theta^\star)\Omega}\big],\]

so that

\[{\mathrm{E}}_{\mathbb{Q}} Z = \left.\frac{\mathrm{d}}{\mathrm{d}\theta}{\mathrm{E}}_{\mathbb{Q}} \,\mathrm{e}^{\theta Z}\right|_{\theta=0}=-\varphi'(\!-\theta^\star)\,{\mathrm{E}}\,\big[\Omega\,\mathrm{e}^{\varphi(-\theta^\star)\Omega}\big] >0.\]

This means that $T(u)<\infty$ almost surely under ${\mathbb Q}$ (i.e. bankruptcy is not rare under ${\mathbb Q}$ ), entailing that

\[{\mathrm{P}}(T(u)<\infty) = {\mathrm{E}}_{\mathbb{Q}}\big[\ell(Z_1)\cdots\ell(Z_{T(u)})\big].\]

Due to the fact that $\ell(Z_i) = \mathrm{e}^{-\theta^\star Z_i}$ , upon combining the above, we thus find that

\[{{\mathfrak p}(u\,|\,\Omega)} = {\mathrm{E}}_{\mathbb{Q}}\big[\ell(Z_1)\cdots\ell(Z_{T(u)})\big] = {\mathrm{E}}_{\mathbb{Q}}\exp\left(-\theta^\star\sum_{i=1}^{T(u)} Z_i\right) = {\mathrm{E}}_{\mathbb{Q}}\,\mathrm{e}^{-\theta^\star Y(\tau(u))}.\]

We have arrived at the following result.

Theorem 5. For any $u\geqslant 0$ ,

\[{{\mathfrak p}(u\,|\,\Omega)} = {\mathrm{E}}_{\mathbb Q}\, \mathrm{e}^{-\theta^\star Y(\tau(u))}.\]

A consequence of this result is the following uniform bound on ${{\mathfrak p}(u\,|\,\Omega)}$ , which can be seen as a Lundberg-type inequality for our model with i.i.d. inter-inspection times. It follows from the obvious property $Y(\tau(u))> u$ . Remarkably, this upper bound does not depend on the inter-inspection time distribution (and will therefore be a loose bound in case of infrequent inspections).

Corollary 1. For any $u\geqslant 0$ , irrespective of the inter-inspection time $\Omega$ ,

\[{{\mathfrak p}(u\,|\,\Omega)}\leqslant \mathrm{e}^{-\theta^\star u}.\]

6.2. Exact asymptotics

In the remainder of this section our objective is to use Theorem 5 to identify the exact asymptotics of ${{\mathfrak p}(u\,|\,\Omega)}$ . More concretely, we aim to identify the constant $\gamma[\Omega]$ featuring in

(23) \begin{equation}\lim_{u\to\infty}{{\mathfrak p}(u\,|\,\Omega)}\, \mathrm{e}^{\theta^\star u}\,=\!:\, \gamma[\Omega]\end{equation}

for the inter-inspection times $\Omega$ studied in Sections 3 and 4. In view of Theorem 5, this means that, with R(u) denoting the overshoot $Y(\tau(u))-u$ , we wish to evaluate the object

\[\gamma[\Omega] = \lim_{u\to\infty} {\mathrm{E}}_{\mathbb{Q}} \mathrm{e}^{-\theta^\star R(u)}.\]

Following [Reference van Kreveld, Mandjes and Dorsman25], the idea is to evaluate this quantity using, with $U_\alpha$ denoting an exponentially distributed random variable with parameter $\alpha$ (i.e. with mean $\alpha^{-1}$ , having probability density function $\alpha \mathrm{e}^{-\alpha u}$ ),

(24) \begin{equation}\gamma[\Omega] = \lim_{\alpha\downarrow 0} {\mathrm{E}}_{\mathbb{Q}}\mathrm{e}^{-\theta^\star R(U_\alpha)}= \lim_{\alpha\downarrow0}\alpha\,\pi^\circ(\alpha\,|\,\Omega),\end{equation}

where the transform

\[\pi^\circ(\alpha\,|\,\Omega) \,:\!=\, \int_0^\infty \mathrm{e}^{-\alpha u}\,{\mathrm{E}}_{\mathbb{Q}} \mathrm{e}^{-\theta^\star R(u)}\,\mathrm{d}u\]

will be evaluated essentially relying on the techniques developed in Sections 3 and 4. In this derivation we heavily use the ‘master equation’, with Z distributed as $Y(T_\omega)$ ,

(25) \begin{align}\pi^\circ(\alpha\,|\,\Omega) =\:& \int_0^\infty \mathrm{e}^{-\alpha u} \int_u^\infty{\mathbb Q}(Z\in\textrm{ d}v)\,\mathrm{e}^{-\theta^\star(v-u)}\,\mathrm{d}u\notag\\&+\int_0^\infty \mathrm{e}^{-\alpha u} \int_{-\infty}^u {\mathbb Q}(Z\in\mathrm{d}v)\,{\mathrm{E}}_{\mathbb{Q}} \mathrm{e}^{-\theta^\star R(u-v)}\,\textrm{ d}u,\end{align}

where the first term corresponds to the scenario of the first Z directly exceeding level u, and the second term to the scenario that the first Z is still below u; cf. (3).

