2. The dependent renewal risk model

In a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ , we consider the surplus process of the form

(2.1) \begin{equation} U_{t}\,:\!=\, u+ct-\sum _{j=1}^{N_{t}}X_{j},\,\,\,t\geq 0, \end{equation}

where $u\geq 0$ denotes the initial surplus, $N=(N_{t},t\geq 0)$ is a renewal process, $X_{j},j=1,2,\ldots$ denote the claim sizes and $c\gt 0$ is the constant premium rate. We also denote by $T_{1},T_{2},\ldots$ the inter-claim times of the process $N,$ and by $Y_{k}\,:\!=\,\sum _{i=1}^{k}T_{i}$ the arrival time of the $k$ -th claim. The non-negative random variables (RVs) $T_{i}$ and $X_{i}$ may be dependent, while the pairs $(T_{1},X_{1})$ , $(T_{2},X_{2})$ ,… are independent and identically distributed (IID) with common joint cumulative distribution function $F_{T,X}$ . Since the ruin event occurs only at claim arrival times $Y_{1},Y_{2},\ldots$ , the process $(U_{t},t\geq 0)$ can be considered only at claim arrival times, and thus, for our exposition, it suffices to study the discrete time process $S=(S_{k},k=1,2,\ldots )$ , with

(2.2) \begin{equation} S_{k}\,:\!=\, U_{Y_{k}}=u+cY_{k}-\sum _{j=1}^{k}X_{j}=u+\sum _{j=1}^{k}\left (cT_{j}-X_{j}\right )=u+\sum _{j=1}^{k}Z_{j} \end{equation}

where $Z_{j}\,:\!=\, cT_{j}-X_{j}$ , for every $j=1,2,\ldots$ . Let $\nu \,:\!=\,\inf \left \{ k\,:\,S_{k}\lt 0\right \}$ be the stopping time that counts the number of claims until the process $S$ drops below zero (ultimate ruin event) and $\tau \,:\!=\, Y_{\nu }$ the total time until ruin. Obviously, $S_{\nu }=U_{\tau },\nu =N_{\tau }$ . Focusing on the process $S$ , we define a new measure on the measurable space $(\Omega ,\mathcal{F})$ , to be denoted by $\mathbb{P}_{h,\delta }$ , via the following martingale likelihood process

\begin{equation*} L_{k}\,:\!=\,\frac {e^{\sum _{i=1}^{k}\left (\left (ch-\delta \right )T_{i}-hX_{i}\right )}}{\mathbb{M}_{T,X}\left (ch-\delta ,-h\right )^{k}},\;\;\;k=1,2,\ldots \end{equation*}

for two real parameters $\delta ,h$ such that $\mathbb{M}_{T,X}\left (ch-\delta ,-h\right )\lt \infty$ , where $\mathbb{M}_{T,X}\left (t,x\right )\,:\!=\,\mathbb{E}(e^{tT_{i}+xX_{i}})$ is the joint moment generating function (MGF) of $\left (T,X\right ).$ In particular, for every $A\in \mathcal{F}_{n}\,:\!=\,\sigma \left (\left \{ \left (T_{i},X_{i}\right ),i\leq n\right \} \right )\subseteq \mathcal{F}$ , we set

\begin{equation*} \mathbb{P}_{h,\delta }\left (A\right )\,:\!=\,\mathbb{E}\left (1_{A}L_{n}\right ). \end{equation*}

The measures $\mathbb{P}$ and $\mathbb{P}_{h,\delta }$ are equivalent, and under $\mathbb{P}_{h,\delta }$ , the MGF of $Z_{1}$ is given by

(2.3) \begin{equation} \mathbb{E}_{h,\delta }\big(e^{tZ_{1}}\big)=\mathbb{E}\left (e^{t\left (cT_{1}-X_{1}\right )}\frac {e^{\left (\left (ch-\delta \right )T_{1}-hX_{1}\right )}}{\mathbb{M}_{T,X}\left (ch-\delta ,-h\right )}\right )=\frac{\mathbb{M}_{T,X}(c(t+h)-\delta ,-(t+h))}{\mathbb{M}_{T,X}(ch-\delta ,-h)}. \end{equation}

It is worth stressing that this new measure can be considered as an exponential tilting measure affecting the joint distribution of $T_{i},X_{i}$ (i.e., it is a bivariate Esscher transform), while it is easy to verify that the pairs $(T_{1},X_{1})$ , $(T_{2},X_{2})$ , … remain IID under $\mathbb{P}_{h,\delta }$ . A fundamental result of this work is the following theorem, where, by exploiting this change of measure technique, we offer an identity that can be used to generalize existing GS functions. An RV $W$ will be called $\mathcal{F}_{\nu }$ -measurable if, for every $k\in \mathbb{N},$ the RV $W1_{[\nu =k]}$ is $\mathcal{F}_{k}$ -measurable.

Theorem 1. If there exists a real function $\gamma =\gamma _{z,\delta }$ such that

(2.4) \begin{equation} z\mathbb{M}_{T,X}\left (c\gamma -\delta ,-\gamma \right )=1, \end{equation}

then for every $\mathcal{F}_{\nu }$ -measurable RV $W$ , the following identity holds

(2.5) \begin{equation} \mathbb{E}\left (z^{\nu }e^{-\delta \tau }W1_{[\tau \lt \infty ]}\right )=e^{\gamma u}\mathbb{\mathbb{E}}_{\gamma ,\delta }\left (e^{-\gamma U_{\tau }}W1_{[\tau \lt \infty ]}\right ), \end{equation}

for all $\gamma ,\delta ,z\in \mathcal{R}$ such that the above expectations exist.

Proof. Let $V$ be any $\mathcal{F}_{\nu }$ -measurable RV. We have that

\begin{equation*} \mathbb{E}\left(Ve^{h\left (S_{\nu }-u\right )-\delta Y_{\nu }}1_{[\nu \lt \infty ]}\right )=\sum _{k=1}^{\infty }\mathbb{E}\Big(V1_{[\nu =k]}e^{h\sum _{j=1}^{k}Z_{j}-\delta \sum _{j=1}^{k}T_{j}}\Big). \end{equation*}

For every $k\geq 1$ , the RV $V_{k}=V1_{[\nu =k]}$ is $\mathcal{F}_{k}$ -measurable and therefore

\begin{equation*} \frac{\mathbb{E}\big(V1_{[\nu =k]}e^{h\sum _{j=1}^{k}Z_{j}-\delta \sum _{j=1}^{k}T_{j}}\big)}{\mathbb{M}_{T,X}\left (ch-\delta ,-h\right )^{k}}=\frac {\mathbb{E}\big(V_{k}e^{\sum _{i=1}^{k}\left (\left (ch-\delta \right )T_{i}-hX_{i}\right )}\big)}{\mathbb{M}_{T,X}(ch-\delta ,-h)^{k}}=\mathbb{E}\left (V_{k}L_{k}\right )=\mathbb{\mathbb{E}}_{h,\delta }\left (V_{k}\right ). \end{equation*}

Hence,

(2.6) \begin{eqnarray}\mathbb{E}\left (Ve^{h\left (S_{\nu }-u\right )-\delta Y_{\nu }}1_{[\nu \lt \infty ]}\right ) & = &\sum _{k=1}^{\infty }\mathbb{E}\Big(V1_{[\nu =k]}e^{h\sum _{j=1}^{k}Z_{j}-\delta \sum _{j=1}^{k}T_{j}}\Big)\nonumber\\& =&\sum _{k=1}^{\infty }\mathbb{\mathbb{E}}_{h,\delta }\big(V1_{[\nu =k]}\mathbb{M}_{T,X}\left (ch-\delta ,-h\right )^{k}\big)\nonumber \\ & = &\mathbb{\mathbb{E}}_{h,\delta }\left (V\mathbb{M}_{T,X}\left (ch-\delta ,-h\right )^{\nu }1_{[\nu \lt \infty ]}\right ). \end{eqnarray}

For $h=\gamma$ such that $z\mathbb{M}_{T,X}\left (c\gamma -\delta ,-\gamma \right )=1$ , we get

\begin{equation*} \mathbb{E}\big(Ve^{\gamma \left (S_{\nu }-u\right )-\delta Y_{\nu }}1_{[\nu \lt \infty ]}\big)=\mathbb{\mathbb{E}}_{\gamma ,\delta }\left (Vz^{-\nu }1_{[\nu \lt \infty ]}\right ). \end{equation*}

Applying the above to the $\mathcal{F}_{\nu }$ -measurable RV $V=z^{\nu }e^{-\gamma \left (S_{\nu }-u\right )}W$ and taking into account the fact that $S_{\nu }=U_{\tau },Y_{\nu }=\tau$ and $1_{[\nu \lt \infty ]}=1_{[\tau \lt \infty ]}$ a.s., we get the desired result.

We can now exploit Theorem1 in order to reduce the form of Gerber–Shiu functions (see Gerber & Shiu, Reference Gerber and Shiu1998), which also contain the number of claims until ruin, $\nu ,$ by expressing them under the new measure $\mathbb{P}_{\gamma ,\delta }$ . We offer the following corollary for any penalty function $w\left (U_{\tau -},-U_{\tau }\right )$ of the surplus prior to ruin and the deficit at ruin.

Corollary 2. The generalized GS function satisfies the following identity

\begin{equation*} \phi (u)\,:\!=\,\mathbb{E}\Big(z^{\nu }e^{-\delta \tau }\,w\left (U_{\tau -},-U_{\tau }\right )1_{[\tau \lt \infty ]}\Big)=e^{\gamma u}\mathbb{E}_{\gamma ,\delta }\left (e^{-\gamma U_{\tau }}w\left (U_{\tau -},-U_{\tau },\right )1_{[\tau \lt \infty ]}\right ), \end{equation*}

where $\gamma =\gamma _{z,\delta }$ is a real root of the generalized Lundberg equation ( 2.4 ) and $\delta$ represents the constant discount rate.

The above form of the GS function (in the special case where $z=1$ ) is considered by Badescu et al. (Reference Badescu, Cheung and Landriault2009) and Cossette et al. (Reference Cossette, Marceau and Marri2008).

It is worth mentioning that the above corollary could similarly be formulated employing alternative penalty functions, for example, depending on the last minimum of the surplus before ruin, $\underline {U}_{\tau -}\,:\!=\,\inf \{U_{t},t\lt \tau \}$ , the surplus on the second to last claim before ruin, $U_{Y_{\nu -1}}$ , etc. (e.g., see Woo, Reference Woo2010). We refer to the review paper of He et al. (Reference He, Kawai, Shimizu and Yamazaki2023) for a list of cases for GS penalty functions.

In what follows, we denote by

\begin{equation*} \mathbb{B}_{h,\delta }\left (t\right )\,:\!=\,\mathbb{E}_{h,\delta }\big(e^{t(-U_{\tau })}\mid \tau \lt \infty\big) \end{equation*}

the MGF of the deficit at ruin, under the measure $\mathbb{P}_{h,\delta }$ . In the following proposition, we present an identity that enables us to explicitly express a special case of the GS function, namely, the joint GF of the number of claims until ruin, the deficit at ruin, and the time until ruin. We further denote by $\psi _{h,\delta }\,:\!=\,\mathbb{P}_{h,\delta }\left (\tau \lt \infty \right )$ the ruin probability under $\mathbb{P}_{h,\delta }$ .

Proposition 3. If there exists a real function $\gamma =\gamma _{z,\delta }$ such that $z\mathbb{M}_{T,X}\left (c\gamma -\delta ,-\gamma \right )=1$ , then

(2.7) \begin{equation} \phi _{z,\delta ,s}(u)\,:\!=\,\mathbb{E}(z^{\nu }e^{-sU_{\tau }-\delta \tau }1_{[\tau \lt \infty ]})=\psi _{\gamma ,\delta }\,e^{\gamma u}\mathbb{B}_{\gamma ,\delta }\left (\gamma +s\right ), \end{equation}

for all $s,\delta ,z\in \mathcal{R}$ such that the above expectations exist. In addition, the probability of the ultimate ruin $\psi \,:\!=\,\mathbb{P}\left (\tau \lt \infty \right )$ is given by

\begin{equation*} \psi =\psi _{\gamma ^{*},0}\,e^{\gamma ^{*}u}\mathbb{B}_{\gamma ^{*},0}\left (\gamma ^{*}\right ), \end{equation*}

where $\gamma ^{*}\,:\!=\,\gamma _{1,0}$ is a real root of the equation $\mathbb{M}_{T,X}\left (c\gamma ^{*},-\gamma ^{*}\right )=1$ .

