2. General Results

Due to the presence of the dividend threshold b, the IDEs in u satisfied by $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ are different depending on whether 0≤u<b or u≥b (and hence the solution forms will also be different as in section 3). Therefore, we shall denote $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ by $\phi _{{L,\delta _{{123}} ,n,m}} (u;b)$ for 0≤u<b and $\phi _{{U,\delta _{{123}} ,n,m}} (u;b)$ for u≥b, where “L” and “U” stand for “Lower” and “Upper” layers, respectively. Furthermore, we shall use $\phi '_{{\delta _{{123}} ,n,m}} (u;b)=(d/du)\phi _{{\delta _{{123}} ,n,m}} (u;b)$ to denote the derivative of $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ with respect to the first argument u. Similar notations will be applied to $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ and other related functions as well. The proofs of the theorems and lemmas in this section are given in the Appendix.

2.1. IDEs and continuity condition for $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ when $n \, = \, 0$

First, we consider $\phi _{{\delta _{{123}} ,0,m}} (u;b)$ (i.e., n=0) so that the dividend component $D_{{\delta _{2} }} (\tau _{b} )$ does not appear in the definition (1.5). For notational convenience, we write

(2.1) $$\hskip -3pt \phi _{{\delta _{{13}} ,m}} (u;b)=\phi _{{\delta _{{123}} ,0,m}} (u;b)=E[e^{{{\minus} \delta _{1} \tau _{b} }} Z_{{\delta _{3} }}^{m} (\tau _{b} )w(U_{b} (\tau _{b}^{\,{\minus}\,} ),\! \mid \! \! U_{b} (\tau _{b} )\! \! \mid \! )1_{{\{ \tau _{b} \,\lt \, \infty\} }} \! \! \mid \! \! U_{b} (0)=u],\,\,\,u\, \geq \, 0$$

The IDEs and continuity condition for $\phi _{{\delta _{{13}} ,m}} (u;b)$ are stated in the following theorem, where $\phi _{{\delta _{{13}} ,m}} (u;b)=\phi _{{L,\delta _{{13}} ,m}} (u;b)$ for 0≤u<b and $\phi _{{\delta _{{13}} ,m}} (u;b)=\phi _{{U,\delta _{{13}} ,m}} (u;b)$ for u≥b according to our afore-mentioned convention.

Theorem 1 For $$m\in {\Bbb N}$$ , the Gerber–Shiu function $\phi _{{\delta _{{13}} ,m}} (u;b)$ in (2.1) satisfies the IDEs, for 0<u<b

(2.2) $$\eqalignno{ &#x0026; c_{1} \phi '_{{L,\delta _{{13}} ,m}} (u;b)\,{\minus}\,(\lambda {\plus}\delta _{1} {\plus}m\delta _{3} )\phi _{{L,\delta _{{13}} ,m}} (u;b){\plus}\lambda \mathop{\sum}\limits_{i=0}^m {\left( {\matrix{ m \cr i \cr } } \right)\mathop{\int}\nolimits_0^u {f^{{m\,{\minus}\,i}} (y)\phi _{{L,\delta _{{13}} ,i}} (u\,{\minus}\,y;b)p(y)\, dy} } \cr &#x0026; \quad {\plus}\lambda \mathop{\int}\nolimits_u^\infty {f^{m} (y)w(u,y\,{\minus}\,u)p(y)\, dy=0} $$

and for u>b

(2.3) $$\eqalignno{ &#x0026; c_{2} \phi '_{{U,\delta _{{13}} ,m}} (u;b)\,{\minus}\,(\lambda {\plus}\delta _{1} {\plus}m\delta _{3} )\phi _{{U,\delta _{{13}} ,m}} (u;b){\plus}\lambda \mathop{\sum}\limits_{i=0}^m {\left( {\matrix{ m \cr i \cr } } \right)} \mathop{\int}\nolimits_0^{u\,{\minus}\,b} {f^{{m\,{\minus}\,i}} (y)\phi _{{U\!,\delta _{{13}}\! ,i}} (u\,{\minus}\,y;b)p(y)\, dy} \cr &#x0026; \quad {\plus}\lambda \mathop{\sum}\limits_{i=0}^m {\left( {\matrix{ m \cr i \cr } } \right)} \mathop{\int}\nolimits_{u\,{\minus}\,b}^u {f^{{m\,{\minus}\,i}} (y)\phi _{{L,\delta _{{13}} ,i}} (u\,{\minus}\,y;b)p(y)\,dy} {\plus}\lambda \mathop{\int}\nolimits_u^\infty {f^{m} (y)w(u,y\,{\minus}\,u)p(y)\, dy=0} $$

In addition, $\phi _{{\delta _{{13}} ,m}} (u;b)$ is continuous at u=b, i.e.

(2.4) $$\phi _{{L,\delta _{{13}} ,m}} (b^{{\minus}\,} ;b)=\phi _{{U,\delta _{{13}} ,m}} (b^\!{{\plus}} ;b)$$

□

Remark 1 As $\phi _{{\delta _{{13}} ,m}} (u;b)$ reduces to the classical Gerber–Shiu function $$\phi _{{\delta _{1} }} (u;b)$$ when m=0, it is noted that the results in Lin & Pavlova (Reference Lin and Pavlova2006, theorem 3.1) can be retrieved from the above theorem by putting m=0. Note also that the determination of $\phi _{{\delta _{{13}} ,m}} (u;b)$ is recursive in m, with the starting point given by $\phi _{{\delta _{1} }} (u;b)$ . Assuming that the lower-order Gerber–Shiu functions $\phi _{{\delta _{{13}} ,i}} ( \cdot ;b)$ for i=0, 1, … , m−1 are known, it is observed that the IDE (2.3) involves both $\phi _{{L,\delta _{{13}} ,m}} ( \cdot ;b)$ in the lower layer and $\phi _{{U,\delta _{{13}} ,m}} ( \cdot ;b)$ in the upper layer as unknown functions, while (2.2) only involves $\phi _{{L,\delta _{{13}} ,m}} ( \cdot ;b)$ . Therefore, the typical procedure is to first utilise (2.2) to determine the solution form of $\phi _{{L,\delta _{{13}} ,m}} ( \cdot ;b)$ , and then attempt to find $\phi _{{U,\delta _{{13}} ,m}} ( \cdot ;b)$ in (2.3) by treating $\phi _{{L,\delta _{{13}} ,m}} ( \cdot ;b)$ as known (see the proofs of theorems in section 3).□

Remark 2 Having established the continuity of $\phi _{{\delta _{{13}} ,i}} ( \cdot ;b)$ for $$i\in {\Bbb N}$$ in the proof of Theorem 1, we observe from (2.2) that for each $$m\in {\Bbb N}$$ the derivative $\phi '_{{L,\delta _{{13}} ,m}} (u;b)$ is continuous in u in the layer 0<u<b if $\mathop{\int}\nolimits_u^\infty {f^{m} (y)w(u,y\,{\minus}\,u)p(y)\, dy} $ is continuous in u. A sufficient condition for $\mathop{\int}\nolimits_u^\infty {f^{m} (y)w(u,y\,{\minus}\,u)p(y)\, dy} $ to be continuous in u is that the penalty w(·, ·) is a continuous function. For the same reason, $\phi '_{{U,\delta _{{13}} ,m}} (u;b)$ is continuous in u for u>b under the same sufficient condition. However, although $\phi _{{\delta _{{13}} ,m}} (u;b)$ is continuous at u=b, the derivative $\phi '_{{\delta _{{13}} ,m}} (u;b)$ is generally not continuous at u=b. To see this, letting u→b ⁻ in (2.2) and u→b ⁺ in (2.3) (assuming $\mathop{\int}\nolimits_u^\infty {f^{m} (y)w(u,y\,{\minus}\,u)p(y)\, dy} $ is continuous at u=b) and comparing the two equations gives rise to

$$c_{1} \phi '_{{L,\delta _{{13}} ,m}} (b^{{\minus}\,} ;b)=c_{2} \phi '_{{U,\delta _{{13}} ,m}} (b\!^{{\plus}} ;b)$$

which generalises the result at the end of section 3 in Lin & Pavlova (Reference Lin and Pavlova2006) (see also Gerber & Shiu, Reference Gerber and Shiu2006, equation (10.5)). Therefore, $\phi '_{{L,\delta _{{13}} ,m}} (b^{\,{\minus}\,} ;b)\,\ne\,\phi '_{{U,\delta _{{13}} ,m}} (b^{{\plus}} ;b)$ unless c ₁=c ₂ (or equivalently α=0).□

2.2. IDEs and continuity condition for $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ when $$n\in {\Bbb N}^{{\plus}} $$

Next, we look at the Gerber–Shiu function $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ defined in (1.5) when $$n\in {\Bbb N}^{{\plus}} $$ and $$m\in {\Bbb N}$$ (where $${\Bbb N}^{\!{\plus}} $$ is the set of positive integers). The following theorem gives the associated IDEs and continuity condition.