6.2.1. Exponentially distributed inter-inspection times.

Our first goal is to evaluate $\pi^\circ(\alpha\,|\,\Omega)$ for exponentially distributed inter-inspection times with parameter $\omega$ , essentially following the approach that we developed in Section 3.1. The underlying idea is to separate Z into the independent parts $Z^+\,:\!=\, \bar Y(T_\omega)$ and $Z^-\,:\!=\,-\underline Y(T_\omega)$ , the latter random variable being exponentially distributed with parameter $\theta\,:\!=\, \psi_{\mathbb Q}(\omega)$ . As a consequence, the first term on the right-hand side of (25) can be rewritten as

\begin{align*}&\int_0^\infty \mathrm{e}^{-\alpha u} \int_u^\infty {\mathbb Q}(Z\in\mathrm{d}v)\,\mathrm{e}^{-\theta^\star(v-u)}\,\mathrm{d}u\\&\qquad=\int_{u=0}^\infty \mathrm{e}^{-\alpha u} \int_{v=u}^\infty \int_{w=0}^\infty\theta \mathrm{e}^{-\theta w}\,{\mathbb Q}(Z^+-w\in\textrm{d}v)\,\mathrm{e}^{-\theta^*(v-u)}\,\mathrm{d}w\,\mathrm{d}u\\&\qquad=\int_{u=0}^\infty \mathrm{e}^{-\alpha u} \int_{v=u}^\infty \int_{x=v}^\infty\theta \mathrm{e}^{-\theta (x-v)}\,{\mathbb Q}(Z^+\in\textrm{d}x)\,\mathrm{e}^{-\theta^*(v-u)}\,\mathrm{d}v\,\mathrm{d}u,\end{align*}

where in the last equality the change of variable $x\,:\!=\, v+w$ has been performed. Swapping the inner two integrals and computing the integral over v yields that the expression in the previous display reads

\[\int_{u=0}^\infty \mathrm{e}^{-\alpha u} \int_{x=u}^\infty\frac{\theta}{\theta^\star-\theta}\big(\mathrm{e}^{-\theta(x-u)}-\mathrm{e}^{-\theta^\star(x-u)}\big) \,{\mathbb Q}(Z^+\in\mathrm{d}x)\mathop{}\!\mathrm{d} u.\]

After interchanging the two remaining integrals and integrating over u, we readily obtain

\[\frac{\theta}{\theta^\star-\theta}\int_0^\infty\left(\frac{\mathrm{e}^{-\theta x}-\mathrm{e}^{-\alpha x}}{\alpha-\theta}-\frac{\mathrm{e}^{-\theta^\star x}-\mathrm{e}^{-\alpha x}}{\alpha-\theta^\star}\right)\,{\mathbb Q}(Z^+\in\mathrm{d}x).\]

We thus end up with the first integral on the right-hand side of (25) equalling

\[\int_0^\infty \mathrm{e}^{-\alpha u} \int_u^\infty {\mathbb Q}(Z\in\mathrm{d}v)\,\mathrm{e}^{-\theta^\star(v-u)}\,\textrm{ d}u=F^\circ(\alpha,\omega),\]

where, recalling that $\theta=\psi_{\mathbb Q}(\omega)$ ,

(26) \begin{equation}F^\circ(\alpha,\omega)\,:\!=\, \frac{\theta}{\theta^\star-\theta}\left(\frac{\xi_{\mathbb Q}(\theta,\omega)-\xi_{\mathbb Q}(\alpha,\omega)}{\alpha-\theta}- \frac{\xi_{\mathbb Q}(\theta^\star,\omega)-\xi_{\mathbb Q}(\alpha,\omega)}{\alpha-\theta^\star}\right)\end{equation}

and, using the fact that $\psi_{\mathbb Q}(\beta)=\psi(\beta)+\theta^\star$ for any $\beta\geqslant 0$ ,

\begin{align*}\xi_{\mathbb Q}(\alpha,\beta)&\,:\!=\, {\mathrm{E}}_{\mathbb Q}\mathrm{e}^{-\alpha\bar Y(T_\beta)}= {\mathrm{E}}_{\mathbb Q}\,\mathrm{e}^{-\alpha Z^+}=\frac{\alpha-\psi_{\mathbb Q}(\beta)}{\varphi_{\mathbb Q}(\alpha)-\beta}\frac{\beta}{\psi_{\mathbb Q}(\beta)}=\frac{\psi(\beta)+\theta^\star-\alpha}{\psi(\beta)+\theta^\star}\frac{\beta}{\beta-\varphi(\alpha-\theta^\star)};\end{align*}

cf. (1). Regarding the second integral on the right-hand side of (25), the procedure mimics step by step the one developed in Section 3.1, thus leading to

\[\frac{\theta}{\theta-\alpha}\ \big({\pi}^\circ(\alpha\,|\,T_\omega)\,\xi_{\mathbb Q}(\alpha,\omega) -{\pi}^\circ(\theta\,|\,T_\omega)\,\xi_{\mathbb Q}(\theta,\omega)\big).\]

Combining the above, we obtain the equation

\[{\pi}^\circ(\alpha\,|\,T_\omega)\left(1-\frac{\theta}{\theta-\alpha}\,\xi_{\mathbb Q}(\alpha,\omega)\right)=F^\circ(\alpha,\omega)-G^\circ(\alpha,\omega),\quad G^\circ(\alpha,\omega) \,:\!=\, {\theta}\,\xi_{\mathbb Q}(\theta,\omega)\frac {{\pi}^\circ(\theta\,|\,T_\omega)}{\theta-\alpha}.\]

We are left with determining the unknown constant ${\pi}^\circ(\theta\,|\,T_\omega)$ . To this end, it is first observed that $\alpha = \theta^\star$ is a root of the left-hand side, and hence, it is a root of the right-hand side as well. This leads to

\[{\pi}^\circ(\theta\,|\,T_\omega) = \frac{\theta-\theta^\star}{\theta}\frac{F^\circ(\theta^\star,\omega)}{\xi_{\mathbb Q}(\theta,\omega)}.\]

We conclude that

(27) \begin{equation}{\pi}^\circ(\alpha\,|\,T_\omega) =\frac{(\theta-\alpha)F^\circ(\alpha,\omega)-\theta\,{\pi}^\circ(\theta\,|\,T_\omega)\,\xi_{\mathbb Q}(\theta,\omega) }{\theta-\alpha-\theta\,\xi_{\mathbb Q}(\alpha,\omega)}.\end{equation}