Proof. Using Theorem1 for $W=e^{-sU_{\tau }}$ , the r.h.s. of Equation (2.5) becomes

\begin{align*} e^{\gamma u}\mathbb{\mathbb{E}}_{\gamma ,\delta }\left (e^{-\gamma U_{\tau }}e^{-sU_{\tau }}1_{[\tau \lt \infty ]}\right ) & =e^{\gamma u}\mathbb{\mathbb{E}}_{\gamma ,\delta }\left (\left .e^{\left (\gamma +s\right )\left (-U_{\tau }\right )}\right |\tau \lt \infty \right )\mathbb{P}_{\gamma ,\delta }\left (\tau \lt \infty \right )\\ & =\psi _{\gamma ,\delta }e^{\gamma u}\mathbb{B}_{\gamma ,\delta }\left (\gamma +s\right ), \end{align*}

which gives us the first equation. Setting $\delta =s=0$ and $z=1$ , we get the ultimate ruin probability $\psi ,$ but in this case, the change of measure parameter $\gamma _{1,0}$ corresponds to the solution of the equation $\mathbb{M}_{T,X}\left (c\gamma ,-\gamma \right )=1$ with respect to $\gamma$ .

Remark 4. In the aforementioned ruin model, we have assumed, primarily for analytical tractability, that the premium collected over a time interval of length $t$ is equal to $ct$ . Extending to a more general and realistic setup, stochastic premium income may be integrated to account for the influence of market conditions by assuming that the premium collected over the time interval $[0,t]$ is equal to $C_{t}$ , where $(C_{t},t\geq 0)$ is an appropriate stochastic process with $\mathbb{E}(C_{t})=ct,\,t\geq 0$ . To maintain the validity of our approach, this process must necessarily possess independent and identical increments (i.e., must be a Lévy process), and in addition, it must be non-decreasing a.s., ensuring that ruin can occur only upon the occurrence of a claim. Hence, the process $(C_{t},t\geq 0)$ must be a subordinator with Lévy–Khintchine representation of the form $\mathbb{E}(e^{-sC_{t}})=e^{-\phi (s)t},s\geq 0,$ and Laplace exponent

\begin{equation*} \phi (s)=as+\int _{0}^{\infty }(1-e^{-sx})\Pi (dx), \end{equation*}

where $a\geq 0$ is the drift coefficient and $\Pi$ denotes the Lévy measure on $(0,\infty )$ satisfying $a+\int _{0}^{\infty }x\Pi (dx)=c\lt \infty$ . It is noteworthy that the validity of our general results (cf. Theorem1, Corollary2, Proposition3) can be confirmed even when $(C_{t},t\geq 0)$ is a subordinator (e.g., gamma or inverse Gaussian process or compound Poisson process with positive jumps). The only modification that needs to be made is that the root $\gamma =\gamma _{z,\delta }$ must now satisfy the equation

\begin{equation*} z\mathbb{M}_{T,X}\left (\phi (\gamma )-\delta ,-\gamma \right )=1. \end{equation*}

However, in order to obtain closed-form formulae for the quantities of interest, in what follows, we proceed under our initial assumption $C_{t}=ct$ (constant, deterministic premium rate), which is also commonly adopted by most authors. The use of subordinators to incorporate stochastic premium rates will be further investigated in our future research since, under this framework, the relevant analysis becomes much more complex and may require numerical methods or approximation techniques.

The joint transform in Proposition 3 can now be explicitly computed when the distribution of the deficit at ruin (undershoot) under the measure $\mathbb{P}_{h,\delta }$ is known. In order to demonstrate how this approach can be used in the simplest case where $T_{i}$ , $X_{i}$ are independent, we apply (2.7) in the following example to a ruin model with phase-type distributed claims that have also been studied (in a more general setup) by Frostig et al. (Reference Frostig, Pitts and Politis2012).

Example. We assume that $(N_{t},t\geq 0)$ is a Poisson process with constant intensity $\lambda$ , while the claim sizes, $X_{1},X_{2},\ldots$ are IID RVs, independent of $(N_{t},t\geq 0)$ , following a phase-type distribution $PH\left (\boldsymbol{\alpha },\boldsymbol{S}\right )$ , with density

\begin{equation*} f_{X}(x)=\boldsymbol{\alpha }e^{x\boldsymbol{S}}(-\boldsymbol{S}\textbf{1}),\,\,x\geq 0, \end{equation*}

where $\boldsymbol{\alpha }=\left (\alpha _{1},\ldots ,\alpha _{p}\right )$ is a row vector, $\textbf{1}=\left (1,\ldots ,1\right )^{\prime }$ is $p\times 1$ column vector, and $\boldsymbol{S}$ is a $p\times p$ sub-intensity matrix. We assume that there exists a real solution $\gamma =\gamma _{z,\delta }$ of the equation

\begin{equation*} z\mathbb{M}_{T,X}\left (c\gamma -\delta ,-\gamma \right )=1\iff z\mathbb{M}_{T}\left (c\gamma -\delta \right )\mathbb{M}_{X}\left (-\gamma \right )=1, \end{equation*}

such that the MGFs of $T_{i},X_{i}$ under $\mathbb{P}_{\gamma ,\delta }$ are well defined (cf. 2.3 ). Under this new measure, the inter-claim times have MGF

\begin{equation*} \mathbb{E}_{\gamma ,\delta }\big(e^{tT_{1}}\big)=\mathbb{E}\left (\frac {e^{\left (t+c\gamma -\delta \right )T_{1}}}{\mathbb{M}_{T}\left (c\gamma -\delta \right )}\right )=\frac {\frac {\lambda }{\lambda -(t+c\gamma -\delta )}}{\frac {\lambda }{\lambda -(c\gamma -\delta )}}=\frac {\lambda -c\gamma +\delta }{\lambda -c\gamma +\delta -t}, \end{equation*}

and therefore $T_{i}$ are exponentially distributed with parameter $\lambda ^{*}=\lambda -c\gamma +\delta \gt 0$ . In addition, under $\mathbb{P}_{\gamma ,\delta }$ , the RVs $X_{1},X_{2},\ldots$ are again independent of $T_{1},T_{2},\ldots$ following the $-\gamma$ exponentially tilted $PH\left (\boldsymbol{\alpha },\boldsymbol{S}\right )$ distribution, which is again PH. More specifically, from Lemma 5 of Asmussen et al. (Reference Asmussen, Avram and Pistorius2004), we have that $X_{i}\sim PH\left (\boldsymbol{\alpha }^{*},\boldsymbol{S}^{*}\right )$ under $\mathbb{P}_{\gamma ,\delta }$ , with

\begin{equation*} \boldsymbol{\alpha }^{*}=\frac {\boldsymbol{\alpha }\boldsymbol{\Delta }}{\mathbb{M}_{X}\left (-\gamma \right )},\;\;\;\boldsymbol{S}^{*}=\boldsymbol{\Delta }^{-1}\boldsymbol{S}\boldsymbol{\Delta }-\gamma \boldsymbol{I}, \end{equation*}

where $\boldsymbol{I}$ is the $p\times p$ identity matrix and $\boldsymbol{\Delta }$ is a $p\times p$ diagonal matrix with elements on the diagonal given by $\left (\boldsymbol{S}-\gamma \boldsymbol{I}\right )^{-1}\boldsymbol{S}\textbf{1}$ . Since the claim sizes are phase-type distributed under the measure $\mathbb{P}_{\gamma ,\delta }$ , the deficit at ruin will again be phase-type distributed as described in Drekic et al. (Reference Drekic, Dickson, Stanford and Willmot2004), with the same intensity matrix $\boldsymbol{S}^{*}$ but different vector $\tilde {\boldsymbol{\alpha }}$ . Hence, the MGF of the deficit at ruin takes on the form

\begin{equation*} \mathbb{B}_{\gamma ,\delta }\left (s\right )=\mathbb{\mathbb{E}}_{\gamma ,\delta }\left (\left .e^{s\left (-U_{\tau }\right )}\right |\tau \lt \infty \right )=\tilde {\boldsymbol{\alpha }}\left (s\boldsymbol{I}+\boldsymbol{S}^{*}\right )^{-1}\left (\boldsymbol{S}^{*}\textbf{1}\right ). \end{equation*}

From Drekic et al. (Reference Drekic, Dickson, Stanford and Willmot2004), we know that $\tilde {\boldsymbol{\alpha }}$ can be written as $\tilde {\boldsymbol{\alpha }}=\boldsymbol{\beta }/\psi _{\gamma ,\delta }$ where $\boldsymbol{\beta }$ is a $1\times p$ known vector (see the main theorem of Drekic et al. (Reference Drekic, Dickson, Stanford and Willmot2004) along with relations (1) and (2) of that paper). Exploiting this equality, the formula in Proposition 3 now reduces to

\begin{align*} \mathbb{E}\left (z^{\nu }e^{-sU_{\tau }-\delta \tau }1_{[\tau \lt \infty ]}\right ) & =\psi _{\gamma ,\delta }\,e^{\gamma u}\mathbb{B}_{\gamma ,\delta }\left (\gamma +s\right )=\psi _{\gamma ,\delta }\,e^{\gamma u}\tilde {\boldsymbol{\alpha }}\left ((\gamma +s)\boldsymbol{I}+\boldsymbol{S}^{*}\right )^{-1}\left (\boldsymbol{S}^{*}\textbf{1}\right )\\ & =e^{\gamma u}\boldsymbol{\beta }\left ((\gamma +s)\boldsymbol{I}+\boldsymbol{S}^{*}\right )^{-1}\boldsymbol{S}^{*}\textbf{1}. \end{align*}

In the case when $T$ and $X$ are dependent, the distribution of the undershoot (deficit at ruin) is generally more complicated. Compared with the previous example, even when $X$ is phase-type distributed, there is no guarantee that the distribution of the undershoot falls within the same class. Even if this is true, the intensity matrix of the undershoot may not be the same as the intensity matrix of the claim sizes; see Sections 3 and 4. In the following two sections, we choose the joint distribution of the pair $\left (T_{i},X_{i}\right )$ so that the distribution of the increments of the random walk $S$ can be expressed via exponential or Erlang mixtures. Thus, by invoking the (generalized) memoryless property, we can use the fact that the distribution of the undershoot falls within the same class. Hence, we offer explicit formulae for the above general results in two cases of bivariate distributions for the pair $\left (T_{i},X_{i}\right )$ . In what follows, we shall investigate the case of Downton–Moran bivariate exponential distribution proposed by Downton (Reference Downton1970), and as a generalization, we shall also consider the Kibble–Moran bivariate Erlang distribution. The Downton–Moran bivariate distribution weakens the independence assumption of the Cramer–Lundberg classical risk model with exponentially distributed claim sizes, making it possible to conduct a sensitivity analysis with respect to the correlation coefficient parameter between $T,X$ . The Kibble–Moran distribution further generalizes this model, allowing the use of Erlang as a marginal distribution for the inter-claim times and claim sizes. These bivariate distributions admit positive correlation between the claim sizes and the inter-claim times, which is a common scenario for insurance companies to expect bigger claim sizes in longer report periods and smaller claim sizes as the reports are more frequent (see also a brief related discussion in the introduction). Note that the Erlang model has been extensively employed by many authors (see, e.g., Dickson (Reference Dickson1998), Dickson & Hipp (Reference Dickson and Hipp2001), Li & Garrido (Reference Li and Garrido2004) in the independent case or Ambagaspitiya (Reference Ambagaspitiya2009), Chadjiconstantinidis & Vrontos (Reference Chadjiconstantinidis and Vrontos2014) for the dependent case). The Downton–Moran and Kibble–Moran distributions have the ability to model high positive correlation between $T,X$ , that is, $\rho \in \left [0,1\right )$ , while other bivariate models constructed using copulas have the disadvantage of bounded range for the correlation coefficient; for example, the classical FGM copula has $\rho \in \left [-1/3,1/3\right ]$ , while generalized FGM copulas can achieve a slightly wider bounded range (see, e.g., some cases in Zachariah et al. (Reference Zachariah, Arshad and Pathak2024)). An additional critical advantage of these bivariate distributions lies in the simplicity of their GF, which aligns well with our methodological framework and allows for the explicit derivation of closed-form expressions. Note that, for these bivariate distributions, closed-form formulae have been derived by Ambagaspitiya (Reference Ambagaspitiya2009) only for the ultimate ruin probability.