Theorem 2 For $$n\in {\Bbb N}^{\!{\plus}} $$ and $$m\in {\Bbb N}$$ , the Gerber–Shiu function $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ in (1.5) satisfies the IDEs, for 0<u<b

(2.5) $$\eqalignno{ &#x0026;c_{1} \phi '_{{L,\delta _{{123}} ,n,m}} (u;b)\,{\minus}\,(\lambda {\plus}\delta _{1} {\plus}n\delta _{2} {\plus}m\delta _{3} )\phi _{{L,\delta _{{123}} ,n,m}} (u;b) \cr &#x0026; \quad {\plus}\,\lambda \mathop{\sum}\limits_{i\,=\,0}^m {\left( {\matrix{ m \cr i \cr } } \right)} \mathop{\int}\nolimits_0^u {f^{{m\,{\minus}\,i}} (y)\phi _{{L,\delta _{{123}} ,n,i}} (u\,{\minus}\,y;b)p(y)\, dy=0} $$

and for u>b

(2.6) $$\eqalignno{ &#x0026; c_{2} \phi '_{{U,\delta _{{123}} ,n,m}} (u;b)\,{\minus}\,(\lambda {\plus}\delta _{1} {\plus}n\delta _{2} {\plus}m\delta _{3} )\phi _{{U,\delta _{{123}} ,n,m}} (u;b){\plus}\alpha n\phi _{{U,\delta _{{123}} ,n\,{\minus}\,1,m}} (u;b) \cr &#x0026; \quad{\plus}\,\lambda \mathop{\sum}\limits_{i\,=\,0}^m {\left( {\matrix{ m \cr i \cr } } \right)} \mathop{\int}\nolimits_0^{u\,{\minus}\,b} {f^{{m\,{\minus}\,i}} (y)\phi _{{U,\delta _{{123}} ,n,i}} (u\,{\minus}\,y;b)p(y)\, dy} \cr &#x0026;\quad{\plus}\,\lambda \mathop{\sum}\limits_{i\,=\,0}^m {\left( {\matrix{ m \cr i \cr } } \right)} \mathop{\int}\nolimits_{u\,{\minus}\,b}^u {f^{{m\,{\minus}\,i}} (y)\phi _{{L,\delta _{{123}} ,n,i}} (u\,{\minus}\,y;b)p(y)\, dy=0} $$

In addition, $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ is continuous at u=b, i.e.

(2.7) $$\phi _{{L,\delta _{{123}} ,n,m}} (b^{\,{\minus}\,} ;b)=\phi _{{U,\delta _{{123}} ,n,m}} (b\!^{{\plus}} ;b)$$

□

Remark 3 It is instructive to note from (2.5) and (2.6) that one requires a double recursion in both n and m to determine $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ (see similar comments in Remark 1 concerning $\phi _{{\delta _{{13}} ,m}} (u;b)$ ). Furthermore, in parallel to Remark 2 (assuming $\mathop{\int}\nolimits_u^\infty {f^{m} (y)w(u,y\,{\minus}\,u)p(y)\, dy} $ is continuous at u=b) it is clear that, for $$n\in {\Bbb N}^{\!{\plus}} $$

$$c_{1} \phi '_{{L,\delta _{{123}} ,n,m}} (b^{{\minus}\,} ;b)=c_{2} \phi '_{{U,\delta _{{123}} ,n,m}} (b\!^{{\plus}} ;b){\plus}\alpha n\phi _{{U,\delta _{{123}} ,n\,{\minus}\,1,m}} (b;b)$$

i.e. the derivative of $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ is not necessarily continuous at u=b.□

Remark 4 Note that the IDEs (2.2) and (2.5) for $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ in the lower layer are the same as those in theorems 1 and 2 in Cheung et al. (Reference Cheung, Liu and Woo2015) concerning the dividend barrier strategy. This is because these IDEs are obtainable by considering an infinitesimal time interval, for which the dynamics of the surplus process are identical as no dividend is payable in the lower layer regardless of whether a barrier or a threshold strategy is implemented. Such an observation will allow us to reuse some of the intermediate results in Cheung et al. (Reference Cheung, Liu and Woo2015) in section 3.□

2.3. IDEs and continuity condition for $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$

In the next theorem, the IDEs and continuity condition concerning $\varphi _{{\delta _{{23}}\! ,n,m}} (u;b)$ (i.e., without ruin occurring) for $$n,m\in {\Bbb N}$$ will be provided. Because the dividend value $D_{{\delta _{2} }} (\tau _{b} )$ cannot be 0 on the set $\{ \tau _{b} =\infty\} $ , here we do not need to separate the analysis into the cases n=0 or $$n\in {\Bbb N}^{\!{\plus}} $$ (unlike the Gerber–Shiu function $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ in the previous two subsections).

Theorem 3 For $$n,m\in {\Bbb N}$$ , the joint moment $$\varphi _{{\delta _{{23}} ,n,m}} (u;b)$$ in (1.6) satisfies the IDEs, for 0<u<b

(2.8) $$c_{1} \varphi \prime _{{L,\delta _{{23}} ,n,m}} (u;b)\,{\minus}\,(\lambda {\plus}n\delta _{2} {\plus}m\delta _{3} )\varphi _{{L,\delta _{{23}} ,n,m}} (u;b){\plus}\lambda \mathop{\sum}\limits_{i\,=\,0}^m {\left( {\matrix{ m \cr i \cr } } \right)} \mathop{\int}\nolimits_0^u {f^{{m\,{\minus}\,i}} (y)\varphi _{{L,\delta _{{23}} ,n,i}} (u\,{\minus}\,y;b)p(y)\, dy=0} $$

and for u>b

(2.9) $$\eqalignno{ &#x0026; c_{2} \varphi '_{{U,\delta _{{23}} ,n,m}} (u;b)\,{\minus}\,(\lambda {\plus}n\delta _{2} {\plus}m\delta _{3} )\varphi _{{U,\delta _{{23}} ,n,m}} (u;b){\plus}\alpha n\varphi _{{U,\delta _{{23}} ,n\,{\minus}\,1,m}} (u;b) \cr &#x0026; \quad{\plus}\,\lambda \mathop{\sum}\limits_{i\,=\,0}^m {\left( {\matrix{ m \cr i \cr } } \right)} \mathop{\int}\nolimits_0^{u\,{\minus}\,b} {f^{{m\,{\minus}\,i}} (y)\varphi _{{U,\delta _{{23}} ,n,i}} (u\,{\minus}\,y;b)p(y)\, dy} \cr &#x0026;\quad{\plus}\,\lambda \mathop{\sum}\limits_{i\,=\,0}^m {\left( {\matrix{ m \cr i \cr } } \right)\mathop{\int}\nolimits_{u\,{\minus}\,b}^u {f^{{m\,{\minus}\,i}} (y)\varphi _{{L,\delta _{{23}} ,n,i}} (u\,{\minus}\,y;b)p(y)\, dy=0} } $$

In addition, $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ is continuous at u=b, i.e.

(2.10) $$\varphi _{{L,\delta _{{23}} ,n,m}} (b^{{\minus}\,} ;b)=\varphi _{{U,\delta _{{23}} ,n,m}} (b\!^{{\plus}} ;b)$$

It is understood that $\varphi _{{U,\delta _{{23}} ,n\,{\minus}\,1,m}} (u;b)$ appearing in (2.9) is regarded as 0 when n=0.□

Remark 5 For $$n\in {\Bbb N}^{\!{\plus}} $$ and $$m\in {\Bbb N}$$ , it is observed that the IDEs (2.5) and (2.6) in Theorem 2 satisfied by $\phi _{{\delta _{{123}} ,n,m}} (u;b) \!\! \mid _{{\delta _{1} \,=\,0}} $ in the case of ruin are identical to the IDEs (2.8) and (2.9) in Theorem 3 for $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ concerning the case of survival. However, the full solutions to $\phi _{{\delta _{{123}} ,n,m}} (u;b) \! \! \mid _{{\delta _{1} \, =\, 0}} $ and $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ are generally different due to different limiting conditions as u→∞ (which will be discussed in the next subsection) and the fact that the lower-order moments appearing in the two sets of IDEs are different.□

2.4. Limits of $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ and $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ as $u \rightarrow \infty $

From Theorems 1 and 2, the Gerber–Shiu function $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ satisfies two different IDEs in the lower and upper layers, and each IDE contains a derivative term. Therefore, the determination of the full solution of $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ from the IDEs typically requires one more piece of information apart from the continuity condition. Similar comments are applicable to $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ as well. In this subsection, we shall consider the limits ${{\rm lim}}_{u\to\infty} \, \phi _{{\delta _{{123}} ,n,m}} (u;b)$ and ${{\rm lim}}_{u\to\infty} \, \varphi _{{\delta _{{23}} ,n,m}} (u;b)$ . It is not our objective here to discuss the existence and/or uniqueness of solution to the IDEs given the continuity and limiting conditions in general, but we point out that these are sufficient to yield a unique solution in section 3 when each claim is distributed as a combination of exponentials (see Remark 9). Interested readers are referred to e.g. Mihálykó & Mihálykó (Reference Mihálykó and Mihálykó2011) where conditions for the uniqueness of the solution to an integral equation satisfied by the classical Gerber–Shiu function are analysed. Before providing the limits in Lemmas 2 and 3, we state the following lemma which is a special case of Léveillé & Garrido (Reference Léveillé and Garrido2001, corollary 2.1).