Now that we have found the transform ${\pi}^\circ(\alpha\,|\,T_\omega)$ , we are in the position to identify the constant $\gamma[{T_\omega}]$ . Indeed, appealing to relation 24, we find as a direct application of L’Hôpital’s rule, that the expression for ${\pi}^\circ(\alpha\,|\,T_\omega)$ displayed in (27) leads to

\begin{align*}\gamma[{T_\omega}] &= \lim_{\alpha\downarrow 0} \alpha\,{\pi}^\circ(\alpha\,|\, T_\omega)= \lim_{\alpha\downarrow 0}\frac{\alpha\,\kappa}{{\theta-\alpha-\theta\,\xi_{\mathbb Q}(\alpha,\omega)}}=\frac{\kappa}{-\big(1+\theta f(0,\omega)\big)},\end{align*}

where

\begin{gather*}f(0,\omega)\,:\!=\, \left.\frac{\mathrm{d}}{\mathrm{d}\alpha} \xi_{\mathbb Q} (\alpha,\omega)\right|_{\alpha=0}=\frac{1}{\theta}\frac{\theta\,\varphi'_{\mathbb Q}(0)-\omega}{\omega},\\\kappa\,:\!=\, \theta\big( F^\circ(0,\omega)-\pi^\circ(\theta\,|\,T_\omega)\,\xi_{\mathbb Q}(\theta,\omega)\big)=\theta\,F^\circ(0,\omega)+(\theta^\star-\theta)\,F^\circ(\theta^\star,\omega).\end{gather*}

Observe that, using $\theta=\psi_{\mathbb Q}(\omega)=\psi(\omega)+\theta^\star$ ,

\[1+\theta f(0,\omega) =\frac{\theta\,\varphi_{\mathbb Q}'(0)}{\omega}.\]

Our next task is to write the constant $\kappa$ in a more explicit form. By computing $F^\circ(0,\omega)$ and $F^\circ(\theta^\star,\omega)$ , it is directly seen that

\begin{align*}\kappa =\frac{\theta}{\psi(\omega)}\Big(&\xi_{\mathbb Q}(\theta,\omega)-1-\frac{\theta}{\theta^\star}\left(\xi_{\mathbb Q}(\theta^\star,\omega)-1\right)-\xi_{\mathbb Q}(\theta,\omega)+\xi_{\mathbb Q}(\theta^\star,\omega)+\psi(\omega)\,f(\theta^\star,\omega)\Big),\end{align*}

where

\[\xi_{\mathbb Q}(\theta^\star,\omega)=\frac{\psi(\omega)}{\theta},\qquad f(\theta^\star,\omega)\,:\!=\, \left.\frac{\textrm{ d}}{\mathrm{d}\alpha}\xi_{\mathbb Q}(\alpha,\omega)\right|_{\alpha=\theta^\star}=\frac{1}{\theta}\frac{\psi(\omega)\,\varphi'(0)-\omega}{\omega}.\]

Putting the above results together, standard algebra yields that $\kappa=-\psi(\omega)\,\varphi'(0)/\omega$ . This leads us to the following conclusion, in line with [Reference Boxma and Mandjes14, Proposition 4.1].

Theorem 6. As $u\to\infty$ , we have ${{\mathfrak p}(u\,|\,T_\omega)}\, \mathrm{e}^{\theta^\star u}\to \gamma[{T_\omega}]$ , where

\[\gamma[{T_\omega}] =-\frac{\varphi'(0)}{\varphi'_{\mathbb Q}(0)}\frac{\psi(\omega)}{\theta}= -\frac{\varphi'(0)}{\varphi'_{\mathbb Q}(0)}\frac{\psi(\omega)}{\psi(\omega)+\theta^\star}.\]

As the inspection rate $\omega$ grows large, we recover the asymptotics under permanent inspection. Indeed, in agreement with the findings of [Reference Doney21], revealing the exact asymptotics for spectrally positive Lévy processes, we obtain $\gamma[{T_\omega}]\to-{\varphi'(0)}/{\varphi'_{\mathbb Q}(0)}$ as $\omega\to\infty$ .

6.2.2. Hyperexponentially distributed inter-inspection times.

Denoting as before $\theta_i\,:\!=\, \psi_{\mathbb Q}(\omega_i)$ , following the same approach as in the case of exponentially distributed inter-inspection times, we obtain the relation

\begin{align*}{\pi}^\circ(\alpha\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})&\left(1-\sum_{i=1}^d p_i \frac{\theta_i}{\theta_i-\alpha}\,\xi_{{\mathbb Q}}(\alpha,\omega_i) \right)=\sum_{i=1}^dp_i\left(F^\circ(\alpha,\omega_i)- G^\circ(\alpha,\omega_i)\right),\end{align*}

with $F^\circ(\cdot,\cdot)$ as defined in (26) with $\theta$ replaced by $\theta_i$ and

\[G^\circ(\alpha,\omega_i)\,:\!=\,{\theta_i}\,\xi_{{\mathbb Q}}(\theta_i,\omega_i) \frac{{\pi}^\circ(\theta_i\,|\,T_{{\boldsymbol\omega},{\boldsymbol p}})} {\theta_i-\alpha}.\]

Inserting the expression for $\xi_{{\mathbb Q}}(\alpha,\omega_i)$ , we thus obtain