3. Downton–Moran bivariate exponential analysis

We consider the Moran–Downton bivariate exponential distribution (see Downton, Reference Downton1970) for the pairs $\left (T_{i},X_{i}\right )$ , with MGF

(3.1) \begin{equation} \mathbb{M}_{T,X}\left (t,x\right )=\frac {\lambda _{1}\lambda _{2}}{\left (\lambda _{1}-t\right )\left (\lambda _{2}-x\right )-\rho tx}, \end{equation}

where $\lambda _{1},\lambda _{2}\gt 0$ and $\rho \in \left [0,1\right )$ . From (3.1), we see that the marginal distributions of $T_{i},X_{i}$ are exponential with parameters $\lambda _{1},\lambda _{2}$ , respectively . The parameter $\rho$ is easily verified to be the correlation coefficient between $T$ and $X$ . Using this bivariate distribution, we naturally assume positive correlation between the claim sizes and the inter-claim times in the sense that we expect bigger claim sizes in longer report periods and smaller claim sizes as the reports are more frequent. From (3.1), the MGF of the steps $Z_{i}\,:\!=\, cT_{i}-X_{i}$ takes on the form

\begin{align*} \mathbb{M}_{Z}\left (t\right ) & =\mathbb{M}_{T,X}\left (ct,-t\right )=\frac {\lambda _{1}\lambda _{2}}{\left (\lambda _{1}-ct\right )\left (\lambda _{2}+t\right )+c\rho t^{2}}=\frac {r_{2}}{r_{1}+r_{2}}\cdot \frac {r_{1}}{r_{1}+t}+\frac {r_{1}}{r_{1}+r_{2}}\cdot \frac {r_{2}}{r_{2}-t}, \end{align*}

where

(3.2) \begin{equation} r_{i}\,:\!=\,\frac {\left (-1\right )^{i+1}\left (c\lambda _{2}-\lambda _{1}\right )+\sqrt {\left (c\lambda _{2}-\lambda _{1}\right )^{2}+4c\lambda _{1}\lambda _{2}\left (1-\rho \right )}}{2c\left (1-\rho \right )}\gt 0,\;\;\;\;i=1,2, \end{equation}

and therefore, the distribution of $Z_{i}$ is a mixture of a positive exponential distribution with parameter $r_{2}$ and a negative exponential distribution with parameter $r_{1}$ . Combining (2.3) and (3.1), we determine the MGF of $Z_{1}$ under the measure $\mathbb{P}_{h,\delta }$ , which is

(3.3) \begin{equation} \mathbb{E}_{h,\delta }\big(e^{tZ_{1}}\big)=\frac {\left (-ch+\delta +\lambda _{1}\right )\left (h+\lambda _{2}\right )+\rho h\left (ch-\delta \right )}{\left (-c\left (t+h\right )+\delta +\lambda _{1}\right )\left (t+h+\lambda _{2}\right )+\rho \left (t+h\right )\left (c\left (t+h\right )-\delta \right )}. \end{equation}

In order to compute the GS function of Proposition 3, we are in need of a real root $\gamma \,:\!=\,\gamma _{z,\delta }$ of the equation

(3.4) \begin{equation} z\mathbb{M}_{T,X}\left (c\gamma -\delta ,-\gamma \right )=1\Leftrightarrow z\frac {\lambda _{1}\lambda _{2}}{\left (\lambda _{1}-\left (c\gamma -\delta \right )\right )\left (\lambda _{2}+\gamma \right )+\rho \left (c\gamma -\delta \right )\gamma }=1. \end{equation}

The above equation can be simplified to the following quadratic equation

(3.5) \begin{equation} c\left (1-\rho \right )\gamma ^{2}-\left (\lambda _{1}-c\lambda _{2}+\delta \left (1-\rho \right )\right )\gamma -\left (\delta \lambda _{2}+\left (1-z\right )\lambda _{1}\lambda _{2}\right )=0 \end{equation}

where, for every $z\in \left (0,1\right ]$ and $\delta \geq 0$ , there exist two real solutions $\gamma ^{(1)},\gamma ^{(2)}$ of the form

(3.6) \begin{eqnarray} \gamma ^{(i)}&=&\frac {\lambda _{1}-c\lambda _{2}+\delta \left (1-\rho \right )+(-1)^{i}\sqrt {\left (\lambda _{1}-c\lambda _{2}+\delta \left (1-\rho \right )\right )^{2}+4c\lambda _{2}\left (1-\rho \right )\left (\delta +\left (1-z\right )\lambda _{1}\right )}}{2c\left (1-\rho \right )},\nonumber\\i&=&1,2. \end{eqnarray}

If $c\lambda _{2}\gt \lambda _{1}$ , the real roots $\gamma ^{(1)},\gamma ^{(2)}$ satisfy the inequalities $\gamma ^{(1)}\lt 0\leq \gamma ^{(2)}$ while $\gamma ^{(2)}=0$ when $z=1$ and $\delta =0$ . If $c\lambda _{2}\lt \lambda _{1}$ , then $\gamma ^{(1)}\leq 0\lt \gamma ^{(2)}$ and $\gamma ^{(1)}=0$ when $z=1,\delta =0$ . In the last case where $c\lambda _{2}=\lambda _{1}$ , we verify that $\gamma ^{(1)}\leq 0\leq \gamma ^{(2)}$ with $\gamma ^{(1)}=-\gamma ^{(2)}$ and $\gamma ^{(1)}=\gamma ^{(2)}=0$ when $z=1,\delta =0$ . Obviously, the above cases depend on the sign of the loading factor $\eta =c\mathbb{E}\left (T_{1}\right )/\mathbb{E}\left (X_{1}\right )-1=c\lambda _{2}/\lambda _{1}-1,$ since $c\lambda _{2}\gt \lambda _{1}\Leftrightarrow \eta \gt 0$ . The condition $\eta \gt 0$ is the net profit condition that an insurance company usually sets in order to avoid the ruin event, which is certain if $\eta \leq 0$ .

By choosing the smaller root $\gamma ^{(1)}$ in (3.6) as the parameter of the tilted measure, we conveniently get that ${{\mathbb{P}_{\gamma ^{(1)},\delta }\left (\tau \lt \infty \right )}}=1$ (cf. Lemma 5 below). Therefore, in what follows, we shall proceed our analysis with $\gamma =\gamma _{z,\delta }\,:\!=\,\gamma ^{(1)}$ . The distribution, under $\mathbb{P}_{\gamma ,\delta }$ , of the increments of the random walk $S$ is described in the next result.

Lemma 5. For $\gamma =\gamma ^{(1)}$ given by ( 3.6 ), under the measure $\mathbb{P}_{\gamma ,\delta }$ , the increments $Z_{1},Z_{2},\ldots$ are double-exponential or asymmetric Laplace (AL $\left (0,\alpha ,\beta \right )$ ) distributed with location parameter 0 and MGF of the form

(3.7) \begin{equation} \mathbb{E}_{\gamma ,\delta }\left (e^{tZ_{1}}\right )=\frac {1}{1-\alpha t-\frac {1}{2}\beta ^{2}t^{2}}=p\frac {\omega _{1}}{\omega _{1}+t}+\left (1-p\right )\frac {\omega _{2}}{\omega _{2}-t}, \end{equation}

for $t\in (-\omega _{1},\omega _{2})$ , where

(3.8) \begin{eqnarray} \alpha &=&\alpha \left (z,\delta \right )\,:\!=\,-\frac {\sqrt {\left (\lambda _{1}-c\lambda _{2}+\delta \left (1-\rho \right )\right )^{2}+4c\lambda _{2}\left (1-\rho \right )\left (\delta +\left (1-z\right )\lambda _{1}\right )}}{z\lambda _{1}\lambda _{2}},\nonumber\\[8pt]\beta &=&\beta \left (z\right )\,:\!=\,\sqrt {\frac {2c\left (1-\rho \right )}{z\lambda _{1}\lambda _{2}}}, \end{eqnarray}

and

(3.9) \begin{equation} \omega _{i}\,:\!=\,\frac {\left (-1\right )^{i+1}\alpha +\sqrt {\alpha ^{2}+2\beta ^{2}}}{\beta ^{2}}\gt 0,\;i=1,2,\,\,\,\,p\,:\!=\,\frac {\sqrt {\alpha ^{2}+2\beta ^{2}}-\alpha }{2\sqrt {\alpha ^{2}+2\beta ^{2}}}=\frac {\omega _{2}}{\omega _{1}+\omega _{2}}\in \left (0,1\right ). \end{equation}

In addition, $\mathbb{P}_{\gamma ,\delta }\left (\tau \lt \infty \right )=1$ provided that $\eta \neq 0$ or $\delta \gt 0$ or $z\neq 1$ .

Proof. For $\gamma =\gamma ^{(1)}$ , using (3.3), we get

\begin{align*} \mathbb{E}_{\gamma ,\delta }\left (e^{tZ_{1}}\right ) & =\frac {z\lambda _{1}\lambda _{2}}{z\lambda _{1}\lambda _{2}+\left (\lambda _{1}-c\lambda _{2}+\left (1-\rho \right )\left (\delta -2c\gamma \right )\right )t-c\left (1-\rho \right )t^{2}}\\ & =\frac {z\lambda _{1}\lambda _{2}}{z\lambda _{1}\lambda _{2}+\sqrt {\left (\lambda _{1}-c\lambda _{2}+\delta \left (1-\rho \right )\right )^{2}+4c\lambda _{2}\left (1-\rho \right )\left (\delta +\left (1-z\right )\lambda _{1}\right )}t-c\left (1-\rho \right )t^{2}}, \end{align*}

and using the notations (3.8) for $\alpha$ and $\beta$ , we immediately get that $Z_{1}\sim$ AL $\left (0,\alpha ,\beta \right )$ under $\mathbb{P}_{\gamma ,\delta }$ . Since the increments of the random walk are AL $\left (0,\alpha ,\beta \right )$ distributed, then under the assumption $\eta \neq 0$ or $\delta \gt 0$ or $z\neq 1$ , we have $\mathbb{E}_{\gamma ,\delta }\left (Z_{1}\right )=\alpha \lt 0$ , and hence, under $\mathbb{P}_{\gamma ,\delta }$ , the process $\left (S_{k},k=1,2,\ldots \right )$ has negative drift, concluding that $\mathbb{P}_{\gamma ,\delta }\left (\tau \lt \infty \right )=1$ .

The next proposition offers an explicit formula for the joint GF of the number of claims until ruin, the deficit at ruin, and the time until ruin.