Lemma 1 Define, for $$m\in {\Bbb N}$$ and δ>0

(2.11) $$\theta _{{\delta ,m}} =E\left[ {\left( {\mathop{\sum}\limits_{k\,=1\,}^\infty {e^{{{\minus} \delta T_{k} }} f(Y_{k} )} } \right)^{m} } \right]$$

Then, θ _δ,m can be computed recursively using, for $$m\in {\Bbb N}^{\!{\plus}} $$

$$\theta _{{\delta ,m}} ={\lambda \over {m\delta }}\mathop{\sum}\limits_{i\,=\,0}^{m\,{\minus}\,1} \left( {\matrix{ m \cr i \cr } } \right)E[f^{{m\,{\minus}\,i}} (Y_{1} )]\theta _{{\delta ,i}} $$

with the starting value θ _{δ, 0}=1.□

Remark 6 From Lemma 1, it is clear that θ _{δ, 1} is finite if E[f(Y ₁)] is finite. For θ _{δ, 2} to be finite, one requires E[f ²(Y ₁)] to be finite (which implies finiteness of E[f(Y ₁)] and hence θ _{δ, 1}). Recursively, one observes that θ _{δ, m} is finite if E[f ^m (Y ₁)] is finite.□

Next, we shall first state ${{\rm lim}}_{u\to\infty}\, \varphi _{{\delta _{{23}} ,n,m}} (u;b)$ in the following lemma under some mild conditions, and some intermediate results in the proof will be used to identify ${{\rm lim}}_{u\to\infty} \,\phi _{{\delta _{{123}}\! ,n,m}} (u;b)$ in the proof of Lemma 2.

Lemma 2 For a given value of $$m\in {\Bbb N}$$ , if $$\theta _{{\delta _{3} ,m}} $$ defined via (2.11) is finite, then under the positive security loading condition, the limit of $\varphi _{{\delta _{{23}} \!,n,m}} (u;b)$ is finite and is given by, for $$n\in {\Bbb N}$$

(2.12) $$\mathop {{\rm lim}}\limits_{u\to\infty} \varphi _{{\delta _{{23}} ,n,m}} (u;b)=\left( {{\alpha \over {\delta _{2} }}} \right)^{n} \theta _{{\delta _{3} ,m}} $$

□

Lemma 3 For a given value of $m\in {\Bbb N}$ , if the penalty function w(·, ·) is bounded and $$\theta _{{\delta _{3} ,m}} $$ defined via (2.11) is finite, then under the positive security loading condition, the limit of $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ is given by, for $$n\in {\Bbb N}$$

(2.13) $$\mathop {{\rm lim}}\limits_{u\to\infty} \phi _{{\delta _{{123}} ,n,m}} (u;b)=0$$

□

3. Combination of Exponentials Claims

In this entire section, we assume that the distribution of each claim amount Y _k follows a combination of exponentials with density

(3.1) $$p(y)=\mathop{\sum}\limits_{k\,=1\,}^r {q_{k} \mu _{k} e^{{{\minus} \mu _{k} y}} } ,\,\,\,\,\,y\,\gt \,0$$

where $\mathop{\sum}\nolimits_{k\,=1\,}^r {q_{k} =1} $ , and for k=1, 2, … , r the parameters μ _k ’s are positive and distinct whereas q _k ’s are non-zero. The class of combinations of exponentials is known to be dense in the set of distributions on (0,∞), and we refer interested readers to Dufresne (Reference Dufresne2007) for its fitting. Concerning the quantity $Z_{{\delta _{3} }} (\tau _{b} )$ defined via (1.3), we shall focus on its special case $\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} } $ (for δ ₃>0), which represents the aggregate discounted claims until ruin. Thus, it is assumed that f(y)=y throughout this section. Since all the moments of Y ₁ are finite, the quantity $$\theta _{{\delta _{3\!} ,m}} $$ computed via Lemma 1 is also finite for every $$m\in {\Bbb N}$$ according to Remark 6. In particular, the first two moments of Y ₁ are given by $E[Y_{1} ]=\mathop{\sum}\nolimits_{k\,=1\,}^r {q_{k} /\mu _{k} } $ and $E[Y_{1}^{2} ]=\mathop{\sum}\nolimits_{k\,=1\,}^r {2q_{k} /\mu _{k}^{2} } $ . Consequently, the random variable $\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} } $ is integrable as $\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} } \leq \mathop{\sum}\nolimits_{k\,=1\,}^\infty {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} } $ and $E[\mathop{\sum}\nolimits_{k\,=1\,}^\infty {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} ]=\theta _{{\delta _{3} \!,1}} } $ is finite, and hence $\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} } $ is also integrable on the sets $\{ \tau _{b} \,\lt \, \infty\} $ and $\{ \tau _{b} =\infty\} $ . Conditional on ruin occurring, we are interested in correlations involving any two of the aggregate discounted claims until ruin $\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} } $ , the discounted dividends until ruin $D_{{\delta _{2} }} (\tau _{b} )$ (for δ ₂>0), and the ruin time $\tau _{b} $ (see section 4). Although a penalty function of w≡1 is sufficient for our purposes, we shall assume a bounded penalty w(x,y)=w(y) that depends on the deficit $\! \mid \! \!U_{b} (\tau _{b} ) \!\! \mid \! $ but not the surplus before ruin $U_{b} (\tau _{b}^{\,{\minus}\,} )$ , as this does not complicate our analysis. Note that the limiting condition (2.13) is applicable under the above setting, so is the condition (2.12) as far as the joint moments of the aggregate discounted claims and the discounted dividends are concerned without ruin occurring.

The derivations of explicit expressions for $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ and $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ rely on the Lundberg’s equation, for l=1, 2 and $$n,m\in {\Bbb N}$$

(3.2) $$c_{l} s\,{\minus}\,(\lambda {\plus}\delta _{1} {\plus}n\delta _{2} {\plus}m\delta _{3} ){\plus}\lambda \tilde{p}(s)=0$$

where $\tilde{p}(s)=\mathop{\sum}\nolimits_{k\,=1\,}^r {q_{k} \mu _{k} /(\mu _{k} {\plus}s)} $ is the Laplace transform of Y ₁. Let $$\{ \rho _{{n,m,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $\{ \kappa _{{n,m,j}} \} _{{j=1}}^{{r{\plus}1}} $ be the r+1 roots of (3.2) when l=1 and l=2, respectively (i.e., the roots ρ’s correspond to the full premium rate c ₁=c, while $\kappa $ ’s belong to the net premium rate c ₂=c−α). Each of these two sets of roots are assumed to be distinct (see Remark 7). It is well known that (3.2) has a unique root with non-negative real part (and it is a real root), while the other r roots have negative real parts. When l=2, we need to distinguish between these roots, and the non-negative root is denoted by $\kappa _{{n,m,r{\plus}1}} $ . (Note that $\kappa _{{n,m,r{\plus}1}} $ is indeed positive except when δ ₁=n=m=0.) Also, $$\rho _{{n,m,j}} $$ and $\kappa _{{n,m,j}} $ are denoted by $$\rho _{{n,m,j}}^{{\asterisk}} $$ and $\kappa _{{n,m,j}}^{{\asterisk}} $ , respectively, when δ ₁=0. We shall see that the solutions to $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ and $$\varphi _{{\delta _{{23}} ,n,m}} (u;b)$$ admit the representations

(3.3) $$\phi _{{L,\delta _{{123}} ,n,m}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^m {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{n,m,i,j}} e^{{\rho _{{n,i,j}} u}} } } ,\,\,\,\,\,0\leq u\leq b$$

(3.4) $$\phi _{{U,\delta _{{123}} ,n,m}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^n {\mathop{\sum}\limits_{j=0}^m {\mathop{\sum}\limits_{k\,=1\,}^r {A_{{n,m,i,j,k}}^{{\asterisk}} e^{{\kappa _{{i,j,k}} u}} } } } ,\,\,\,\,\,u\, \geq \, b$$

(3.5) $$\varphi _{{L,\delta _{{23}} ,n,m}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^m {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{n,m,i,j}} e^{{\rho _{{n,i,j}}^{{\asterisk}} u}} } } ,\,\,\,\,\,0\leq u\leq b$$

and

(3.6) $$\varphi _{{U,\delta _{{23}} ,n,m}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^n {\mathop{\sum}\limits_{j=0}^m {\mathop{\sum}\limits_{k\,=1\,}^r {C_{{n,m,i,j,k}}^{{\asterisk}} e^{{\kappa _{{i,j,k}}^{{\asterisk}} u}} } } } {\plus}\left( {{\alpha \over {\delta _{2} }}} \right)^{n} \theta _{{\delta _{3} ,m}}, \,\,\,\,\,u\, \geq \, b$$

As it has been shown that $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ and $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ are continuous for u≥0, we shall use the domain 0≤u≤b and u≥b for the lower and upper layers, respectively, in the upcoming lemmas and theorems.