(28) \begin{equation}{\pi}^\circ(\alpha\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})=\frac{\displaystyle \sum_{i=1}^d p_i(F^\circ(\alpha,\omega_i)-({\theta_i}/({\theta_i-\alpha})) \,\xi_{{\mathbb Q}}(\theta_i,\omega_i)\,z_i)}{\displaystyle 1-\sum_{i=1}^d p_i({\omega_i}/({\omega_i-\varphi(\alpha-\theta^\star)}))},\end{equation}

where the d unknowns $z_i\,:\!=\,{\pi}^\circ(\theta_i\,|\,T_{{\boldsymbol \omega},{\boldsymbol p}})$ are yet to be determined. The denominator on the right-hand side of 28 (and, hence, the numerator as well) has precisely d roots in the right half of the complex plane, say $\alpha_1,\ldots,\alpha_d$ . This follows by the same line of reasoning as in the proof of Lemma 1. We consider the same MAP, but now we have $\varphi_1(\alpha)=\ldots=\varphi_d(\alpha)=\varphi_{\mathbb{Q}}(\alpha)$ and $\boldsymbol{\eta}=0$ (i.e. no killing). From [Reference Ivanovs, Boxma and Mandjes23, Theorem 2] we know that, since we have positive drift under $\mathbb{Q}$ , the equation has exactly d roots in the right half of the complex plane. Applying the same reasoning as in Remark 2, the roots $\alpha_1,\ldots,\alpha_d$ are real-valued, and hence, the solution ${\boldsymbol z}$ of the linear system, with $j=1,\ldots,d$ ,

\[\sum_{i=1}^d p_iF^\circ(\alpha_j,\omega_i)=\sum_{i=1}^d p_i\frac{\theta_i}{\theta_i-\alpha_j}\,\xi_{{\mathbb Q}}(\theta_i,\omega_i)\,z_i\]

is real-valued as well. Defining

\[\Phi(\alpha)\,:\!=\, \prod_{i=1}^d (\omega_i-\varphi(\alpha-\theta^\star)), \qquad \Phi_i(\alpha)\,:\!=\,\frac{\Phi(\alpha)}{\omega_i-\varphi(\alpha-\theta^\star)},\]

and applying the same reasoning as in the exponential case, we have that the expression for $\pi^{\circ}(\alpha\,|\,T_{\omega,\boldsymbol{p}})$ displayed in (28) leads to

\[\gamma[T_{\omega,\boldsymbol{p}}]= \lim_{\alpha\downarrow 0} \alpha\,{\pi}^\circ(\alpha\,|\,T_{\omega,\boldsymbol{p}})= \lim_{\alpha\downarrow0}\frac{\alpha K(\alpha)\,\Phi(\alpha)}{\Phi(\alpha)-\sum_{i=1}^d p_i\omega_i \Phi_i(\alpha)},\]

where

\[K(\alpha)\,:\!=\, \sum_{i=1}^d p_i\big(F^\circ(\alpha,\omega_i) -\frac{\theta_i}{\theta_i-\alpha}\xi_{\mathbb Q}(\theta_i ,\omega_i)\,z_i\big).\]

We obtain the following result.

Theorem 7. As $u\to\infty$ , we have ${{\mathfrak p}(u\,|\,T_{{\boldsymbol\omega},{\boldsymbol p}})}\, \mathrm{e}^{\theta^\star u}\to \gamma[{T_{{\boldsymbol\omega},{\boldsymbol p}}}]$ , where

\[\gamma[{T_{{\boldsymbol \omega},{\boldsymbol p}}}] = \frac{\displaystyle\Phi(0)\sum_{i=1}^d p_i\big(F^\circ(0,\omega_i) -\xi_{\mathbb Q}(\theta_i,\omega_i)\,z_i\big)}{\displaystyle\Phi'(0)-\sum_{i=1}^dp_i\Phi_i'(0)\,\omega_i}.\]

6.2.3. Erlang distributed inter-inspection times.

In the Erlang case the computations become somewhat more involved. It is readily verified that the first term on the right-hand side of (25) can be rewritten as

(29) \begin{align}\int_{u=0}^\infty \mathrm{e}^{-\alpha u} \int_{x=u}^\infty\int_{v=u}^x \mathrm{e}^{-\theta(x-v)}\,\frac{\theta^k{(x-v)^{k-1}}}{(k-1)!}\,{\mathbb Q}(Z_k^+\in\mathrm{d}x)\,\mathrm{e}^{-\theta^*(v-u)}\,\mathrm{d}v\,\textrm{d}u.\end{align}

Then we use the fact that

\[\int_u^x \mathrm{e}^{-(\theta^\star-\theta)v}\frac{\theta^k(x-v)^{k-1}}{(k-1)!}\mathrm{d}v =\left(\frac{\theta}{\theta-\theta^\star}\right)^k\left(\mathrm{e}^{-(\theta^\star-\theta)x}-\sum_{n=0}^{k-1}\mathrm{e}^{-(\theta^\star-\theta)u}\frac{((\theta^\star-\theta)(x-u))^n}{n!}\right),\]

so that (29) reads, after changing the order of integration,

\begin{align*}&\int_{x=0}^\infty \int_{u=0}^x \left(\mathrm{e}^{-\theta^\star x}\mathrm{e}^{-(\alpha-\theta^\star)u}\textrm{d}u-\sum_{n=0}^{k-1}\int_{u=0}^x\mathrm{e}^{-\alpha x}\mathrm{e}^{-(\theta-\alpha)u}\frac{((\theta^\star-\theta)u)^n}{n!}\textrm{d}u\right)\\&\times\left(\frac{\theta}{\theta-\theta^\star}\right)^k {\mathbb Q}(Z_k^+\in\mathrm{d}x).\end{align*}

After straightforward calculus we conclude that the first term on the right-hand side of 25 equals, with $\xi_{\mathbb Q,k}(\alpha,\beta)$ defined in the evident manner,

\begin{align*}I^\circ(\alpha,\omega)\,:\!=\,&\left(\frac{\theta}{\theta-\theta^\star}\right)^k\frac{\xi_{{\mathbb Q},k}(\theta^\star,\omega)-\xi_{{\mathbb Q},k}(\alpha,\omega) }{\alpha-\theta^\star}\\&-\left(\frac{\theta}{\theta-\theta^\star}\right)^k\sum_{n=0}^{k-1}\frac{(\theta-\theta^\star)^n}{(\theta-\alpha)^{n+1}}\left(\xi_{\mathbb Q,k}(\alpha,\omega)-\sum_{m=0}^n\xi_{\mathbb Q,k}^{(m)}(\theta,\omega)\frac{(\alpha-\theta)^m}{m!}\right).\end{align*}