Proposition 6. The joint GF of $\left (\nu ,-U_{\tau },\tau \right )$ , is given by

(3.10) \begin{equation} \phi _{z,\delta ,s}\left (u\right )\,:\!=\,\mathbb{E}\left (z^{\nu }e^{s\left (-U_{\tau }\right )-\delta \tau }1_{[\tau \lt \infty ]}\right )=e^{\gamma u}\frac {\omega _{1}}{\omega _{1}-\gamma -s}, \end{equation}

for every $z\in \left (0,1\right ],\delta \geq 0$ and $s\lt \omega _{1}-\gamma$ , where $\gamma =\gamma ^{(1)}$ is given by ( 3.6 ) and $\omega _{1}$ by ( 3.9 ). In addition, the ruin probability is given by

(3.11) \begin{equation} \psi =\mathbb{P}(\tau \lt \infty )=e^{\gamma ^{*}u}\frac {r_{1}+\gamma ^{*}}{r_{1}}, \end{equation}

where $r_{1}$ is given by ( 3.2 ) and $\gamma ^{*}\,:\!=\,\frac {\lambda _{1}-c\lambda _{2}-|\lambda _{1}-c\lambda _{2}|}{2c\left (1-\rho \right )}.$

Proof. Using the representation (3.7) for the distribution of $Z_{1},Z_{2},\ldots$ , under $\mathbb{P}_{\gamma ,\delta }$ , it is evident that the random walk $S$ makes exponentially distributed negative steps (with parameter $\omega _{1}$ ) with probability $p$ or exponentially distributed positive steps (with parameter $\omega _{2}$ ) with probability $1-p$ . Hence, from the memoryless property of the exponential distribution, the deficit at ruin will also be exponentially distributed with parameter $\omega _{1}$ . Therefore, we have that

(3.12) \begin{equation} \mathbb{B}_{\gamma ,\delta }\left (t\right )=\mathbb{E}_{\gamma ,\delta }\left (e^{t(-U_{\tau })}\mid \tau \lt \infty \right )=\frac {\omega _{1}}{\omega _{1}-t}. \end{equation}

The proof of (3.10) is now a straightforward application of Proposition 3 exploiting the fact $\psi _{\gamma ,\delta }=\mathbb{P}_{\gamma ,\delta }\left (\tau \lt \infty \right )=1$ (cf. Lemma 5). Finally, the relation (3.11) follows readily from (3.10) by setting $z=1,s=0,\delta =0$ and taking also into account that, in this case, $\omega _{1}=r_{1}+\gamma ^{*}$ .

Remark 7. If $\eta \leq 0$ , then $\gamma ^{*}=0$ , and hence, $\psi =1$ (cf. (3.11)). If $\eta \gt 0$ , we verify that $\gamma ^{*}=r_{2}-r_{1}=\frac {\lambda _{1}-c\lambda _{2}}{c\left (1-\rho \right )}$ , and hence, we have

(3.13) \begin{equation} \psi =\frac {r_{2}}{r_{1}}e^{\left (r_{2}-r_{1}\right )u}. \end{equation}

Note that in the last case, the formula for $\psi$ agrees with the one obtained by Ambagaspitiya (Reference Ambagaspitiya2009) using a completely different methodology.

Furthermore, utilizing the explicit form of (3.10), our analysis will be focused on the (defective) joint density, $f_{\nu ,-U_{\tau },\tau }$ , of $\left (\nu ,-U_{\tau },\tau \right )$ . The corresponding GS function reads

(3.14) \begin{equation} \phi _{z,\delta ,s}\left (u\right )=\mathbb{E}\left (z^{\nu }e^{s\left (-U_{\tau }\right )-\delta \tau }1_{[\tau \lt \infty ]}\right )=\sum _{m=1}^{\infty }z^{m}\intop _{0}^{\infty }\intop _{0}^{\infty }e^{sy}e^{-\delta t}f_{\nu ,-U_{\tau },\tau }\left (m,y,t\right )dydt, \end{equation}

and the density $f_{\nu ,-U_{\tau },\tau }$ can be derived by the inversion of $\phi _{z,\delta ,s}\left (u\right )$ with respect to $z,\delta$ and $s$ . In general, simultaneous inversions that involve three variables are difficult to derive analytically. As another option, the joint density of $\left (\nu ,-U_{\tau },\tau \right )$ can be determined using a single variable inversion of the transform

(3.15) \begin{equation} \ell _{\delta }\left (m,y,u\right )\,:\!=\,\intop _{0}^{\infty }e^{-\delta t}f_{\nu ,-U_{\tau },\tau }\left (m,y,t\right )dt, \end{equation}

where numerical inversion can be easily applied; see, for example, Abate and Whitt (Reference Abate and Whitt1992) for a detailed list of algorithms. Alternatively, for fixed discounted factor $\delta$ , the transform (3.15) can be seen as the discounted defective density of $\left (\nu ,-U_{\tau }\right )$ with corresponding Gerber–Shiu measure

\begin{equation*} \ell _{\delta }\left (m,dy,u\right )\,:\!=\,\mathbb{E}\left (e^{-\delta \tau };\nu =m,-U_{\tau }\in dy\right ),\;\;\;y\geq 0,m=1,2,\ldots \end{equation*}

which can be used to derive the distribution of the number of claims until ruin. In some cases, $\ell _{\delta }\left (m,y,u\right )$ can be expressed in terms of scale functions (see Biffis & Kyprianou, Reference Biffis and Kyprianou2010) or by renewal arguments (see Cheung & Woo, Reference Cheung and Woo2016). By utilizing the analytic formula (3.10), we shall be able to apply a straightforward approach. The inversion of (3.15) with respect to $\delta$ leads to $f_{\nu ,-U_{\tau },\tau }$ , while setting $\delta =0$ induces the joint density of the pair $\left (\nu ,-U_{\tau }\right )$ . Moreover, we can derive the (defective) probability function of the number of claims until ruin $f_{\nu }\left (m\right )\,:\!=\,\mathbb{P}\left (\nu =m\right )$ from $f_{\nu }\left (m\right )=\intop _{0}^{\infty }\ell _{0}\left (m,y,u\right )dy$ . In order to find the LT (3.15), we present a connection between $\phi _{z,\delta ,s}\left (u\right )$ and $\phi _{z,\delta ,s}\left (0\right )$ . If $u=0$ , then $\phi _{z,\delta ,s}\left (0\right )=\frac {\omega _{1}}{\omega _{1}-\gamma -s}$ , and by (3.10), we observe that

(3.16) \begin{equation} \phi _{z,\delta ,s}\left (u\right )=e^{\gamma u}\frac {\theta +\gamma }{\theta -s}=e^{-\theta u}\phi _{z,\delta ,s}\left (0\right )e^{\phi _{z,\delta ,s}\left (0\right )(\theta -s)u},\,\,u\gt 0, \end{equation}

where, for convenience, we have denoted by $\theta$ the quantity

(3.17) \begin{equation} \theta \,:\!=\,\omega _{1}-\gamma =\frac {\sqrt {\left (\lambda _{1}-c\lambda _{2}+\delta \left (1-\rho \right )\right )^{2}+4c\lambda _{2}\left (1-\rho \right )\left (\delta +\lambda _{1}\right )}-\left (\lambda _{1}-c\lambda _{2}+\delta \left (1-\rho \right )\right )}{2c\left (1-\rho \right )}, \end{equation}

which does not depend on $z$ (whereas $\omega _{1},\gamma$ are functions of $z$ ). Note also that $\theta =\lambda _{2}$ when $\rho =0$ . In the case where $T$ and $X$ are independent (i.e., $\rho =0$ ), a version of (3.16) was derived by Landriault et al. (Reference Landriault, Shi and Willmot2011) for $s=0$ and also by Frostig et al. (Reference Frostig, Pitts and Politis2012) for $\delta =s=0$ . Because of the connection offered by (3.16), we shall first investigate the case when $u=0$ . The next proposition offers an explicit formula for $\ell _{\delta }\left (m,y,0\right )$ .

Proposition 8. The defective discounted density of $\left (\nu ,-U_{\tau }\right )$ when $u=0$ is given by

\begin{equation*} \ell _{\delta }\left (m,y,0\right )=\frac {\left (2m-2\right )!}{m!\left (m-1\right )!}\frac {e^{-\theta y}\left (\lambda _{2}\lambda _{1}\right )^{m}\left (c\left (1-\rho \right )\right )^{m-1}}{\left (4c\lambda _{2}\left (1-\rho \right )\left (\delta +\lambda _{1}\right )+\left (c\lambda _{2}-\lambda _{1}-\delta \left (1-\rho \right )\right )^{2}\right )^{m-1/2}}, \end{equation*}

for every $y\gt 0,m=1,2,\ldots$ .

Proof. From (3.14) and (3.15), we express $\phi _{z,\delta ,s}\left (0\right )$ as

(3.18) \begin{equation} \phi _{z,\delta ,s}\left (0\right )=\sum _{m=1}^{\infty }\ell _{s,\delta }(m)z^{m}, \end{equation}

where $\ell _{s,\delta }(m)\,:\!=\,\intop _{0}^{\infty }e^{sy}\ell _{\delta }\left (m,y,0\right )dy,m=1,2,\ldots ,$ are the coefficients of the above power series. Using (3.14) and (3.10), we have

\begin{equation*} \phi _{z,\delta ,s}\left (0\right )=\frac {\omega _{1}}{\omega _{1}-\gamma -s}=\frac {\omega _{1}-\gamma }{\omega _{1}-\gamma -s}+\frac {1}{\omega _{1}-\gamma -s}\gamma =\frac {\theta }{\theta -s}+\frac {1}{\theta -s}\gamma , \end{equation*}

where the quantity $\theta =\omega _{1}-\gamma$ is constant w.r.t. $z$ . Therefore,

\begin{equation*} \ell _{s,\delta }(m)=\frac {1}{m!}\left .\frac {\partial ^{m}\phi _{z,\delta ,s}\left (0\right )}{\partial z^{m}}\right |_{z=0}=\frac {1}{m!(\theta -s)}\cdot \left .\frac {\partial ^{m}}{\partial z^{m}}\gamma \right |_{z=0},\;\;\;m=1,2,\ldots \end{equation*}

and using induction, we readily obtain,

\begin{equation*} \left .\frac {\partial ^{m}}{\partial z^{m}}\gamma \right |_{z=0}=\frac {\left (2m-2\right )!}{\left (m-1\right )!}\frac {\left (\lambda _{2}\lambda _{1}\right )^{m}\left (c\left (1-\rho \right )\right )^{m-1}}{\left (4c\lambda _{2}\left (1-\rho \right )\left (\delta +\lambda _{1}\right )+\left (c\lambda _{2}-\lambda _{1}-\delta \left (1-\rho \right )\right )^{2}\right )^{m-1/2}}. \end{equation*}

We can verify that the expression of $\phi _{z,\delta ,s}\left (0\right )$ in (3.18) is proper, since

\begin{equation*} R^{\left (\delta \right )}\,:\!=\,\lim _{m\rightarrow \infty }\frac {\ell _{s,\delta }(m)}{\ell _{s,\delta }(m+1)}=\frac {4c\lambda _{2}\left (1-\rho \right )\left (\delta +\lambda _{1}\right )+\left (c\lambda _{2}-\lambda _{1}-\delta \left (1-\rho \right )\right )^{2}}{4\lambda _{2}\lambda _{1}c\left (1-\rho \right )}\gt 0, \end{equation*}

for every $s,\delta \geq 0$ , and hence, the power series converges within the radius $R^{\left (\delta \right )}$ . Finally, the transform $\ell _{\delta }\left (m,y,0\right )$ follows by a direct inversion of $\ell _{s,\delta }(m)$ with respect to $s$ .

In the following proposition, we derive $\ell _{\delta }\left (m,y,u\right )$ for every $u\gt 0$ with the help of the connection (3.16) and some properties of the generalized binomial series.