3.1. $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ and $$\varphi _{{\delta _{{23}} ,n,m}} (u;b)$$ when $n = m =0$

When n=m=0, $\phi _{{\delta _{{123}} ,0,0}} (u;b)=\phi _{{\delta _{{13}} ,0}} (u;b)=\phi _{{\delta _{1} }} (u;b)$ is simply the classical Gerber–Shiu function defined in (1.4). Its solution can be obtained from Gerber & Shiu (Reference Gerber and Shiu2006, appendix B) with minor adjustments (as they considered w≡1). Since this will be used as a starting point to compute higher moments, the result is stated in the following lemma.

Lemma 4 The classical Gerber–Shiu function $\phi _{{\delta _{1} }} (u;b)$ is given by

(3.7) $$\phi _{{\delta _{1} }} (u;b)=\phi _{{L,\delta _{1} }} (u;b)=\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{0,0,0,j}} e^{{\rho _{{0,0,j}} u}} } ,\,\,\,\,\,0\leq u\leq b$$

and

(3.8) $$\phi _{{\delta _{1} }} (u;b)=\phi _{{U,\delta _{1} }} (u;b)=\mathop{\sum}\limits_{j=1}^r {A_{{0,0,0,0,j}}^{{\asterisk}} e^{{\kappa _{{0,0,j}} u}} } ,\,\,\,\,\,u\, \geq \, b$$

where $$\{ \rho _{{0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $\{ \kappa _{{0,0,j}} \} _{{j=1}}^{r} $ are Lundberg’s roots defined via (3.2). The coefficients $$\{ A_{{0,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,0,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ satisfy the 2r+1 linear equations which consist of

(3.9) $$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{0,0,0,j}} } \over {\mu _{k} {\plus}\rho _{{0,0,j}} }}} =\tilde{w}(\mu _{k} ),\,\,\,\,\,k=1,2,\,\ldots\,,r$$

(3.10) $$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{0,0,0,j}} } \over {\mu _{k} {\plus}\rho _{{0,0,j}} }}e^{{\rho _{{0,0,j}} b}} } =\mathop{\sum}\limits_{j=1}^r {{{A_{{0,0,0,0,j}}^{{\asterisk}} } \over {\mu _{k} {\plus}\kappa _{{0,0,j}} }}e^{{\kappa _{{0,0,j}} b}} } ,\,\,\,\,\,k=1,2,\,\ldots\,,r$$

and

(3.11) $$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{0,0,0,j}} e^{{\rho _{{0,0,j}} b}} } =\mathop{\sum}\limits_{j=1}^r {A_{{0,0,0,0,j}}^{{\asterisk}} e^{{\kappa _{{0,0,j}} b}} } $$

where $\tilde{w}(s)=\mathop{\int}\nolimits_0^\infty {e^{{{\minus} sy}} w(y)\, dy} $ is the Laplace transform of w(·).□

When n=m=0, it is clear from the definition (1.6) that $\varphi _{{\delta _{{23}} ,0,0}} (u;b)=\varphi (u;b)$ is the survival probability (i.e., probability that ruin does not occur). Therefore, one has that $\varphi (u;b)=1\,{\minus}\,\phi _{{\delta _{1} }} (u;b)\!\! \mid \! _{{\delta _{1} \,=\,0,w\,\equiv\,1}} $ , where $\phi _{{\delta _{1} }} (u;b)\!\! \mid \! \,_{{\delta _{1}\, = \, 0,w\,\equiv\,1}} $ can be computed using Lemma 4. This leads to the following lemma.

Lemma 5 The survival probability $\varphi (u;b)$ is given by

$$\varphi (u;b)=\varphi _{L} (u;b)=\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{0,0,0,j}} e^{{\rho _{{0,0,j}}^{{\asterisk}} u}} } ,\,\,\,\,\,0\leq u\leq b$$

and

(3.12) $$\varphi (u;b)=\varphi _{U} (u;b)=\mathop{\sum}\limits_{j=1}^r {C_{{0,0,0,0,j}}^{{\asterisk}} e^{{\kappa _{{0,0,j}}^{{\asterisk}} u}} {\plus}1} ,\,\,\,\,\,u\, \geq \, b$$

where $$\{ \rho _{{0,0,j}}^{{\asterisk}} \} _{{j\,=\,1}}^{{r\,{\plus}1}} $$ and $$\{ \kappa _{{0,0,j}}^{{\asterisk}} \} _{{j\,=\,1}}^{r} $$ are Lundberg’s roots defined via (3.2). The coefficients $$\{ C_{{0,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{0,0,0,0,j}}^{{\asterisk}} \} _{{j\,=\,1}}^{r} $$ satisfy the 2r+1 linear equations which consist of

$$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{0,0,0,j}} } \over {\mu _{k} {\plus}\rho _{{0,0,j}}^{{\asterisk}} }}=0} ,\,\,\,\,\,k=1,2,\,\ldots\,,r$$

$$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{0,0,0,j}} \mu _{k} } \over {\mu _{k} {\plus}\rho _{{0,0,j}}^{{\asterisk}} }}e^{{\rho _{{0,0,j}}^{{\asterisk}} b}} } =\mathop{\sum}\limits_{j=1}^r {{{C_{{0,0,0,0,j}}^{{\asterisk}} \mu _{k} } \over {\mu _{k} {\plus}\kappa _{{0,0,j}}^{{\asterisk}} }}e^{{\kappa _{{0,0,j}}^{{\asterisk}} b}} } {\plus}1,\,\,\,\,\,k=1,2,\,\ldots\,,r$$

and

$$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{0,0,0,j}} e^{{\rho _{{0,0,j}}^{{\asterisk}} b}} } =\mathop{\sum}\limits_{j=1}^r {C_{{0,0,0,0,j}}^{{\asterisk}} e^{{\kappa _{{0,0,j}}^{{\asterisk}} b}} } {\plus}1$$

□

3.2. $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ and $$\varphi _{{\delta _{{23}} ,n,m}} (u;b)$$ when $$n\in {\Bbb N}^{\!{\plus}} $$ and $$m = 0$$

We start with the special cases of (1.5) and (1.6) where $$n\in {\Bbb N}^{\!{\plus}} $$ and m=0, so that the aggregate claims component $Z_{{\delta _{3} }} (\tau _{b} )=\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} } $ is absent in $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ and $$\varphi _{{\delta _{{23}\!} ,n,m}} (u;b)$$ . These will be denoted by $\phi _{{\delta _{{123}} ,n,0}} (u;b)=\phi _{{\delta _{{12}} ,n}} (u;b)$ and $$\varphi _{{\delta _{{23}} ,n,0}} (u;b)=\varphi _{{\delta _{2} ,n}} (u;b)$$ , respectively, and given in the following two theorems. The proofs are provided in the Appendix.