The second term on the right-hand side of (25) can be found analogously to Lemma 3. Defining, recalling the definition of $\bar J_i(\alpha,\beta,\omega)$ from (12),

\begin{align*}\bar J_i^\circ(\alpha,\omega)&\,:\!=\, \sum_{n=i}^{k-1}\delta^{\mathbb Q}_{k-1-n,k}\frac{(\alpha-\theta)^i}{i!}\left(\frac{\theta}{\theta-\alpha}\right)^{n+1},\qquad \delta_{n,k}^{\mathbb Q}\,:\!=\,\frac{(\!-\theta)^n}{n!}\,\xi_{\mathbb Q,k}^{(n)}(\theta,\omega),\end{align*}

we end up with, writing $J^{\circ}(\alpha,\omega)\,:\!=\,\sum_{i=0}^{k-1}\bar{J}_i^{\circ}(\alpha,\omega)\,z_i$ with $z_i\,:\!=\,(\pi^\circ)^{(i)}(\theta\,|\,E_{k,\omega})$ ,

\[\pi^\circ(\alpha\,|\,E_{k,\omega}) = I^\circ(\alpha,\omega) -J^{\circ}(\alpha,\omega)+\left(\frac{\omega}{\omega-\varphi(\alpha-\theta^\star)}\right)^k\pi^\circ(\alpha\,|\,E_{k,\omega}),\]

or equivalently

\[\pi^\circ(\alpha\,|\,E_{k,\omega})=\frac{\displaystyle I^\circ(\alpha,\omega) - J^\circ(\alpha,\omega)}{\displaystyle 1-({\omega}/({\omega-\varphi(\alpha-\theta^\star))})^k},\]

where we still need to identify, for $i=0,\ldots,k-1$ , the constants $z_i$ .

As before, an application of [Reference Ivanovs, Boxma and Mandjes23, Theorem 2] reveals that the denominator has precisely k roots in the right half of the complex plane, say $\alpha_1,\ldots,\alpha_k,$ and, hence, the corresponding numerator as well. It implies that ${\boldsymbol z}$ can be determined by solving, for $j=1,\ldots,k$ ,

\[I^\circ(\alpha_j,\omega) = \sum_{i=0}^{k-1}\bar J_i^\circ(\alpha_j,\omega)\,z_i.\]

We obtain the following result, by following the same procedure as before; observe that $\varphi'(-\theta^\star)<0.$

Theorem 8. As $u\to\infty$ , we have ${{\mathfrak p}(u\,|\,E_{k,\omega})\,\mathrm{e}^{\theta^\star u}}\to \gamma[{E_{k,\omega}}]$ , where

\[\gamma[E_{k,\omega}] =\frac{\displaystyle I^\circ(0,\omega)-J^\circ(0,\omega)} {-k\,\varphi'(\!-\theta^\star)/\omega}.\]

6.2.4. Hyper-Erlang distributed inter-inspection times.

Following the same approach as in the case of Erlang distributed inter-inspection times, we obtain the relation

\begin{align*}\pi^\circ(\alpha\,|\,E_{\boldsymbol{k,p},\omega}) = \frac{\displaystyle\sum_{i=1}^d p_i\big( I_i^\circ(\alpha,\omega) -J_{i}^\circ(\alpha,\omega)\big)}{\displaystyle 1-\sum_{i=1}^dp_i({\omega}/({\omega-\varphi(\alpha-\theta^\star))})^{k_{i}}},\end{align*}

where

\begin{align*}I_i^\circ(\alpha,\omega)\,:\!=\,&\left(\frac{\theta}{\theta-\theta^\star}\right)^{k_i}\frac{\xi_{{\mathbb Q},k_i}(\theta^\star,\omega)-\xi_{{\mathbb Q},k_i}(\alpha,\omega)}{\alpha-\theta^\star}\\&-\left(\frac{\theta}{\theta-\theta^\star}\right)^{k_i}\sum_{n=0}^{k_i-1}\frac{(\theta-\theta^\star)^n}{(\theta-\alpha)^{n+1}}\left(\xi_{\mathbb Q,k_i}(\alpha,\omega)-\sum_{m=0}^n\xi_{\mathbb Q,k_i}^{(m)}(\theta,\omega)\frac{(\alpha-\theta)^m}{m!}\right),\end{align*}

and, with $z_j\,:\!=\,(\pi^\circ)^{(j)}(\theta\,|\,E_{\boldsymbol{k,p},\omega})$ ,

\begin{align*}\bar{J}^\circ_{i,j}(\alpha,\omega)\,:\!=\,\sum_{n=j}^{k_i-1}\delta^{\mathbb{Q}}_{k_i-1-n,k_i}\frac{(\alpha-\theta)^j}{j!}\left(\frac{\theta}{\theta-\alpha}\right)^{n+1},\qquad J_i^\circ(\alpha,\omega)\,:\!=\,\sum_{j=0}^{k_i-1}\bar{J}^\circ_{i,j}(\alpha, \omega)\,z_j.\end{align*}

As before, assuming (without loss of generality) that $k_1 < \cdots < k_d$ , we are left with the task to identify the constants $z_j$ for $j=0,\ldots,k_d-1$ . Again, application of [Reference Ivanovs, Boxma and Mandjes23, Theorem 2] reveals that the denominator has exactly $k_d$ roots in the right half of the complex plane, say $\alpha_1,\ldots,\alpha_{k_d}$ , which are then roots of the numerator as well. Hence, the vector ${\boldsymbol z}$ can be determined by solving the linear system

\begin{align*}\sum_{i=1}^d p_iI_i^\circ (\alpha_\ell, \omega)= \sum_{i=1}^dp_i\sum_{j=0}^{k_i-1}\bar{J}^\circ_{i,j}(\alpha_\ell, \omega)\,z_j,\end{align*}

for $\ell=1,\ldots,k_d$ . We obtain the following result.