Proposition 9. The defective discounted density, $\ell _{\delta }\left (m,y,u\right ),$ has the following explicit form

(3.19) \begin{equation} \ell _{\delta }\left (m,y,u\right )=e^{-\theta \left (u+y\right )}\left (\lambda _{2}\lambda _{1}\right )^{m}\sum _{\kappa =0}^{m-1}\frac {u^{\kappa }}{\kappa !}\left (\begin{array}{c} 2m-\kappa -1\\ m-\kappa -1 \end{array}\right )\frac {\kappa +1}{2m-\kappa -1}\frac {\left (c(1-\rho )\right )^{m-\kappa -1}}{\left (\sqrt {G\left (\delta \right )}\right )^{2m-\kappa -1}}, \end{equation}

for every $y\gt 0,\;m=1,2,\ldots$ , where

(3.20) \begin{equation} G\left (\delta \right )\,:\!=\,4c\lambda _{2}\left (1-\rho \right )\left (\delta +\lambda _{1}\right )+\left (c\lambda _{2}-\lambda _{1}-\delta \left (1-\rho \right )\right )^{2}. \end{equation}

Proof. From (3.16), using the power series expansion of the exponential function, we have

\begin{equation*} \phi _{z,\delta ,s}\left (u\right )=e^{-\theta u}\sum _{m=1}^{\infty }\frac {u^{m-1}(\theta -s)^{m-1}}{\left (m-1\right )!}\left (\phi _{z,\delta ,s}\left (0\right )\right )^{m}. \end{equation*}

We write $\phi _{z,\delta ,s}\left (0\right )$ as (3.18), taking advantage of the closed form of the coefficients $\ell _{s,\delta }(m)$ , where we observe that

\begin{align*} \phi _{z,\delta ,s}\left (0\right ) & =\sum _{k=1}^{\infty }z^{k}\cdot \frac {\left (2k-2\right )!}{k!(\theta -s)\left (k-1\right )!}\cdot \frac {\left (\lambda _{2}\lambda _{1}\right )^{k}\left (c\left (1-\rho \right )\right )^{k-1}}{\left (4c\lambda _{2}\left (1-\rho \right )\left (\delta +\lambda _{1}\right )+\left (c\lambda _{2}-\lambda _{1}-\delta \left (1-\rho \right )\right )^{2}\right )^{k-1/2}}\\[6pt] & =\frac {\sqrt {G\left (\delta \right )}}{c\left (1-\rho \right )(\theta -s)}\sum _{k=1}^{\infty }\frac {\left (2k-2\right )!}{k!\left (k-1\right )!}\cdot \left (\frac {zc\lambda _{2}\lambda _{1}\left (1-\rho \right )}{G\left (\delta \right )}\right )^{k}\\[6pt] & =\frac {z\lambda _{2}\lambda _{1}}{(\theta -s)\sqrt {G\left (\delta \right )}}\sum _{k=0}^{\infty }\left (\begin{array}{c} 2k+1\\[6pt] k \end{array}\right )\frac {1}{2k+1}\cdot \left (\frac {zc\lambda _{2}\lambda _{1}\left (1-\rho \right )}{G\left (\delta \right )}\right )^{k}\\[6pt] & =\frac {z\lambda _{2}\lambda _{1}}{(\theta -s)\sqrt {G\left (\delta \right )}}\mathcal{B}_{2}\left (\frac {zc\lambda _{2}\lambda _{1}\left (1-\rho \right )}{G\left (\delta \right )}\right ), \end{align*}

where the function $\mathcal{B}_{n}\left (z\right )\,:\!=\,\sum _{k=0}^{\infty }\left (\begin{array}{c} nk+1\\ k \end{array}\right )\frac {z^{k}}{nk+1}$ is known as the generalized binomial series (e.g., see Dickson, Reference Dickson2012). Hence, we have

\begin{equation*} \phi _{z,\delta ,s}\left (u\right )=e^{-\theta u}\sum _{m=1}^{\infty }\frac {u^{m-1}}{\left (m-1\right )!(\theta -s)}\left (\frac {z\lambda _{2}\lambda _{1}}{\sqrt {G\left (\delta \right )}}\right )^{m}\left (\mathcal{B}_{2}\left (\frac {zc\lambda _{2}\lambda _{1}\left (1-\rho \right )}{G\left (\delta \right )}\right )\right )^{m}, \end{equation*}

and using the following property for the $m$ -th power of the generalized binomial series,

\begin{equation*} \left (\mathcal{B}_{n}\left (x\right )\right )^{m}=\sum _{k=0}^{\infty }\left (\begin{array}{c} nk+m\\ k \end{array}\right )\frac {mx^{k}}{nk+m} \end{equation*}

we get

\begin{align*} \phi _{z,\delta ,s}\left (u\right ) & =\frac {e^{-\theta u}}{(\theta -s)}\sum _{m=1}^{\infty }\frac {u^{m-1}}{\left (m-1\right )!}\left (\frac {z\lambda _{2}\lambda _{1}}{\sqrt {G\left (\delta \right )}}\right )^{m}\sum _{k=0}^{\infty }\left (\begin{array}{c} 2k+m\\ k \end{array}\right )\frac {m}{2k+m}\left (\frac {zc\lambda _{2}\lambda _{1}\left (1-\rho \right )}{G\left (\delta \right )}\right )^{k}\\ & =\frac{e^{-\theta u}}{\left(\theta -s\right)}\!\sum _{m=1}^{\infty }\!\sum _{k=0}^{\infty }\frac{u^{m-1}}{\left (m-1\right )!}\!\left(\begin{array}{c} 2k+m\\ k \end{array}\!\right)\!\frac{m}{2k+m}\!\left(\frac {\sqrt {G\left (\delta \right )}}{c(1-\rho )}\!\right)^{m}\!\left(\frac {c\lambda _{2}\lambda _{1}(1-\rho )}{G\left (\delta \right )}\!\right)^{k+m}z^{m+k}. \end{align*}

By applying the transformation $i=k+m,\kappa =m-1\,(k=i-\kappa -1,m=\kappa +1)$ , the above double sum can equivalently be written in the form

\begin{equation*} \phi _{z,\delta ,s}\left (u\right )=\frac {e^{-\theta u}}{(\theta -s)}\sum _{i=1}^{\infty }\sum _{\kappa =0}^{i-1}\frac {u^{\kappa }}{\kappa !}\left (\begin{array}{c} 2i-\kappa -1\\ i-\kappa -1 \end{array}\right )\frac {\kappa +1}{2i-\kappa -1}\left (\frac {\sqrt {G\left (\delta \right )}}{c(1-\rho )}\right )^{\kappa +1}\left (\frac {c\lambda _{2}\lambda _{1}(1-\rho )}{G\left (\delta \right )}\right )^{i}z^{i}, \end{equation*}

and since (cf. (3.14) and(3.15))

\begin{equation*} \phi _{z,\delta ,s}\left (u\right )=\sum _{m=1}^{\infty }z^{m}\intop _{0}^{\infty }e^{sy}\ell _{\delta }\left (m,y,u\right )dy. \end{equation*}

It follows that

\begin{equation*} \intop _{0}^{\infty }e^{sy}\ell _{\delta }(m,y,u)dy=\frac {e^{-\theta u}}{(\theta -s)}\left (\frac {c\lambda _{2}\lambda _{1}(1-\rho )}{G\left (\delta \right )}\right )^{m}\sum _{\kappa =0}^{m-1}\frac {u^{\kappa }}{\kappa !}\left (\begin{array}{c} 2m-\kappa -1\\ m-\kappa -1 \end{array}\right )\frac {\kappa +1}{2m-\kappa -1}\left (\frac {\sqrt {G\left (\delta \right )}}{c(1-\rho )}\right )^{\kappa +1}\!, \end{equation*}

and the inversion with respect to $s$ leads to the desired result.

The analytical inversion of transform $\ell _{\delta }\left (m,y,u\right )$ with respect to $\delta$ is not an easy task. However, in the case when $\rho =0$ , we see that $\theta =\lambda _{2},G\left (\delta \right )=\left (c\lambda _{2}+\lambda _{1}+\delta \right )^{2}$ (cf. (3.17) and (3.20)) and $\ell _{\delta }\left (m,y,u\right )$ (cf. (3.19)) reduces to

\begin{align*} \ell _{\delta }\left (m,y,u\right ) & =e^{-\lambda _{2}\left (u+y\right )}\left (\lambda _{1}\lambda _{2}\right )^{m}\sum _{k=0}^{m-1}\frac {\left (2m-k-2\right )!\left (k+1\right )u^{k}}{m!k!\left (m-k-1\right )!}\frac {c^{m-k-1}}{\left (c\lambda _{2}+\lambda _{1}+\delta \right )^{2m-k-1}}, \end{align*}

which can now be easily inverted with respect to $\delta$ (cf. (3.15)) leading to

(3.21) \begin{equation} f_{\nu ,-U_{\tau },\tau }\left (m,y,t\right )=e^{-\lambda _{2}\left (u+y\right )}\left (\lambda _{1}\lambda _{2}\right )^{m}\sum _{k=0}^{m-1}\frac {\left (k+1\right )u^{k}c^{m-k-1}}{m!k!\left (m-k-1\right )!}t^{2\left (m-1\right )-k}e^{-\left (c\lambda _{2}+\lambda _{1}\right )t}, \end{equation}

for every $t\gt 0,y\gt 0,m=1,2,\ldots$ . The above density can be considered as an explicit solution for the recursive formula of Dickson (Reference Dickson2012). In addition, integrating (3.21) with respect to $y$ over $\left (0,\infty \right )$ , we get the joint density of $\left (\nu ,\tau \right )$ . A more general result regarding the joint distribution of $T$ and $X$ was derived by Zhao and Zhang (Reference Zhao and Zhang2013) via an alternative methodology. Despite the difficulty inverting $\ell _{\delta }\left (m,y,u\right )$ w.r.t. $\delta$ when $\rho \gt 0$ , Proposition 9 allows us to explicitly derive the joint density of $\left (\nu ,-U_{\tau }\right )$ and the (defective) probability function of $\nu$ as we see in the next corollary.

Corollary 10. The (defective) probability function of the number of claims, $f_{\nu }(m)$ is given by

(3.22) \begin{equation} f_{\nu }(m)=\frac {e^{-r_{1}u}}{r_{1}}\sum _{k=0}^{m-1}\frac {\left (2m-k-2\right )!\left (k+1\right )u^{k}}{m!k!\left (m-k-1\right )!}\frac {\left (\lambda _{1}\lambda _{2}\right )^{m}\left (c\left (1-\rho \right )\right )^{m-k-1}}{\left (\sqrt {\left (c\lambda _{2}+\lambda _{1}\right )^{2}-4c\lambda _{2}\lambda _{1}\rho }\right )^{2m-k-1}}, \end{equation}

for every $m=1,2,\ldots ,$ where $r_{1}$ is given by ( 3.2 ), while $\sum _{m=1}^{\infty }f_{\nu }(m)=\psi$ . In addition, the joint (defective) density of $\left (\nu ,-U_{\tau }\right )$ is given by

(3.23) \begin{equation} f_{\nu ,-U_{\tau }}\left (m,y\right )=f_{\nu }(m)f_{-U_{\tau }|\tau \lt \infty }(y), \end{equation}

for every $y\gt 0,\;m=1,2,\ldots$ , where $f_{-U_{\tau }|\tau \lt \infty }(y)=r_{1}e^{-r_{1}y},y\geq 0$ .

Proof. From (3.15), we deduce that $f_{\nu ,-U_{\tau }}\left (m,y\right )=\ell _{0}\left (m,y,u\right )$ . Employing (3.19) for $\delta =0$ and invoking the fact that $f_{\nu }(m)=\intop _{0}^{\infty }\ell _{0}\left (m,y,u\right )dy$ , we get (3.22) since, in this case, $\theta =r_{1}$ and $G\left (0\right )=\left (c\lambda _{2}+\lambda _{1}\right )^{2}-4c\lambda _{2}\lambda _{1}\rho$ . The expression (3.23) follows again from (3.19) by verifying that $\ell _{0}\left (m,y,u\right )=f_{\nu }(m)r_{1}e^{-r_{1}y}$ . Finally (cf. (3.12)), when $\eta \neq 0,$

\begin{eqnarray*} \sum _{m=1}^{\infty }f_{\nu }(m)&=&\sum _{m=1}^{\infty }\intop _{0}^{\infty }\ell _{0}\left (m,y,u\right )dy\\&=&\sum _{m=1}^{\infty }\intop _{0}^{\infty }\intop _{0}^{\infty }f_{\nu ,-U_{\tau },\tau }\left (m,y,t\right )dtdy=\mathbb{E}\left (1_{[\tau \lt \infty ]}\right )=\mathbb{P}(\tau \lt \infty )=\psi . \end{eqnarray*}

Moment-based approximation algorithms have been widely used in order to estimate the distribution of the variables of interest, when a number of moments are finite, see, for example, Cossette et al. (Reference Cossette, Landriault, Marceau and Moutanabbir2016). Hence, employing Proposition 6, we can immediately derive the (defective) probability generating function (PGF) of the number of claims until ruin, the MGF of the deficit at ruin, the LT of the ruin time, and the MGF of total claim amount until ruin (see the corollary that follows). From these transforms, we can easily get the moments for the RVs of our interest.