Theorem 4 For $$n\in {\Bbb N}^{\!{\plus}} $$ , the Gerber–Shiu function $\phi _{{\delta _{{12}}\! ,n}} (u;b)=\phi _{{\delta _{{123}} \!,n,0}} (u;b)$ is given by

(3.13) $$\phi _{{\delta _{{12}\!} ,n}} (u;b)=\phi _{{L,\delta _{{12}} \!,n}} (u;b)=\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{n,0,0,j}} e^{{\rho _{{n,0,j}} u}} } ,\,\,\,\,\,0\leq u\leq b$$

and

(3.14) $$\phi _{{\delta _{{12}} \!,n}} (u;b)=\phi _{{U,\delta _{{12}\!} ,n}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^n {\mathop{\sum}\limits_{j=1}^r {A_{{n,0,i,0,j}}^{{\asterisk}} e^{{\kappa _{{i,0,j}} u}} } } ,\,\,\,\,\,u\, \geq \, b$$

where $\{ \rho _{{n,0,j}} \} _{{j=1}}^{{r{\plus}1}} $ and $\{ \kappa _{{i,0,j}} \} _{{j=1}}^{r} $ are Lundberg’s roots defined via (3.2). The coefficients $\{ A_{{n,0,i,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $ (for i=0, 1, … , n−1) can be obtained from

(3.15) $$A_{{n,0,i,0,j}}^{{\asterisk}} ={{\alpha n} \over {(n\,{\minus}\,i)\delta _{2} }}A_{{n\,{\minus}\,1,0,i,0,j}}^{{\asterisk}} ,\,\,\,\,\,i=0,1,\,\ldots\,,n\,{\minus}\,1;j=1,2,\,\ldots\,,r$$

while the coefficients $\{ A_{{n,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $ and $\{ A_{{n,0,n,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $ satisfy the 2r+1 linear equations which consist of

(3.16) $$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{n,0,0,j}} } \over {\mu _{k} {\plus}\rho _{{n,0,j}} }}} =0,\,\,\,\,\,k=1,2,\,\ldots\,,r$$

(3.17) $$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{n,0,0,j}} } \over {\mu _{k} {\plus}\rho _{{n,0,j}} }}e^{{\rho _{{n,0,j}} b}} } =\mathop{\sum}\limits_{i\,=\,0}^n {\mathop{\sum}\limits_{j=1}^r {{{A_{{n,0,i,0,j}}^{{\asterisk}} } \over {\mu _{k} {\plus}\kappa _{{i,0,j}} }}e^{{\kappa _{{i,0,j}} b}} } } ,\,\,\,\,\,k=1,2,\,\ldots\,,r$$

and

(3.18) $$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{n,0,0,j}} e^{{\rho _{{n,0,j}} b}} } =\mathop{\sum}\limits_{i\,=\,0}^n {\mathop{\sum}\limits_{j=1}^r {A_{{n,0,i,0,j}}^{{\asterisk}} e^{{\kappa _{{i,0,j}} b}} } } $$

The coefficients $$\{ A_{{0,0,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ which form the starting point of the recursion in n can be evaluated using Lemma 4.□

Theorem 5 For $$n\in {\Bbb N}^{\!{\plus}} $$ , the nth moment of the discounted dividends without ruin occurring $$\varphi _{{\delta _{2} ,n}} (u;b)=\varphi _{{\delta _{{23}} ,n,0}} (u;b)$$ is given by

(3.19) $$\varphi _{{\delta _{2} ,n}} (u;b)=\varphi _{{L,\delta _{2} ,n}} (u;b)=\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{n,0,0,j}} e^{{\rho _{{n,0,j}}^{{\asterisk}} u}} } ,\,\,\,\,\,0\leq u\leq b$$

and

(3.20) $$\varphi _{{\delta _{2} ,n}} (u;b)=\varphi _{{U,\delta _{2} ,n}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^n {\mathop{\sum}\limits_{j=1}^r {C_{{n,0,i,0,j}}^{{\asterisk}} e^{{\kappa _{{i,0,j}}^{{\asterisk}} u}} } } {\plus}\left( {{\alpha \over {\delta _{2} }}} \right)^{n} ,\,\,\,\,\,u\, \geq \, b$$

where $$\{ \rho _{{n,0,j}}^{{\asterisk}} \} _{{j=1}}^{{r{\plus}1}} $$ and $\{ \kappa _{{i,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $ are Lundberg’s roots defined via (3.2). The coefficients $$\{ C_{{n,0,i,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (for i=0, 1, … , n−1) can be obtained from

(3.21) $$C_{{n,0,i,0,j}}^{{\asterisk}} ={{\alpha n} \over {(n\,{\minus}\,i)\delta _{2} }}C_{{n\,{\minus}\,1,0,i,0,j}}^{{\asterisk}} ,\,\,\,\,\,i=0,1,\,\ldots\,,n\,{\minus}\,1;j=1,2,\,\ldots\,,r$$

while the coefficients $$\{ C_{{n,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{n,0,n,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ satisfy the 2r+1 linear equations which consist of

(3.22) $$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{n,0,0,j}} } \over {\mu _{k} {\plus}\rho _{{n,0,j}}^{{\asterisk}} }}} =0,\,\,\,\,\,k=1,2,\,\ldots\,,r$$

(3.23) $$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{n,0,0,j}} } \over {\mu _{k} {\plus}\rho _{{n,0,j}}^{{\asterisk}} }}e^{{\rho _{{n,0,j}}^{{\asterisk}} b}} } =\mathop{\sum}\limits_{i\,=\,0}^n {\mathop{\sum}\limits_{j=1}^r {{{C_{{n,0,i,0,j}}^{{\asterisk}} } \over {\mu _{k} {\plus}\kappa _{{i,0,j}}^{{\asterisk}} }}e^{{\kappa _{{i,0,j}}^{{\asterisk}} b}} } } {\plus}{1 \over {\mu _{k} }}\left( {{\alpha \over {\delta _{2} }}} \right)^{n} ,\,\,\,\,\,k=1,2,\,\ldots\,,r$$

and

(3.24) $$\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{n,0,0,j}} e^{{\rho _{{n,0,j}}^{{\asterisk}} b}} } =\mathop{\sum}\limits_{i\,=\,0}^n {\mathop{\sum}\limits_{j=1}^r {C_{{n,0,i,0,j}}^{{\asterisk}} e^{{\kappa _{{i,0,j}}^{{\asterisk}} b}} } } {\plus}\left( {{\alpha \over {\delta _{2} }}} \right)^{n} $$

The coefficients $$\{ C_{{0,0,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ , which form the starting point of the recursion in n, can be evaluated using Lemma 5.□

3.3. $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ and $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ when $n =0$ and $m =1, 2$

We now look at $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ when n=0, which is denoted by $\phi _{{\delta _{{123}} ,0,m}} (u;b)=\phi _{{\delta _{{13}} ,m}} (u;b)$ in section 2.1. The following theorem gives the explicit expression for $\phi _{{\delta _{{13}} ,1}} (u;b)$ . The proof is given in the Appendix. It will be seen that the analysis is more involved when m is now non-zero.

Theorem 6 The Gerber–Shiu function $\phi _{{\delta _{{13}} ,1}} (u;b)=\phi _{{\delta _{{123}} ,0,1}} (u;b)$ is given by

(3.25) $$\phi _{{\delta _{{13}} ,1}} (u;b)=\phi _{{L,\delta _{{13}} ,1}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{0,1,i,j}} e^{{\rho _{{0,i,j}} u}} } } ,\,\,\,\,\,0\leq u\leq b$$

and

(3.26) $$\phi _{{\delta _{{13}} ,1}} (u;b)=\phi _{{U,\delta _{{13}} ,1}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {A_{{0,1,0,i,j}}^{{\asterisk}} e^{{\kappa _{{0,i,j}} u}} } } ,\,\,\,\,\,u\, \geq \, b$$

where $$\{ \rho _{{0,i,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $\{ \kappa _{{0,i,j}} \} _{{j=1}}^{r} $ are Lundberg’s roots defined via (3.2). The coefficients $$\{ A_{{0,1,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,1,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be computed directly using

(3.27) $$A_{{0,1,0,j}} ={{\lambda A_{{0,0,0,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r{\plus}1$$

and

(3.28) $$A_{{0,1,0,0,j}}^{{\asterisk}} ={{\lambda A_{{0,0,0,0,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r$$

where $$\{ A_{{0,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,0,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ are obtainable from Lemma 4. Then the coefficients $$\{ A_{{0,1,1,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,1,0,1,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be solved from the 2r+1 linear equations which consist of

(3.29) $$\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{0,1,i,j}} } \over {\mu _{k} {\plus}\rho _{{0,i,j}} }}} } \,{\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{0,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}} )^{2} }}} ={\cal T}_{{\mu _{k} }}^{2} w(0),\,\,\,\,\,k=1,2,\,\ldots\,,r$$

(3.30) $$\eqalignno{ &#x0026; \mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{0,1,i,j}} } \over {\mu _{k} {\plus}\rho _{{0,i,j}} }}e^{{\rho _{{0,i,j}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{0,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}} )^{2} }}e^{{\rho _{{0,0,j}} b}} } =\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {{{A_{{0,1,0,i,j}}^{{\asterisk}} } \over {\mu _{k} {\plus}\kappa _{{0,i,j}} }}e^{{\kappa _{{0,i,j}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^r {{{A_{{0,0,0,0,j}}^{{\asterisk}} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}} )^{2} }}e^{{\kappa _{{0,0,j}} b}} } , \cr &#x0026; \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad k=1,2,\,\ldots\,,r $$

and

(3.31) $$\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{0,1,i,j}} e^{{\rho _{{0,i,j}} b}} } } =\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {A_{{0,1,0,i,j}}^{{\asterisk}} e^{{\kappa _{{0,i,j}} b}} } } $$

where ${\cal T}_{{\mu _{k} }}^{2} w(0)=\mathop{\int}\nolimits_0^\infty {ye^{{{\minus} \mu _{k} y}} w(y)\, dy} $ in (3.29) is the notation of a double Dickson–Hipp operator (see Dickson & Hipp, Reference Dickson and Hipp2001; Li & Garrido, Reference Li and Garrido2004).□

The next theorem gives the result for $\phi _{{\delta _{{13}} ,2}} (u;b)$ . Since the logic of the derivation is identical to that of Theorem 6 (although it is more tedious), the proof is omitted. Note that (3.32)–(3.35) concerning the Gerber–Shiu function $\phi _{{L,\delta _{{13}} ,2}} (u;b)$ in the lower layer are direct consequences of (39)–(42) in theorem 7 of Cheung et al. (Reference Cheung, Liu and Woo2015) (see Remark 4).