Theorem 9. As $u\to\infty$ , we have ${{\mathfrak p}(u\,|\,E_{\boldsymbol{k,p},\omega})} \, \mathrm{e}^{\theta^\star u}\to\gamma[{E_{\boldsymbol{k,p},\omega}}]$ , where

\[\gamma[{E_{\boldsymbol{k,p},\omega}}]=\frac{\displaystyle \sum_{i=1}^d p_i\big( I_i^\circ(0,\omega)-J_i^\circ(0,\omega)\big)}{ \displaystyle-\sum_{i=1}^d p_ik_i \,\varphi'(-\theta^\star)/\omega}.\]

7.1. Importance sampling

As we observed before, the process $(Y(\bar\Omega_n))_{n\in{\mathbb N}}$ constitutes a random walk with i.i.d. increments distributed as the random variable Z. The results of [Reference Korshunov24] reveal that in the light-tailed regime introduced in Section 6, we have that (6) holds, suggesting the approximation

(30) \begin{equation}{{\mathfrak p}(u\,|\,\Omega)}\approx \gamma[\Omega]\,\mathrm{e}^{-\theta^\star u}.\end{equation}

This approach, however, has two intrinsic difficulties. In the first place, the exact asymptotics relate to the regime that u grows large, so that the approximation is expected to be accurate for large u. Hence, in the absence of error bounds, it is not a priori clear whether for the value of u under consideration, (30) provides an accurate approximation. In the second place, while we have succeeded to compute $\gamma[\Omega]$ for various types of inter-inspection times $\Omega$ , no expression is available for general $\Omega$ . As detailed in [Reference Korshunov24, Theorem 3], with

\[T\,:\!=\, \inf\left\{n\in{\mathbb N}:\sum_{i=1}^n Z_i>0\right\}, \qquad a\,:\!=\, {\mathrm{E}}\big[Z\,\mathrm{e}^{\theta^\star Z}1\{T<\infty\}\big],\]

we do have that

\[\gamma[\Omega] = \frac{1-{{\mathfrak p}(0\,|\,\Omega)}}{\theta^\star\,a},\]

but it is not clear how to compute, for an arbitrarily distributed inter-inspection time $\Omega$ , the quantities ${{\mathfrak p}(0\,|\,\Omega)}$ and a.

The above considerations show that there is a need to develop computational tools to efficiently and accurately evaluate ${{\mathfrak p}(u\,|\,\Omega)}$ for any nonnegative random variable $\Omega$ and $u>0$ . The numerical procedure we discuss in this subsection is based on simulation: we devise an importance sampling algorithm that applies the change of measure that we worked with in Section 6. It succeeds in efficiently generating precise estimates for ${{\mathfrak p}(u\,|\,\Omega)}$ , even in the regime that u grows large (and ruin is rare).

The starting point of the algorithm is Theorem 5. In each (independent) run, the model is simulated under ${\mathbb Q}$ until the hitting time $\tau(u)$ , where we recall that u is reached almost surely. Let $W_i$ denote the value of $Y(\tau(u))$ in the ith run. Our Siegmund-type estimator [Reference Asmussen and Glynn10, Section VI.2a], based on n runs, is

(31) \begin{equation}{{\mathfrak p}^{(n)}(u)} \,:\!=\, \frac{1}{n} \sum_{i=1}^n \mathrm{e}^{-\theta^\star W_i},\end{equation}

where the sample variance of this estimator is

\[v_n(u) \,:\!=\, \frac{1}{n-1}\left(\sum_{i=1}^n (\mathrm{e}^{-\theta^\star W_i}-{{\mathfrak p}^{(n)}(u)})^2\right).\]

Noting that $W_i\geqslant u$ , from

\[{\mathbb V}\textrm{ar}_{\mathbb Q}\, \mathrm{e}^{-\theta^\star W_i} = {\mathrm{E}}_{\mathbb Q} \mathrm{e}^{-2\theta^\star W_i}- ({\mathrm{E}}_{\mathbb Q}\mathrm{e}^{-\theta^\star W_i})^2\leqslant {\mathrm{E}}_{\mathbb Q} \mathrm{e}^{-2\theta^\star W_i} \leqslant \mathrm{e}^{-2\theta^\star u}\]

the following result follows directly; see [Reference Asmussen and Glynn10, Section VI.1] for the definition of logarithmic efficiency.

7.2. Change of measure

We proceed by discussing how to sample the process $Y(\cdot)$ according to the alternative measure ${\mathbb Q}.$ As pointed out in Section 6, under ${\mathbb Q}$ the distribution of Z should follow 22. We have

\begin{align*}\mathrm{E}[\mathrm{e}^{- \alpha Z}] &= \int_0^{\infty}\mathrm{e}^{\varphi(\alpha)\,t}\,{\mathrm{P}}(\Omega\in\mathrm{d}{t}) = {\mathrm{E}}\,\mathrm{e}^{\varphi(\alpha)\,\Omega},\end{align*}

so that

\[\mathrm{E}_{\mathbb Q}[\mathrm{e}^{- \alpha Z}] = {\mathrm{E}}\,\mathrm{e}^{\varphi(\alpha-\theta^\star)\,\Omega}.\]

We conclude that under this new measure the net cumulative claim process $Y(\cdot)$ has Laplace exponent $\varphi(\alpha - \theta^\star )$ instead of $\varphi(\alpha)$ , while the distribution of $\Omega$ remains unaltered. This leaves us with the question of how to sample a Lévy process with Laplace exponent $\varphi(\alpha - \theta^\star )$ ? We consider the practically relevant case of $Y(\cdot)$ corresponding to the sum of two independent processes: a compound Poisson process $C(\cdot)$ and a Brownian motion $B(\cdot)$ ; it is noted that any spectrally positive Lévy process can be approximated arbitrarily closely by such a sum [Reference Asmussen and Rosiński11].