Corollary 11. The (defective) marginal GF of $\nu$ , $-U_{\tau }$ , $\tau$ and $\sum _{i=1}^{\nu }X_{i}$ are the following

\begin{equation*} \mathbb{E}\left (z^{\nu }1_{[\tau \lt \infty ]}\right )=e^{\gamma _{z,0}u}\frac {r_{1}+\gamma _{z,0}}{r_{1}},\;\;\;\mathbb{E}\left (e^{s\left (-U_{\tau }\right )}1_{[\tau \lt \infty ]}\right )=e^{\gamma ^{*}u}\frac {r_{1}+\gamma ^{*}}{r_{1}-s}=\psi \frac {r_{1}}{r_{1}-s}, \end{equation*}

\begin{equation*} \mathbb{E}\left (e^{-\delta \tau }1_{[\tau \lt \infty ]}\right )=e^{\gamma _{1,\delta }u}\frac {R(\delta )}{R(\delta )-\gamma _{1,\delta }},\,\,\,\mathbb{E}\left (e^{s\sum _{i=1}^{\nu }X_{i}}1_{[\tau \lt \infty ]}\right )=e^{(\gamma _{1,-sc}+s)u}\frac {R(-sc)}{R(-sc)-\gamma _{1,-sc}-s}, \end{equation*}

where $R(\delta )=\omega _{1}$ with $z=1,$ that is,

\begin{equation*} R(\delta )\,:\!=\,\frac {\sqrt {G\left (\delta \right )}-\sqrt {G\left (\delta \right )-\lambda _{1}\lambda _{2}4c\left (1-\rho \right )}}{2c\left (1-\rho \right )}, \end{equation*}

and $G(\delta )$ is given in ( 3.20 ), while $\gamma ^{*}=\gamma _{1,0},\gamma _{z,0},\gamma _{1,\delta }$ and $\gamma _{1,-sc}$ are special cases of ( 3.6 ) ( $r_{1},\omega _{1}$ are given by ( 3.2 ) and ( 3.9 )).

Remark 12. Another representation of the GS function in (3.10), using the notation $\theta =\omega _{1}-\gamma$ , (cf. (3.17)), is the following

\begin{equation*} \phi _{z,\delta ,s}\left (u\right )=e^{\gamma u}\frac {\omega _{1}}{\omega _{1}-\gamma -s}=e^{\gamma u}\frac {\omega _{1}}{\omega _{1}-\gamma }\frac {\theta }{\theta -s}=\phi _{z,\delta ,0}\left (u\right )\frac {\theta }{\theta -s}=\mathbb{E}\left (z^{\nu }e^{-\delta \tau }1_{[\tau \lt \infty ]}\right )\frac {\theta }{\theta -s}. \end{equation*}

When $\rho =0$ , we verify that $\theta =\lambda _{2}$ and hence

\begin{equation*} \phi _{z,\delta ,s}\left (u\right )=\phi _{z,\delta ,0}\left (u\right )\frac {\lambda _{2}}{\lambda _{2}-s}=\mathbb{E}\left (z^{\nu }e^{-\delta \tau }1_{[\tau \lt \infty ]}\right )\mathbb{E}\left (e^{s\left (-U_{\tau }\right )}1_{[\tau \lt \infty ]}\right )\frac {1}{\psi }, \end{equation*}

which implies that the deficit at ruin, $-U_{\tau },$ and the pair $\left (\nu ,\tau \right )$ are conditionally independent on the ruin event. This observation does not hold true when $\rho \gt 0$ because, in this case, $\theta$ depends on $\delta$ .

It is remarkable that, exploiting the special form of the MGF in the Downton–Moran bivariate exponential model (cf. (3.1)), we were able to derive closed-form formulae for the distributions of interest, a task which is not always easy to accomplish in other bivariate models. In the next application, we shall exploit once more the form of the MGF in (3.1) and proceed to the derivation of the GS function from Proposition 3, assuming a direct generalization of the Downton–Moran model.

4. Kibble–Moran bivariate Erlang analysis

In this section, we assume that the pair $\left (T,X\right )$ follows a Kibble–Moran bivariate Erlang distribution, with MGF

\begin{equation*} \mathbb{M}_{T,X}\left (t,x\right )=\left (\frac {\lambda _{1}\lambda _{2}}{\left (\lambda _{1}-t\right )\left (\lambda _{2}-x\right )-\rho tx}\right )^{b}, \end{equation*}

where the marginal distribution of $T$ is Erlang with parameters $b\in \{1,2,\ldots \},\lambda _{1}\gt 0$ ( ${\mathcal{ER}}\left (b,\lambda _{1}\right )$ ), while the marginal distribution of the claim sizes $X$ is ${\mathcal{ER}}\left (b,\lambda _{2}\right )$ . This distribution can be seen as a natural generalization of the distribution used in Section 3. The MGF of the increments $Z_{1},Z_{2},\ldots$ now is

(4.1) \begin{align} \mathbb{M}_{Z}\left (t\right ) & =\mathbb{M}_{T,X}\left (ct,-t\right )=\left (\frac {\lambda _{1}\lambda _{2}}{\left (\lambda _{1}-ct\right )\left (\lambda _{2}+t\right )+c\rho t^{2}}\right )^{b}=\left (\frac {r_{1}}{r_{1}+t}\right )^{b}\left (\frac {r_{2}}{r_{2}-t}\right )^{b}, \end{align}

where $r_{i},i=1,2,$ are as in (3.2). Clearly, every $Z_{i}$ has the same distribution as the difference between two independent Erlang distributed RVs $W_{2}\sim {\mathcal{ER}}\left (b,r_{2}\right )$ and $W_{1}\sim {\mathcal{ER}}\left (b,r_{1}\right ),$ that is, $Z_{1}\stackrel {d}{=}W_{2}-W_{1}$ . In order to apply Proposition 3, it is essential to find a real root, to be denoted by $\tilde {\gamma }$ , of the equation $z\mathbb{M}_{T,X}\left (c\tilde {\gamma }-\delta ,-\tilde {\gamma }\right )=1$ . We observe that $\tilde {\gamma }$ is also a root of the equation

\begin{equation*} \left (z\mathbb{M}_{T,X}\left (c\tilde {\gamma }-\delta ,-\tilde {\gamma }\right )\right )^{1/b}=1\Leftrightarrow \frac {z^{1/b}\lambda _{1}\lambda _{2}}{\left (\lambda _{1}-\left (c\tilde {\gamma }-\delta \right )\right )\left (\lambda _{2}+\tilde {\gamma }\right )+\rho \left (c\tilde {\gamma }-\delta \right )\tilde {\gamma }}=1, \end{equation*}

and using the results of the previous sections, we conclude that a negative solution of $z\mathbb{M}_{T,X}\left (c\tilde {\gamma }-\delta ,-\tilde {\gamma }\right )=1$ is given by $\tilde {\gamma }\,:\!=\,\gamma ^{(1)}$ (cf. (3.6)) with $z$ now replaced by $z^{1/b}.$ With analogous arguments used in Lemma 5, the MGF of $Z_{1}$ under $\mathbb{P}_{\tilde {\gamma },\delta }$ is now

(4.2) \begin{equation} \mathbb{E}_{\tilde {\gamma },\delta }\left (e^{tZ_{1}}\right )=\left (1-\tilde {\alpha }t-\frac {1}{2}\tilde {\beta }^{2}t^{2}\right )^{-b}=\left (\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}+t}\right )^{b}\left (\frac {\tilde {\omega }_{2}}{\tilde {\omega }_{2}-t}\right )^{b},t\in (-\tilde {\omega }_{1},\tilde {\omega }_{2}) \end{equation}

where $\tilde {\alpha }=\alpha \left (z^{1/b},\delta \right ),\tilde {\beta }=\beta \left (z^{1/b}\right )$ are given by (3.8) and $\tilde {\omega }_{i}$ equals $\omega _{i}$ in (3.9), after replacing $z$ with $z^{1/b}$ . This MGF corresponds to a generalized AL distribution with location parameter 0 (GAL $\left (0,\tilde {\alpha },\tilde {\beta },b\right )$ ), which is the sum of $b$ IID AL $\left (0,\tilde {\alpha },\tilde {\beta }\right )$ distributed RVs (e.g., see Kotz et al., Reference Kotz, Kozubowski and Podgorski2012 Chapter 4).

The next lemma offers an alternative way to express the distribution of the increments $Z_{1},Z_{2},\ldots$ under $\mathbb{P}_{\tilde {\gamma },\delta }$ which will be proved to be convenient for the determination of the distribution of the deficit at ruin.

Lemma 13. Under $\mathbb{P}_{\tilde {\gamma },\delta }$ , the increments $Z_{1},Z_{2},\ldots$ are mixtures of two independent mixed-Erlang ( $\mathcal{ME}$ ) distributed RVs with MGF

(4.3) \begin{equation} \mathbb{E}_{\tilde {\gamma },\delta }\left (e^{tZ_{1}}\right )=\sum _{i=1}^{b}h_{i}\left (\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}+t}\right )^{i}+\sum _{i=1}^{b}d_{i}\left (\frac {\tilde {\omega }_{2}}{\tilde {\omega }_{2}-t}\right )^{i},\;\;\;t\in \left (-\tilde {\omega }_{1},\tilde {\omega }_{2}\right ) \end{equation}

where

\begin{equation*} h_{i}=\left (\begin{array}{c} 2b-i-1\\ b-i \end{array}\right )\frac {\tilde {\omega }_{1}^{b-i}\tilde {\omega }_{2}^{b}}{\left (\tilde {\omega }_{1}+\tilde {\omega }_{2}\right )^{2b-i}},\;\;d_{i}=\left (\begin{array}{c} 2b-i-1\\ b-i \end{array}\right )\frac {\tilde {\omega }_{1}^{b}\tilde {\omega }_{2}^{b-i}}{\left (\tilde {\omega }_{1}+\tilde {\omega }_{2}\right )^{2b-i}},\;\;\;i=1,\ldots ,b. \end{equation*}

In addition , $\tilde {\psi }\,:\!=\,\mathbb{P}_{\tilde {\gamma },\delta }\left (\tau \lt \infty \right )=1$ provided that $\eta \neq 0$ or $\delta \gt 0$ or $z\neq 1$ .

Proof. It follows as a special case of Proposition 6.2 in Boutsikas et al. (Reference Boutsikas, Economides and Vaggelatou2024) for $d_{1}=\ldots =d_{b-1}=0,d_{b}=1,h_{1}=\ldots =h_{b-1}=0,h_{b}=1.$ Finally, in view of (4.2), the increments $Z_{i},i=1,\ldots$ are GAL $\left (0,\tilde {\alpha },\tilde {\beta },b\right )$ distributed and hence the drift of the random walk under $\mathbb{P}_{\tilde {\gamma },\delta }$ is negative since

(4.4) \begin{equation} \mathbb{E}_{\tilde {\gamma },\delta }\left (Z_{1}\right )=b\tilde {\alpha }=-b\frac {\sqrt {\left (\lambda _{1}-c\lambda _{2}+\delta \left (1-\rho \right )\right )^{2}+4c\lambda _{2}\left (1-\rho \right )\left (\delta +\left (1-z^{1/b}\right )\lambda _{1}\right )}}{z^{1/b}\lambda _{1}\lambda _{2}}\lt 0, \end{equation}

which implies that $\mathbb{P}_{\tilde {\gamma },\delta }\left (\tau \lt \infty \right )=1$ .

Now we are ready to offer a closed-form formula for the GS function of Proposition 3.