Theorem 7 The Gerber–Shiu function $\phi _{{\delta _{{13}} ,2}} (u;b)=\phi _{{\delta _{{123}} ,0,2}} (u;b)$ is given by

(3.32) $$\phi _{{\delta _{{13}} ,2}} (u;b)=\phi _{{L,\delta _{{13}} ,2}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{0,2,i,j}} e^{{\rho _{{0,i,j}} u}} } } ,\,\,\,\,\,0\leq u\leq b$$

and

$$\phi _{{\delta _{{13}} ,2}} (u;b)=\phi _{{U,\delta _{{13}} ,2}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^r {A_{{0,2,0,i,j}}^{{\asterisk}} e^{{\kappa _{{0,i,j}} u}} } } ,\,\,\,\,\,u\, \geq \, b$$

where $$\{ \rho _{{0,i,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $\{ \kappa _{{0,i,j}} \} _{{j=1}}^{r} $ are Lundberg’s roots defined via (3.2). For i=0, 1, the coefficients $$\{ A_{{0,2,i,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,2,0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be computed directly using

(3.33) $$A_{{0,2,0,j}} ={{\lambda A_{{0,1,0,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}} )^{2} }}} \,{\plus}\,{{\lambda A_{{0,0,0,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}} )^{3} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r{\plus}1$$

(3.34) $$A_{{0,2,1,j}} ={{2\lambda A_{{0,1,1,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{0,1,j}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r{\plus}1$$

$$A_{{0,2,0,0,j}}^{{\asterisk}} ={{\lambda A_{{0,1,0,0,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}} )^{2} }}} {\plus}{{\lambda A_{{0,0,0,0,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}} )^{3} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r$$

and

$$A_{{0,2,0,1,j}}^{{\asterisk}} ={{2\lambda A_{{0,1,0,1,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,1,j}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r$$

where $$\{ A_{{0,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,0,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ are obtainable from Lemma 4, while for i=0, 1, the coefficients $$\{ A_{{0,1,i,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,1,0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ are obtainable from Theorem 6. Then, the coefficients $$\{ A_{{0,2,2,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,2,0,2,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be solved from the 2r+1 linear equations which consist of

(3.35) $$\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{0,2,i,j}} } \over {\mu _{k} {\plus}\rho _{{0,i,j}} }}} } \,{\plus}\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{2A_{{0,1,i,j}} } \over {(\mu _{k} {\plus}\rho _{{0,i,j}} )^{2} }}} }\, {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{2A_{{0,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}} )^{3} }}} =2{\cal T}_{{\mu _{k} }}^{3} w(0),\,\,\,\,\,k=1,2,\,\ldots\,,r$$

$$\eqalignno{ &#x0026; \mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{0,2,i,j}} } \over {\mu _{k} {\plus}\rho _{{0,i,j}} }}e^{{\rho _{{0,i,j}} b}} } } {\plus}\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{2A_{{0,1,i,j}} } \over {(\mu _{k} {\plus}\rho _{{0,i,j}} )^{2} }}e^{{\rho _{{0,i,j}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{2A_{{0,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}} )^{3} }}e^{{\rho _{{0,0,j}} b}} } \cr &#x0026; \quad =\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^r {{{A_{{0,2,0,i,j}}^{{\asterisk}} } \over {\mu _{k} {\plus}\kappa _{{0,i,j}} }}e^{{\kappa _{{0,i,j}} b}} } } {\plus}\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {{{2A_{{0,1,0,i,j}}^{{\asterisk}} } \over {(\mu _{k} {\plus}\kappa _{{0,i,j}} )^{2} }}e^{{\kappa _{{0,i,j}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^r {{{2A_{{0,0,0,0,j}}^{{\asterisk}} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}} )^{3} }}e^{{\kappa _{{0,0,j}} b}} } ,\,\,\,\,\,k=1,2,\,\ldots\,,r $$

and

$$\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{0,2,i,j}} e^{{\rho _{{0,i,j}} b}} } } =\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^r {A_{{0,2,0,i,j}}^{{\asterisk}} e^{{\kappa _{{0,i,j}} b}} } } $$

where ${\cal T}_{{\mu _{k} }}^{3} w(0)=\mathop{\int}\nolimits_0^\infty {(y^{2} e^{{{\minus} \mu _{k} y}} /2)w(y)\, dy} $ in (3.35) is the notation of a triple Dickson–Hipp operator.□

Concerning the first two moments of the aggregate discounted claims without ruin occurring (denoted by $\varphi _{{\delta _{{23}} ,0,m}} (u;b)=\varphi _{{\delta _{3} ,m}} (u;b)$ for m=1, 2), the results are stated in the following theorems. Their proofs follow closely those of Theorems 6 and 7 and are omitted.

Theorem 8 The expected aggregate discounted claims $\varphi _{{\delta _{3} ,1}} (u;b)=\varphi _{{\delta _{{23}} ,0,1}} (u;b)$ without ruin occurring is given by

$$\varphi _{{\delta _{3} ,1}} (u;b)=\varphi _{{L,\delta _{3} ,1}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{0,1,i,j}} e^{{\rho _{{0,i,j}}^{{\asterisk}} u}} } } ,\,\,\,\,\,0\leq u\leq b,$$

and

$$\varphi _{{\delta _{3} ,1}} (u;b)=\varphi _{{U,\delta _{3} ,1}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {C_{{0,1,0,i,j}}^{{\asterisk}} e^{{\kappa _{{0,i,j}}^{{\asterisk}} u}} } } {\plus}{{\lambda E[Y_{1} ]} \over {\delta _{3} }},\,\,\,\,\,u\, \geq \, b$$

where $$\{ \rho _{{0,i,j}}^{{\asterisk}} \} _{{j=1}}^{{r{\plus}1}} $$ and $\{ \kappa _{{0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $ are Lundberg’s roots defined via (3.2). The coefficients $$\{ C_{{0,1,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{0,1,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be computed directly using

$$C_{{0,1,0,j}} ={{\lambda C_{{0,0,0,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}}^{{\asterisk}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r{\plus}1$$

and

$$C_{{0,1,0,0,j}}^{{\asterisk}} ={{\lambda C_{{0,0,0,0,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}}^{{\asterisk}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r$$

where $$\{ C_{{0,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{0,0,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ are obtainable from Lemma 5. Then the coefficients $$\{ C_{{0,1,1,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{0,1,0,1,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be solved from the 2r+1 linear equations which consist of

$$\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{0,1,i,j}} } \over {\mu _{k} {\plus}\rho _{{0,i,j}}^{{\asterisk}} }}} } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{0,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}}^{{\asterisk}} )^{2} }}} =0,\,\,\,\,\,k=1,2,\,\ldots\,,r$$

$$\eqalignno{ &#x0026; \mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{0,1,i,j}} } \over {\mu _{k} {\plus}\rho _{{0,i,j}}^{{\asterisk}} }}e^{{\rho _{{0,i,j}}^{{\asterisk}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{0,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}}^{{\asterisk}} )^{2} }}e^{{\rho _{{0,0,j}}^{{\asterisk}} b}} } \cr &#x0026; =\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {{{C_{{0,1,0,i,j}}^{{\asterisk}} } \over {\mu _{k} {\plus}\kappa _{{0,i,j}}^{{\asterisk}} }}e^{{\kappa _{{0,i,j}}^{{\asterisk}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^r {{{C_{{0,0,0,0,j}}^{{\asterisk}} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}}^{{\asterisk}} )^{2} }}e^{{\kappa _{{0,0,j}}^{{\asterisk}} b}} } {\plus}{{\lambda E[Y_{1} ]} \over {\mu _{k} \delta _{3} }}{\plus}{1 \over {\mu _{k}^{2} }},\,\,\,\,\,k=1,2,\,\ldots\,,r $$

and

$$\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{0,1,i,j}} e^{{\rho _{{0,i,j}}^{{\asterisk}} b}} } } =\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {C_{{0,1,0,i,j}}^{{\asterisk}} e^{{\kappa _{{0,i,j}}^{{\asterisk}} b}} } } {\plus}{{\lambda E[Y_{1} ]} \over {\delta _{3} }}$$