Let the Brownian motion have variance coefficient $\sigma^2$ and let the compound Poisson process have drift r, arrival rate $ \lambda $ , with positive jump sizes characterized by the Laplace–Stieltjes transform $b(\cdot).$ This means that

\[\varphi(\alpha) = \log {\mathrm{E}}\exp(\!-\alpha B(1)-\alpha C(1)) =\frac{\alpha^2\sigma^2}{2}+r\alpha-\lambda(1-b(\alpha)),\]

where we assume that the net profit condition is in place (i.e. ${\mathrm{E}}\,Y(1)<0$ , or equivalently, $-\lambda \,b'(0) < r$ ). To identify $Y(\cdot)$ under ${\mathbb Q}$ , observe that

\begin{align*}\varphi(\alpha - \theta^\star )=\varphi_{\mathbb Q}(\alpha) &=\frac{(\alpha- \theta^\star)^2\sigma^2}{2}+r(\alpha- \theta^\star)-\lambda(1-b(\alpha- \theta^\star)) + \varphi(\!-\theta^\star)\\&=\frac{\alpha^2\sigma^2}{2}+(r-\theta^\star\sigma^2)\alpha-\lambda(b(\!-\theta^\star)-b(\alpha-\theta^\star))\\&=\frac{\alpha^2\sigma^2}{2}+(r-\theta^\star\sigma^2)\alpha-\lambda b(\!-\theta^\star)\left(1-\frac{b(\alpha-\theta^\star)}{b(\!-\theta^\star)}\right),\end{align*}

recalling that $\varphi(\!-\theta^\star)=0.$ This means that under ${\mathbb Q}$ the Brownian part remains unchanged, but that the compound Poisson process now has drift $r-\theta^\star\sigma^2$ , the arrival rate becomes $\lambda b(\!-\theta^\star)$ , and the claim sizes are now characterized by the Laplace–Stieltjes transform $b(\cdot-\theta^\star)/b(\!-\theta^\star).$

In the experiments we let the claim sizes be exponentially distributed with parameter $\mu$ . It is readily verified that (for $\alpha>-\mu$ )

\[\varphi(\alpha) = \alpha\,\frac{\sigma^2\alpha^2 +(\mu\sigma^2 +2r)\alpha+2(r\mu-\lambda)}{2(\mu+\alpha)},\]

so that

\[\theta^\star = \frac{\mu\sigma^2 +2r}{2\sigma^2} -\frac{\sqrt{(\mu\sigma^2+2r)^2 -8\sigma^2(r\mu-\lambda)}}{2\sigma^2}.\]

Under ${\mathbb Q}$ the claim sizes are exponentially distributed with parameter $\mu-\theta^\star.$

7.3. Numerical experiments: two-moment approximation for inter-inspection times

In this seriesof numerical experiments we assess ${{\mathfrak p}(u\,|\,\Omega)}$ for an instance in which the inter-inspection time $\Omega$ does not have one of the phase-type distributions covered in this paper, and compare it to its counterpart in which $\Omega$ is approximated by $\Omega_\textrm{app}$ determined by the two-moment fit that was discussed in Section 5.1. For ease, we use (without losing any generality) the normalization ${\mathrm{E}}\,\Omega = 1$ and vary ${\mathbb V}\textrm{ ar}\,\Omega$ (where we cover values both below and above 1). The inter-inspection time $\Omega$ we consider has a lognormal distribution, with its two parameters chosen such that the mean is 1 and the variance has the target value.

We proceed by pointing out how to numerically evaluate the constant $\gamma[\Omega_\textrm{app}]$ for these $\Omega_\textrm{app}$ as follows.

1. (i) If the coefficient of variation of the inter-inspection times is larger than 1 then the two-moment fit prescribes that we should use a hyperexponential random variable $T_{{\boldsymbol p},{\boldsymbol \omega}}$ with $d=2$ , where the parameter vectors ${\boldsymbol p}$ and ${\boldsymbol\omega}$ are chosen such that the first two moments have the desired values (and in addition the so-called balanced means condition is fulfilled; see e.g. [Reference Kuiper27, Section 3.1]). It requires some elementary calculus to verify that $\alpha_1$ and $\alpha_2$ are, respectively, $\theta^\star$ and the positive root of (with $p\,:\!=\,p_1$ )

   \[\varphi_{\mathbb Q}(\alpha) = (1-p)\omega_1+p\omega_2.\]
   As is readily checked from the expression for $\varphi_{\mathbb Q}(\alpha)$ , finding $\alpha_2$ requires solving an equation involving a third degree polynomial.

2. (ii) If the coefficient of variation is smaller than 1 then the two-moment fit entails that we use an $E_{{\boldsymbol k},{\boldsymbol p},\omega}$ distribution with $d=2$ , $k_1=k$ , $k_2=k+1$ , and $p_1=p$ , where p and k are such that the first two moments have the desired value. In this case $\alpha_1,\ldots,\alpha_{k+1}$ are the roots in the right half of the complex plane of the polynomial equation

   (32) \begin{equation}1-\left(\frac{\omega-\varphi_{\mathbb Q}(\alpha)}{\omega}\right)^{k+1}=\frac{p\varphi_{\mathbb Q}(\alpha)}{\omega},\end{equation}
   where, evidently, $\theta^\star$ is one such root. There are two ways of finding the roots $\alpha_1,\ldots,\alpha_{k+1}$ , both relying on the availability of a procedure to find the, potentially complex, roots of a polynomial equation, which is available in standard numerical packages. An approach to do so would be to insert the expression for $\varphi_{\mathbb Q}(\alpha)$ into (32), and to directly solve the resulting polynomial of order $3k+3$ , after which one selects the roots with a positive real part. There is an alternative, more efficient, approach, though: (i) first, find the $k+1$ roots for $\varphi_{\mathbb Q}(\alpha)$ of (32), say $\varphi_1,\ldots,\varphi_{k+1}$ ; (ii) then per solution $\varphi_i$ find the three solutions (in $\alpha$ ) of the equation $\varphi_{\mathbb Q}(\alpha)=\varphi_i$ ; (iii) after which one selects, from the resulting $3k+3$ solutions, those with a positive real part (where we know that there are precisely $k+1$ of those).