Proposition 14. The joint GF of $\left (\nu ,-U_{\tau },\tau \right )$ is given by

(4.5) \begin{equation} \phi _{z,\delta ,s}(u)=\mathbb{E}\left (z^{\nu }e^{s\left (-U_{\tau }\right )-\delta \tau }1_{[\tau \lt \infty ]}\right )=e^{u\tilde {\gamma }}\sum _{i=1}^{b}e^{\chi _{i}u}\left (\frac {\tilde {\omega }_{1}+\chi _{i}}{\tilde {\omega }_{1}-\tilde {\gamma }-s}\right )^{b}\prod _{\begin{array}{c} j=1,j\ne i\end{array}}^{b}\frac {\left (\chi _{j}+\tilde {\gamma }+s\right )}{\left (\chi _{j}-\chi _{i}\right )}, \end{equation}

for every $z\in \left (0,1\right ],\delta \geq 0,s\lt \tilde {\omega }_{1}-\tilde {\gamma }$ , where $\tilde {\gamma }$ and $\tilde {\omega }_{i}$ are equal to $\tilde {\gamma }\,:\!=\,\gamma ^{(1)}$ (cf. ( 3.6 )) and $\omega _{i}$ in ( 3.9 ), respectively, after replacing $z$ with $z^{1/b}$ , and

\begin{equation*} \chi _{j}=\frac {1}{2}\left (\tilde {\omega }_{2}-\tilde {\omega }_{1}-\sqrt {(\tilde {\omega }_{2}-\tilde {\omega }_{1})^{2}+4\tilde {\omega }_{1}\tilde {\omega }_{2}(1-e^{\frac {2(j-1)\pi }{b}\mathrm{i}})}\right ),\,\,j=1,2,\ldots ,b. \end{equation*}

Proof. We apply Proposition 3 with $\tilde {\psi }=1$ because of Lemma 13. It remains to express the MGF of the deficit at ruin, $\mathbb{B}_{\tilde {\gamma },\delta }\left (t\right )$ . From Lemma 13, we deduce that the increments $Z_{1},Z_{2},\ldots$ of the discrete time process $S=(S_{k},k=1,2,\ldots )$ are either positive, following a $\mathcal{ME}$ distribution, or negative, again following a $\mathcal{ME}$ distribution. Hence, invoking the generalized memoryless property of a $\mathcal{ME}$ RV (e.g., see Boutsikas & Politis, Reference Boutsikas and Politis2017), the deficit at ruin must again follow a $\mathcal{ME}$ distribution, and therefore, its MGF (under $\mathbb{P}_{\tilde {\gamma },\delta }$ ) possesses the following form

(4.6) \begin{equation} \mathbb{B}_{\tilde {\gamma },\delta }\left (t\right )=\mathbb{E}_{\tilde {\gamma },\delta }\left (e^{t(-U_{\tau })}\mid \tau \lt \infty \right )=\sum _{i=1}^{b}\pi _{i}\left (\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}-t}\right )^{i}, \end{equation}

for some unknown vector of probability weights $\boldsymbol{\pi }=\left (\pi _{1},\ldots ,\pi _{b}\right )^{\prime }$ . This vector can now be determined via the roots of Lundberg’s fundamental equation under $\mathbb{P}_{\tilde {\gamma },\delta }$ , that is, the roots with respect to $x$ of the equation

(4.7) \begin{equation} \mathbb{E}_{\tilde {\gamma },\delta }\left (e^{xZ_{1}}\right )=L(x)\,:\!=\,\sum _{i=1}^{b}h_{i}\left (\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}+x}\right )^{i}+\sum _{i=1}^{b}d_{i}\left (\frac {\tilde {\omega }_{2}}{\tilde {\omega }_{2}-x}\right )^{i}=1. \end{equation}

In view of (4.2), the roots of the above equation, $L(x)=1$ , (when $L(x)$ is extended in the whole complex plane) coincide with the roots of the equation

\begin{equation*} \left (\frac {(\tilde {\omega }_{1}+x)(\tilde {\omega }_{2}-x)}{\tilde {\omega }_{1}\tilde {\omega }_{2}}\right )^{b}=1\Longleftrightarrow \frac {(\tilde {\omega }_{1}+x)(\tilde {\omega }_{2}-x)}{\tilde {\omega }_{1}\tilde {\omega }_{2}}=e^{\frac {2k\pi }{b}\mathrm{i}},\,\,k=0,1,\ldots ,b-1. \end{equation*}

( $\mathrm{i}=\sqrt {-1}$ ). Hence, the roots $\chi _{1},\ldots ,\chi _{2b}$ of $L(x)=1$ are

\begin{equation*} \chi _{k+1}=\frac {1}{2}\left (\tilde {\omega }_{2}-\tilde {\omega }_{1}-\sqrt {(\tilde {\omega }_{2}-\tilde {\omega }_{1})^{2}+4\tilde {\omega }_{1}\tilde {\omega }_{2}(1-e^{\frac {2k\pi }{b}\mathrm{i}})}\right ),k=0,1,\ldots ,b-1, \end{equation*}

\begin{equation*} \chi _{b+k+1}=\frac {1}{2}\left (\tilde {\omega }_{2}-\tilde {\omega }_{1}+\sqrt {(\tilde {\omega }_{2}-\tilde {\omega }_{1})^{2}+4\tilde {\omega }_{1}\tilde {\omega }_{2}(1-e^{\frac {2k\pi }{b}\mathrm{i}})}\right ),k=0,1,\ldots ,b-1, \end{equation*}

where, it is obvious that, $\chi _{1}=0$ and $Re(\chi _{k})\lt 0$ for $k=2,3,\ldots ,b$ , while $Re(\chi _{k})\gt 0$ for $k=b+1,\ldots ,2b$ (when $\eta \neq 0$ ).

From the general equality (2.6) in the proof of Theorem1, we get that (for $\delta =0,V=1$ ), $\mathbb{E}\left (e^{h\sum _{i=1}^{\nu }Z_{i}}1_{[\nu \lt \infty ]}\right )=\mathbb{\mathbb{E}}_{h,0}\left (\mathbb{M}_{Z}\left (h\right )^{\nu }1_{[\nu \lt \infty ]}\right )$ . Since this is valid for every initial measure $\mathbb{P}$ , we deduce that it also holds for the ( $\tilde {\gamma },\delta$ )-tilted measure $\mathbb{P}_{\tilde {\gamma },\delta }$ , that is,

\begin{equation*} \mathbb{E}_{\tilde {\gamma },\delta }\left (e^{h\sum _{i=1}^{\nu }Z_{i}}\right )=\mathbb{\mathbb{E}}_{\tilde {\gamma }_{h},\delta }\left (L(h)^{\nu }\right ) \end{equation*}

for $h\,:\,Re(h)\lt 0$ , where the second expectation is under $\mathbb{P}_{\tilde {\gamma },\delta }$ , further tilted by ( $h,0$ ) (having taken also into account that $\mathbb{P}_{\tilde {\gamma },\delta }(\nu \lt \infty )=1$ and $\mathbb{P}_{\tilde {\gamma }_{h},\delta }(\nu \lt \infty )=1$ when $Re(h)\leq 0$ ). For $h$ in a neighborhood of zero (with $Re(h)\leq 0$ ), the above reads

\begin{equation*} e^{-hu}\sum _{i=1}^{b}\pi _{i}\left (\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}+h}\right )^{i}=\sum _{i=1}^{\infty }L(h)^{i}q_{i}. \end{equation*}

By analytic continuation, the above equality holds true also for the roots $\chi _{j},j=1,2,\ldots ,b$ , leading to the following system

(4.8) \begin{equation} \sum _{i=1}^{b}\pi _{i}\left (\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}+\chi _{j}}\right )^{i}=e^{\chi _{j}u},\;\;\;j=1,2,\ldots ,b. \end{equation}

Finally, the above system can be easily solved using the Lagrange interpolation formula (see also the Remark following this proposition). Consider the $b-th$ degree polynomial, $P(x)=\sum _{i=1}^{b}\pi _{i}x^{i}$ which, in view of (4.8), satisfies the equations

\begin{equation*} P\left (x_{j}\right )=y_{j}\text{ where }x_{j}=\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}+\chi _{j}},y_{j}=e^{\chi _{j}u}, \end{equation*}

for all $j=1,2,\ldots ,b,$ and also $P(0)=0$ . Hence,

(4.9) \begin{equation} P(x)=\sum _{i=1}^{b}\pi _{i}x^{i}=\sum _{i=1}^{b}y_{i}\frac {x}{x_{i}}\prod _{\begin{array}{c} j=1,j\ne i\end{array}}^{b}\frac {x-x_{j}}{x_{i}-x_{j}}. \end{equation}

Therefore, from (4.6) and the above, we get,

\begin{equation*} \mathbb{B}_{\tilde {\gamma },\delta }\left (t\right )=P\left (\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}-t}\right )=\sum _{i=1}^{b}e^{\chi _{i}u}\left (\frac {\tilde {\omega }_{1}+\chi _{i}}{\tilde {\omega }_{1}-t}\right )^{b}\prod _{\begin{array}{c} j=1,j\ne i\end{array}}^{b}\frac {\left (\chi _{j}+t\right )}{\left (\chi _{j}-\chi _{i}\right )}, \end{equation*}

from which, invoking Proposition 3, we derive (4.5).

Remark 15. The vector of probability weights $\boldsymbol{\pi }=\boldsymbol{\pi }\left (z,\delta ,u\right )=\left (\pi _{1},\ldots ,\pi _{b}\right )^{\prime }$ that determine the distribution of the deficit at ruin under $\mathbb{P}_{\tilde {\gamma },\delta }$ (cf. (4.6)) can explicitly be derived from Equation (4.9), exploiting the well-known identity,

\begin{align*} \prod _{\begin{array}{c} j=1\end{array}}^{n}(x+a_{j}) & =\sum _{k=0}^{n}x^{k}\sum _{1\leq j_{1}\lt j_{2}\lt \ldots \lt j_{n-k}\leq n}a_{j_{1}}a_{j_{2}}\ldots a_{j_{n-k}}. \end{align*}

More specifically, by equalizing the $i-th$ order coefficients of the two polynomials in (4.9), we get

(4.10) \begin{equation} \pi _{k}=\pi _{k}\left (z,\delta ,u\right )=(-1)^{b-k}\sum _{i=1}^{b}\frac {y_{i}}{x_{i}}\frac {\sum _{\begin{array}{c} 1\leq j_{1}\lt j_{2}\lt \ldots \lt j_{b-k}\leq b\\ j_{1},\ldots ,j_{b-k}\neq i \end{array}}x_{j_{1}}x_{j_{2}}\ldots x_{j_{b-k}}}{\prod _{\begin{array}{c} 1\leq j\leq b\\ j\neq i \end{array}}(x_{i}-x_{j})},k=1,2,\ldots ,b. \end{equation}

(null sums are assumed to be equal to 1) where $x_{j}=\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}+\chi _{j}},y_{j}=e^{\chi _{j}u}$ .

Using the same notations as the previous section, we denote the (defective) discounted density of $-U_{\tau }$ by $\ell _{\delta }\left (y,u\right )=\intop _{0}^{\infty }e^{-\delta t}f_{-U_{\tau },\tau }\left (y,t\right )dt,$ and the corresponding Gerber–Shiu measure by

\begin{equation*} \ell _{\delta }\left (dy,u\right )\,:\!=\,\mathbb{E}\left (e^{-\delta \tau };-U_{\tau }\in dy\right ),\;\;\;y\geq 0. \end{equation*}

The following corollary offers an explicit expression for $\ell _{\delta }\left (y,u\right )$ .