□

Theorem 9 The second moment of the aggregate discounted claims $$\varphi _{{\delta _{3} ,2}} (u;b)=\varphi _{{\delta _{{23}} ,0,2}} (u;b)$$ without ruin occurring is given by

$$\varphi _{{\delta _{3} ,2}} (u;b)=\varphi _{{L,\delta _{3} ,2}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{0,2,i,j}} e^{{\rho _{{0,i,j}}^{{\asterisk}} u}} } } ,\,\,\,\,\,0\leq u\leq b$$

and

(3.36) $$\varphi _{{\delta _{3} ,2}} (u;b)=\varphi _{{U,\delta _{3} ,2}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^r {C_{{0,2,0,i,j}}^{{\asterisk}} e^{{\kappa _{{0,i,j}}^{{\asterisk}} u}} } } {\plus}\theta _{{\delta _{3} ,2}} ,\,\,\,\,\,u\, \geq \, b$$

where $$\{ \rho _{{0,i,j}}^{{\asterisk}} \} _{{j=1}}^{{r{\plus}1}} $$ and $\{ \kappa _{{0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $ are Lundberg’s roots defined via (3.2). The constant term $$\theta _{{\delta _{3} ,2}} $$ in (3.36) can be evaluated by Lemma 1 as $\theta _{{\delta _{3} ,2}} =\lambda (2E[Y_{1} ]\theta _{{\delta _{3} ,1}} {\plus}E[Y_{1}^{2} ])/(2\delta _{3} )$ with $$\theta _{{\delta _{3} ,1}} =\lambda E[Y_{1} ]/\delta _{3} $$ . For i=0, 1, the coefficients $$\{ C_{{0,2,i,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{0,2,0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be computed directly using

$$\eqalignno{ &#x0026; C_{{0,2,0,j}} ={{\lambda C_{{0,1,0,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}}^{{\asterisk}} )^{2} }}} {\,\plus}{{\lambda C_{{0,0,0,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}}^{{\asterisk}} )^{3} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r{\plus}1 \cr &#x0026; \quad \quad \quad \quad \quad \quad C_{{0,2,1,j}} ={{2\lambda C_{{0,1,1,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{0,1,j}}^{{\asterisk}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r{\plus}1 \cr &#x0026; C_{{0,2,0,0,j}}^{{\asterisk}} ={{\lambda C_{{0,1,0,0,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}}^{{\asterisk}} )^{2} }}} \,\,{\plus}{{\lambda C_{{0,0,0,0,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}}^{{\asterisk}} )^{3} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r $$

and

$$C_{{0,2,0,1,j}}^{{\asterisk}} ={{2\lambda C_{{0,1,0,1,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,1,j}}^{{\asterisk}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r$$

where $$\{ C_{{0,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{0,0,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ are obtainable from Lemma 5, while for i=0, 1 the coefficients $$\{ C_{{0,1,i,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{0,1,0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ are obtainable from Theorem 8. Then, the coefficients $$\{ C_{{0,2,2,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{0,2,0,2,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be solved from the 2r+1 linear equations which consist of

$$\eqalignno{ &#x0026; \mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{0,2,i,j}} } \over {\mu _{k} {\plus}\rho _{{0,i,j}}^{{\asterisk}} }}} } {\plus}\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{2C_{{0,1,i,j}} } \over {(\mu _{k} {\plus}\rho _{{0,i,j}}^{{\asterisk}} )^{2} }}} } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{2C_{{0,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}}^{{\asterisk}} )^{3} }}=0} ,\,\,\,\,\,k=1,2,\,\ldots\,,r $$

$$\eqalignno{ &#x0026; \mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{0,2,i,j}} } \over {\mu _{k} {\plus}\rho _{{0,i,j}}^{{\asterisk}} }}e^{{\rho _{{0,i,j}}^{{\asterisk}} b}} } } {\plus}\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{2C_{{0,1,i,j}} } \over {(\mu _{k} {\plus}\rho _{{0,i,j}}^{{\asterisk}} )^{2} }}e^{{\rho _{{0,i,j}}^{{\asterisk}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{2C_{{0,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{0,0,j}}^{{\asterisk}} )^{3} }}e^{{\rho _{{0,0,j}}^{{\asterisk}} b}} } \,{\minus}\,{{\theta _{{\delta _{3} ,2}} } \over {\mu _{k} }}\,{\minus}\,{{2\theta _{{\delta _{3} ,1}} } \over {\mu _{k}^{2} }}\,{\minus}\,{2 \over {\mu _{k}^{3} }} \cr &#x0026; \quad=\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^r {{{C_{{0,2,0,i,j}}^{{\asterisk}} } \over {\mu _{k} {\plus}\kappa _{{0,i,j}}^{{\asterisk}} }}e^{{\kappa _{{0,i,j}}^{{\asterisk}} b}} } } {\plus}\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {{{2C_{{0,1,0,i,j}}^{{\asterisk}} } \over {(\mu _{k} {\plus}\kappa _{{0,i,j}}^{{\asterisk}} )^{2} }}e^{{\kappa _{{0,i,j}}^{{\asterisk}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^r {{{2C_{{0,0,0,0,j}}^{{\asterisk}} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}}^{{\asterisk}} )^{3} }}e^{{\kappa _{{0,0,j}}^{{\asterisk}} b}} } ,\,\,\,\,\,k=1,2,\,\ldots\,,r $$

and

$$\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{0,2,i,j}} e^{{\rho _{{0,i,j}}^{{\asterisk}} b}} } } =\mathop{\sum}\limits_{i\,=\,0}^2 {\mathop{\sum}\limits_{j=1}^r {C_{{0,2,0,i,j}}^{{\asterisk}} e^{{\kappa _{{0,i,j}}^{{\asterisk}} b}} {\plus}\theta _{{\delta _{3} ,2}} } } $$

□

3.4. $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ and $\varphi _{{\delta _{{23}} ,n,m}} (u;b)$ when $n = m = 1$

In the next two theorems, the procedures to find $\phi _{{\delta _{{123}}\! ,1,1}} (u;b)$ and $\varphi _{{\delta _{{23}}\! ,1,1}} (u;b)$ are provided. These two quantities will be useful for computing the covariance (and hence correlation) between the discounted dividends $D_{{\delta _{2} }} (\tau _{b} )$ and the aggregate discounted claims $Z_{{\delta _{3} }} (\tau _{b} )=\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} } $ . Again, (3.37), (3.39), and (3.43) follow directly from Cheung et al. (Reference Cheung, Liu and Woo2015, equations (50)–(52)).

Theorem 10 The Gerber–Shiu function $$\phi _{{\delta _{{123}} ,1,1}} (u;b)$$ is given by

(3.37) $$\phi _{{\delta _{{123}} ,1,1}} (u;b)=\phi _{{L,\delta _{{123}} ,1,1}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{1,1,i,j}} e^{{\rho _{{1,i,j}} u}} } } ,\,\,\,\,\,0\leq u\leq b$$

and

(3.38) $$\phi _{{\delta _{{123}} ,1,1}} (u;b)=\phi _{{U,\delta _{{123}} ,1,1}} (u;b)=\mathop{\sum}\limits_{l=0}^1 {\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {A_{{1,1,l,i,j}}^{{\asterisk}} e^{{\kappa _{{l,i,j}} u}} } } } ,\,\,\,\,\,u\, \geq \, b$$

where $$\{ \rho _{{1,i,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $\{ \kappa _{{l,i,j}} \} _{{j=1}}^{r} $ are Lundberg’s roots defined via (3.2). The coefficients $$\{ A_{{1,1,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ , $$\{ A_{{1,1,0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (for i=0, 1) and $$\{ A_{{1,1,1,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be computed directly using

(3.39) $$A_{{1,1,0,j}} ={{\lambda A_{{1,0,0,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{1,0,j}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r{\plus}1$$

(3.40) $$A_{{1,1,0,0,j}}^{{\asterisk}} ={{\alpha A_{{0,1,0,0,j}}^{{\asterisk}} } \over {\delta _{2} {\plus}\delta _{3} }}{\plus}{{\lambda A_{{1,0,0,0,j}}^{{\asterisk}} } \over {\delta _{2} {\plus}\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r$$

(3.41) $$A_{{1,1,0,1,j}}^{{\asterisk}} ={\alpha \over {\delta _{2} }}A_{{0,1,0,1,j}}^{{\asterisk}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r$$

and

(3.42) $$A_{{1,1,1,0,j}}^{{\asterisk}} ={{\lambda A_{{1,0,1,0,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{1,0,j}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r$$

where $$\{ A_{{1,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{1,0,i,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (for i=0, 1) are obtainable from Theorem 4, while $$\{ A_{{0,1,0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (for i=0, 1) are obtainable from Theorem 6. Then the coefficients $$\{ A_{{1,1,1,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{1,1,1,1,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be solved from the 2r+1 linear equations which consist of