A second complication lies in the computation of $\delta_{n,k}^{\mathbb Q}$ (for $n=0,\ldots,k-1$ ), which requires the numerical evaluation of the higher-order derivatives

\[\xi_{\mathbb Q,k}^{(n)}(\theta,\omega)= \left.\frac{\mathrm{d}^n}{\mathrm{d}\alpha^n}\left(\frac{\alpha -\psi_{\mathbb Q}(\beta)}{\varphi_{\mathbb Q}(\alpha)-\beta}\frac{\beta}{\psi_{\mathbb Q}(\beta)}\right)^k\right|_{\alpha=\theta}.\]

We have streamlined the evaluation of these quantities relying on two ideas. (i) In the first place, it is observed that repeated symbolic derivation of ratios leads to large, unmanageable expressions. To remedy this, first write

\[\xi_{\mathbb Q,k}(\alpha,\omega)\,\big((\varphi_{\mathbb Q}(\alpha)-\beta))\, \psi_{\mathbb Q}(\beta)\big)^k = \big((\alpha-\psi_{\mathbb Q}(\beta))\,\beta\big)^k,\]

and then differentiate n times using Leibniz’ rule. This yields a recursion for the high-order derivatives of $\xi_{\mathbb Q,k}(\alpha,\omega)$ ; this we did symbolically using SymPy, the open-source Python library for symbolic computation. (ii) In the second place, inserting $\alpha=\theta$ into this recursion leads to floating point errors due to division by extremely small numbers. To resolve this, we have inserted $\alpha=\theta\pm \varepsilon$ for a small value of $\varepsilon>0$ , and have taken the average of the resulting two numbers.

As mentioned, in the experiments we consider the case of exponentially distributed claims with parameter $\mu$ . The parameters chosen are $\lambda= 2$ , $r=1.2$ , $\sigma^2 = 0.02$ , $\mu= 2$ , such that $\theta^\star= 0.32875.$

In Figure 1 we display, for various values of the initial surplus u, $\gamma_u[\Omega]\,:\!=\, {{\mathfrak p}(u\,|\,\Omega)}\,\mathrm{e}^{\theta^\star u}$ as a function of ${\mathbb V}\textrm{ar}\,\Omega$ , relying on our importance sampling-based Monte Carlo procedure. In Figure 2 we show the corresponding results for its phase-type counterpart $\gamma_u[\Omega_\textrm{app}]\,:\!=\, {{\mathfrak p}(u\,|\,\Omega_\textrm{app})}\,\mathrm{e}^{\theta^\star u}$ , again using importance sampling. Finally, Figure 3 displays the curve $\gamma[\Omega_\textrm{app}]$ , obtained by performing the numerical computations described above.

Figure 1.

Monte Carlo-based estimation of $\gamma_u[\Omega]$ .

Figure 2.

Monte Carlo-based estimation of $\gamma_u[\Omega_\textrm{app}]$ .

Figure 3.

Numerical computation for $\gamma[\Omega_\textrm{app}]$ .

All computer codes developed for this paper can be accessed in a GitHub repository via the following link: https://github.com/saramorcy/ruin-probabilities. In Figures 1 and 2 we have displayed, for any value of the variance of the inter-inspection time (that is, ${\mathbb V}\textrm{ar}\,\Omega$ and ${\mathbb V}\textrm{ar}\,\Omega_\textrm{app}$ , respectively) in the range $\{0.02,0.04,\ldots,2.98,3.00\}$ , the quantity $\gamma_u[\Omega]$ respectively $\gamma_u[\Omega_\textrm{app}]$ . Each data point is estimated using our importance sampling-based Monte Carlo procedure, with the number of runs being equal to $10^5$ ; we chose this relatively high number of runs to make sure the corresponding standard errors are extremely low, thus making the curves relatively smooth.

The main conclusions from the figures are as follows. (i) The experiments provide empirical evidence that $\gamma_u[\Omega]$ converges to a constant $\gamma[\Omega]$ as u grows large, also for nonphase-type $\Omega$ . (ii) The experiments confirm that $\gamma_u[\Omega_\textrm{app}]$ converges to a constant $\gamma[\Omega_\textrm{app}]$ as u grows large, in line with our theoretical findings. (iii) The curves $\gamma[\Omega]$ and $\gamma[\Omega_\textrm{app}]$ (that is, as functions of ${\mathbb V}\textrm{ar}\,\Omega$ and ${\mathbb V}\textrm{ar}\,\Omega_\textrm{app}$ , respectively) are hardly distinguishable. This provides backing for the use of the (relatively low-dimensional) two-moment approximation of the inter-inspection times. (iv) Already for relatively low values of the initial surplus u, the value of $\gamma_u[\Omega_\textrm{app}]$ is close to its limiting value $\gamma[\Omega_\textrm{app}]$ .

The above observations justify the use of the approximation

\[{{\mathfrak p}(u\,|\,\Omega)} \approx \gamma[\Omega_\textrm{app}]\,\mathrm{e}^{-\theta^\star u}.\]

For the specific parameters chosen in this example, observe that the constant $\gamma[\Omega_\textrm{app}]$ only mildly varies in the range ${\mathbb V}\textrm{ar}\,\Omega\in[0,3]$ , entailing that we could work with the even simpler approximation

\[{{\mathfrak p}(u\,|\,\Omega)} \approx \gamma[T_1] \,\mathrm{e}^{-\theta^\star u},\]

where it is recalled that Theorem 6 provides an explicit expression for $\gamma[T_1]$ .