Corollary 16. The (defective) discounted density of the deficit at ruin, $-U_{\tau },$ is given by

(4.11) \begin{equation} \ell _{\delta }\left (y,u\right )=e^{\left (u+y\right )\gamma ^{\prime }-y\omega _{1}^{\prime }}\sum _{i=1}^{b}\pi _{i}\left (1,\delta ,u\right )\frac {\left (\omega _{1}^{\prime }\right )^{i}}{\left (i-1\right )!}y^{i-1},\;\;\;y\geq 0, \end{equation}

where $\pi _{k}\left (z,\delta ,u\right )$ is given in (4.10), and $\gamma ^{\prime },\omega _{i}^{\prime }$ are equal to $\gamma ^{(1)}$ , $\omega _{i}$ , respectively, (cf. ( 3.6 ) and ( 3.9 )), after replacing $z$ with $1$ . In addition, the discounted moments of the deficit at ruin are

(4.12) \begin{equation} \mathbb{E}\left (e^{-\delta \tau }\left (-U_{\tau }\right )^{k}1_{[\tau \lt \infty ]}\right )=e^{u\gamma ^{\prime }}\sum _{i=1}^{b}\pi _{i}\left (1,\delta ,u\right )\frac {\left (\omega _{1}^{\prime }\right )^{i}\prod _{n=0}^{k-1}\left (i+n\right )}{\left (\omega _{1}^{\prime }-\gamma ^{\prime }\right )^{i+k}}. \end{equation}

Proof. From Proposition 14, by setting $z=1$ we get (see also (4.6))

(4.13) \begin{align} \mathbb{E}\left (e^{s\left (-U_{\tau }\right )-\delta \tau }1_{[\tau \lt \infty ]}\right ) & =\intop _{0}^{\infty }e^{sy}\ell _{\delta }\left (y,u\right )dy=\phi _{1,\delta ,s}\left (u\right )=e^{u\gamma ^{\prime }}\sum _{i=1}^{b}\pi _{i}\left (1,\delta ,u\right )\left (\frac {\omega _{1}^{\prime }}{\omega _{1}^{\prime }-\gamma ^{\prime }-s}\right )^{i}\nonumber \\ & =e^{u\gamma ^{\prime }}\sum _{i=1}^{b}\pi _{i}\left (1,\delta ,u\right )\left (\frac {\omega _{1}^{\prime }}{\omega _{1}^{\prime }-\gamma ^{\prime }}\right )^{i}\mathbb{E}\left (e^{s\tilde {L_{i}}}\right ), \end{align}

where $\tilde {L_{i}}\sim {\mathcal{ER}}\left (i,\omega _{1}^{\prime }-\gamma ^{\prime }\right )$ . Straightforward inversion of $\phi _{1,\delta ,s}\left (u\right )$ with respect to $s$ gives us (4.11). Finally, (4.12) follows easily, for example, by differentiating $k$ times both sides of (4.13) w.r.t. $s$ , taking also into account that $\mathbb{E}\left (\tilde {L_{i}}^{k}\right )=\prod _{n=0}^{k-1}\left (i+n\right )/\left (\omega _{1}^{\prime }-\gamma ^{\prime }\right )^{k}$ .

Finally, the ruin probability follows readily by a direct application of Proposition 14 and the fact that $\tilde {\omega }_{1}=\omega _{1}=r_{1}+\gamma ^{*}$ when $z=1,s=0,\delta =0$ , that is,

\begin{equation*} \psi =e^{u\gamma ^{*}}\sum _{i=1}^{b}e^{\chi _{i}u}\left (\frac {r_{1}+\gamma ^{*}+\chi _{i}}{r_{1}}\right )^{b}\prod _{\begin{array}{c} j=1,j\ne i\end{array}}^{b}\frac {\left (\chi _{j}+\gamma ^{*}\right )}{\left (\chi _{j}-\chi _{i}\right )}, \end{equation*}

where $r_{1}$ is given by (3.2) and $\gamma ^{*}=\frac {\lambda _{1}-c\lambda _{2}-|\lambda _{1}-c\lambda _{2}|}{2c\left (1-\rho \right )}.$

Obviously, all formulas of Section 4 reduce to those of Section 3 when we set $b=1$ and $\pi _{1}\left (z,\delta ,u\right )=1$ . Because of the complexity of vector $\boldsymbol{\pi }\left (z,\delta ,u\right )$ that involves the roots $\chi _{j},j=1,\ldots ,b$ , which are also functions of $z$ and $\delta$ , it seems unfeasible to derive closed-form formulae for the joint discounted densities and probability functions as we were able to derive in some cases in Section 3. However, we can easily numerically compute the (defective) PGF of $\nu$ and the moments of $\nu$ , $\tau$ and $\sum _{i=1}^{\nu }X_{i}$ using numerical differentiation (see, e.g., Boutsikas et al., Reference Boutsikas, Economides and Vaggelatou2024, Section 7).

Finally, it is worth considering the asymptotic behavior of the GS function $\phi _{z,\delta ,s}(u)$ (cf. Proposition 14) as the initial surplus $u\to \infty .$ In the Cramer–Lundberg model, there are many cases where the ruin probability $\psi \left (u\right )\sim Ce^{-Ru}$ as $u\to \infty$ , with $R$ being the adjustment coefficient and $C$ is some positive constant. An analogous asymptotic result can be shown here for $\phi _{z,\delta ,s}(u)$ . More specifically, we observe that $e^{\chi _{1}u}=1$ since $\chi _{1}=0$ , while $e^{\chi _{i}u}\rightarrow 0$ as $u\rightarrow \infty ,$ since $Re\left (\chi _{i}\right )\lt 0,i=2,3,\ldots ,b$ . Thus, from (4.5), we readily get

\begin{align*} \phi _{z,\delta ,s}(u)e^{-u\tilde {\gamma }} & \rightarrow \left (\frac {\tilde {\omega }_{1}}{\tilde {\omega }_{1}-\tilde {\gamma }-s}\right )^{b}\prod _{\begin{array}{c} j=2\end{array}}^{b}\frac {\chi _{j}+\tilde {\gamma }+s}{\chi _{j}}=:C_{z,\delta ,s}^{\left (b\right )}. \end{align*}

Hence $\phi _{z,\delta ,s}(u)\sim C_{z,\delta ,s}^{\left (b\right )}e^{u\tilde {\gamma }},u\rightarrow \infty .$ When $b=1$ (i.e., in the Downton–Moran bivariate exponential case), we have that $C_{z,\delta ,s}^{\left (1\right )}=\frac {\omega _{1}}{\omega _{1}-\gamma -s},$ and the asymptotic result coincides with the exact formula (3.10).

5. Numerical examples

In this section, we numerically illustrate the behavior of GS functions and (defective) discounted densities that were derived in Sections 3 and 4, in relation to the value of the correlation coefficient  $\rho$ . Since the behavior of the ruin probability w.r.t. $\rho$ has been illustrated by Ambagaspitiya (Reference Ambagaspitiya2009), we will focus on the (defective) densities of the deficit at ruin and the time and the number of claims until ruin, along with their discounted versions and respective moments.

5.1 Downton–Moran bivariate exponential risk model

Initially, we assume that $(T_{i},X_{i})$ follow the Downton–Moran bivariate exponential distribution (cf. Section 3). Figure 1 depicts the form of the function $\bar {F}_{-U_{\tau }\left |\tau \lt \infty \right .}\left (x\right )=\int _{x}^{\infty }f_{-U_{\tau }\left |\tau \lt \infty \right .}\left (y\right )dy$ with respect to $x$ and the function $f_{\nu }\left (m\right )$ with respect to $m$ (cf. Corollary 10). Throughout this subsection, we choose $\lambda _{1}=2,\lambda _{2}=2,c=1.1$ so that $\eta =0.1\gt 0$ and $u=2,\delta =0.03$ .

Figure 1

Left: The survival function of the deficit at ruin given that the ruin occurs. Right: The defective probability function of the number of claims until ruin.

In Figure 1, it is evident that the survival function $\bar {F}_{-U_{\tau }\left |\tau \lt \infty \right .}\left (x\right )=e^{-r_{1}x}$ is a decreasing function of $\rho$ , since $r_{1}$ is an increasing function of $\rho$ . It follows that the deficit at ruin decreases stochastically as $T$ and $X$ become more positively correlated. We also observe that the mode of the defective probability function $f_{\nu }\left (m\right )$ increases with $\rho$ .

Figure 2

The defective discounted density $\ell _{\delta }(m,y,u)$ when $\rho =0$ (left), $\rho =0.2$ (right).

Figures 2 and 3 show the graph of the defective discounted density $\ell _{\delta }\left (m,y,u\right )$ (cf. Proposition 9) for $y\in (0,1.5)$ , $m=1,2,\ldots ,25$ and for various values of the correlation coefficient ( $\rho =0,0.2,0.4$ and $0.6$ ). It is worth stressing that the form of the density $\ell _{\delta }\left (m,y,u\right )$ in (3.19) can also be used to compute discounted probabilities of the form

\begin{equation*} \mathbb{P}^{\delta }\left (\nu =m,-U_{\tau }\gt x,\tau \lt \infty \right )\,:\!=\,\int _{x}^{\infty }\ell _{\delta }\left (m,y,u\right )dy=\frac {1}{\theta }\ell _{\delta }\left (m,x,u\right ),\;\;m=1,2,\ldots ,\;,x\geq 0. \end{equation*}

In Table 1, we compute the above probabilities for three scenarios for $\rho$ .

Table 1.

Computing the defective discounted probabilities $\mathbb{P}^{\delta }(\nu =m,-U_{\tau }\gt x,\tau \lt \infty )$

Figure 3

The defective discounted density $\ell _{\delta }(m,y,u)$ when $\rho =0.4$ (left), $\rho =0.6$ (right).

Finally, by dividing (3.10) by $\psi ,$ (cf. Proposition 6) and setting $z=1,s=0$ , we can readily get the LT, $E(e^{-\delta \tau }|\tau \lt \infty )$ , of the time until ruin, $\tau$ . Figure 4 shows the form of the density function of $\ \tau | \tau \lt \infty$ which was extracted via numerical inversion of this LT, by employing the Gaver–Stehfest algorithm (e.g., cf. Abate & Whitt, Reference Abate and Whitt1992).

Figure 4

The density of the time to ruin when $u=2$ (left) and $u=6$ (right).

5.2 Kibble–Moran bivariate Erlang risk model

We now assume that $(T_{i},X_{i})$ follow the Kibble–Moran bivariate Erlang distribution (cf. Section 4). In Figure 5, we depict the form of the (defective) discounted densities $\ell _{\delta }\left (y,u\right )$ with respect to $y$ that were explicitly derived in Corollary 16. The respective values of the parameters are $\lambda _{1}=1.4,\lambda _{2}=0.8,c=2,u=7,\delta =0.01,b=11$ (left graph) and $\lambda _{1}=1.5,\lambda _{2}=0.7,c=2,$ $u=7,\delta =0.01,b=11$ (right graph).

Using the explicit form of the (defective) discounted density $\ell _{\delta }\left (y,u\right )$ in (4.11), we can compute the discounted probabilities of the form

\begin{eqnarray*} &&\mathbb{P}^{\delta }\left (-U_{\tau }\gt x,\tau \lt \infty \right )\,:\!=\,\int _{x}^{\infty }\ell _{\delta }\left (y,u\right )dy\\&=&e^{\left (u+x\right )\gamma ^{\prime }-x\omega _{1}^{\prime }}\sum _{i=1}^{b}\pi _{i}\left (1,\delta ,u\right )\left (\omega _{1}^{\prime }\right )^{i}\sum _{n=0}^{i-1}\left (\omega _{1}^{\prime }-\gamma ^{\prime }\right )^{n-i}\frac {x^{n}}{n!},\;\;x\geq 0 \end{eqnarray*}

and using the same parameters as in Figure 5 ( $b=11,u=7$ , $\delta =0.01,c=2$ ), we construct Tables 2 and 3 where we observe that the discounted probabilities are decreasing functions of $\rho$ .

Note that the first row ( $x=0$ ) in Tables 2 and 3 gives the value of the LT of the time until ruin, that is, $\mathbb{E}\left (e^{-\delta \tau }1_{\left [\tau \lt \infty \right ]}\right )$ , computed at point $\delta =0.01$ . Next, using formula (4.12) in Corollary 16, the discounted moments of $-U_{\tau }$ are shown in Figure 6 (left: $k=1$ , right: $k=3$ ), as functions of the initial surplus $u$ ( $\lambda _{1}=1.4,\lambda _{2}=0.8,c=2,\delta =0.01,b=11$ ).

Table 2.

The discounted $\mathbb{P}^{\delta }(-U_{\tau }\gt x,\tau \lt \infty ), \ \eta \gt0.$

Table 3.

The discounted $\mathbb{P}^{\delta }(-U_{\tau }\gt x,\tau \lt \infty ), \ \eta\lt0.$

Figure 5

The (defective) discounted density $\ell _{\delta }(y,u)$ when $\eta \gt 0$ (left), $\eta \lt 0$ (right).

Figure 6

The expected present value of the deficit at ruin (left). The third discounted moment of the deficit at ruin (right).