(3.43) $$\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{1,1,i,j}} } \over {\mu _{k} {\plus}\rho _{{1,i,j}} }}} } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{1,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{1,0,j}} )^{2} }}=0} ,\,\,\,\,\,k=1,2,\,\ldots\,,r$$

(3.44) $$\eqalignno{ &#x0026; \mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{1,1,i,j}} } \over {\mu _{k} {\plus}\rho _{{1,i,j}} }}e^{{\rho _{{1,i,j}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{A_{{1,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{1,0,j}} )^{2} }}e^{{\rho _{{1,0,j}} b}} } \cr &#x0026; =\mathop{\sum}\limits_{l=0}^1 {\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {{{A_{{1,1,l,i,j}}^{{\asterisk}} } \over {\mu _{k} {\plus}\kappa _{{l,i,j}} }}e^{{\kappa _{{l,i,j}} b}} } } } {\plus}\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {{{A_{{1,0,i,0,j}}^{{\asterisk}} } \over {(\mu _{k} {\plus}\kappa _{{i,0,j}} )^{2} }}e^{{\kappa _{{i,0,j}} b}} } } ,\,\,\,\,\,k=1,2,\,\ldots\,,r $$

and

(3.45) $$\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {A_{{1,1,i,j}} e^{{\rho _{{1,i,j}} b}} } } =\mathop{\sum}\limits_{l=0}^1 {\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {A_{{1,1,l,i,j}}^{{\asterisk}} e^{{\kappa _{{l,i,j}} b}} } } } $$

□

Theorem 11 The first joint moment of the discounted dividends and the aggregate discounted claims $\varphi _{{\delta _{{23}} ,1,1}} (u;b)$ without ruin occurring is given by

$$\varphi _{{\delta _{{23}} ,1,1}} (u;b)=\varphi _{{L,\delta _{{23}} ,1,1}} (u;b)=\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{1,1,i,j}} e^{{\rho _{{1,i,j}}^{{\asterisk}} u}} } } ,\,\,\,\,\,0\leq u\leq b$$

and

$$\varphi _{{\delta _{{23}} ,1,1}} (u;b)=\varphi _{{U,\delta _{{23}} ,1,1}} (u;b)=\mathop{\sum}\limits_{l=0}^1 {\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {C_{{1,1,l,i,j}}^{{\asterisk}} e^{{\kappa _{{l,i,j}}^{{\asterisk}} u}} {\plus}{{\alpha \lambda E[Y_{1} ]} \over {\delta _{2} \delta _{3} }}} } } ,\,\,\,\,\,u\, \geq \, b$$

where $$\{ \rho _{{1,i,j}}^{{\asterisk}} \} _{{j=1}}^{{r{\plus}1}} $$ and $\{ \kappa _{{l,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $ are Lundberg’s roots defined via (3.2). The coefficients $$\{ C_{{1,1,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ , $$\{ C_{{1,1,0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (for i=0, 1) and $$\{ C_{{1,1,1,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be computed directly using

$$\eqalignno{ &#x0026; C_{{1,1,0,j}} ={{\lambda C_{{1,0,0,j}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\rho _{{1,0,j}}^{{\asterisk}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r{\plus}1 \cr &#x0026; C_{{1,1,0,0,j}}^{{\asterisk}} ={{\alpha C_{{0,1,0,0,j}}^{{\asterisk}} } \over {\delta _{2} {\plus}\delta _{3} }}{\plus}{{\lambda C_{{1,0,0,0,j}}^{{\asterisk}} } \over {\delta _{2} {\plus}\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{0,0,j}}^{{\asterisk}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r \cr &#x0026; C_{{1,1,0,1,j}}^{{\asterisk}} ={\alpha \over {\delta _{2} }}C_{{0,1,0,1,j}}^{{\asterisk}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r $$

and

$$C_{{1,1,1,0,j}}^{{\asterisk}} ={{\lambda C_{{1,0,1,0,j}}^{{\asterisk}} } \over {\delta _{3} }}\mathop{\sum}\limits_{k\,=1\,}^r {{{q_{k} \mu _{k} } \over {(\mu _{k} {\plus}\kappa _{{1,0,j}}^{{\asterisk}} )^{2} }}} ,\,\,\,\,\,j=1,2,\,\ldots\,,r$$

where $$\{ C_{{1,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{1,0,i,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (for i=0, 1) are obtainable from Theorem 5, while $$\{ C_{{0,1,0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (for i=0, 1) are obtainable from Theorem 8. Then the coefficients $$\{ C_{{1,1,1,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ C_{{1,1,1,1,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ can be solved from the 2r+1 linear equations which consist of

$$\eqalignno{ &#x0026; \mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{1,1,i,j}} } \over {\mu _{k} {\plus}\rho _{{1,i,j}}^{{\asterisk}} }}} } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{1,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{1,0,j}}^{{\asterisk}} )^{2} }}} =0,\,\,\,\,\,k=1,2,\,\ldots\,,r \cr &#x0026; \mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{1,1,i,j}} } \over {\mu _{k} {\plus}\rho _{{1,i,j}}^{{\asterisk}} }}e^{{\rho _{{1,i,j}}^{{\asterisk}} b}} } } {\plus}\mathop{\sum}\limits_{j=1}^{r{\plus}1} {{{C_{{1,0,0,j}} } \over {(\mu _{k} {\plus}\rho _{{1,0,j}}^{{\asterisk}} )^{2} }}e^{{\rho _{{1,0,j}}^{{\asterisk}} b}} } \cr &#x0026; =\mathop{\sum}\limits_{l=0}^1 {\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {{{C_{{1,1,l,i,j}}^{{\asterisk}} } \over {\mu _{k} {\plus}\kappa _{{l,i,j}}^{{\asterisk}} }}e^{{\kappa _{{l,i,j}}^{{\asterisk}} b}} } } } {\plus}\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {{{C_{{1,0,i,0,j}}^{{\asterisk}} } \over {(\mu _{k} {\plus}\kappa _{{i,0,j}}^{{\asterisk}} )^{2} }}e^{{\kappa _{{i,0,j}}^{{\asterisk}} b}} } } {\plus}{{\alpha \lambda E[Y_{1} ]} \over {\mu _{k} \delta _{2} \delta _{3} }}{\plus}{\alpha \over {\mu _{k}^{2} \delta _{2} }},\,\,\,\,\,k=1,2,\,\ldots\,,r $$

and

$$\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^{r{\plus}1} {C_{{1,1,i,j}} e^{{\rho _{{1,i,j}}^{{\asterisk}} b}} } } =\mathop{\sum}\limits_{l=0}^1 {\mathop{\sum}\limits_{i\,=\,0}^1 {\mathop{\sum}\limits_{j=1}^r {C_{{1,1,l,i,j}}^{{\asterisk}} e^{{\kappa _{{l,i,j}}^{{\asterisk}} b}} } } } {\plus}{{\alpha \lambda E[Y_{1} ]} \over {\delta _{2} \delta _{3} }}$$

□

Remark 8 From the statements of the theorems in this section, it is important to note the recursive nature of the determination of the coefficients involved in the solution forms (3.3)–(3.6). For example, the computational steps required to calculate $$\phi _{{\delta _{{123}} ,1,1}} (u;b)$$ via Theorem 10 involve the use of earlier lemma and theorems, and these are summarised as follows:

1. 1. For each fixed l=1, 2 and n, m=0, 1, we solve the Lundberg’s equation (3.2), which has r+1 roots, i.e. the equation is solved eight times. When l=1 the roots are denoted by $$\{ \rho _{{n,m,j}} \} _{{j=1}}^{{r{\plus}1}} $$ , but when l=2 they are denoted by $$\{ \kappa _{{n,m,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and the non-negative root $\kappa _{{n,m,r{\plus}1}} $ is discarded.

2. 2. Obtain $$\{ A_{{0,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,0,0,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ from Lemma 4 by solving the linear equations (3.9)–(3.11).

3. 3. Apply the special case of Theorem 4 under n=1, where $$\{ A_{{1,0,0,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{1,0,i,0,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (for i=0, 1) are computed from (3.15) and the linear system (3.16)–(3.18).

4. 4. Use Theorem 6 to compute $$\{ A_{{0,1,i,j}} \} _{{j=1}}^{{r{\plus}1}} $$ and $$\{ A_{{0,1,0,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (both for i=0, 1) via (3.27) and (3.28) along with the linear system (3.29)–(3.31). (Although $$A_{{0,1,i,j}} $$ ’s are not needed in the next step, they have to be determined together with $$A_{{0,1,0,1,j}}^{{\asterisk}} $$ ’s.)

5. 5. Utilise Theorem 10 to calculate $$\{ A_{{1,1,i,j}} \} _{{j=1}}^{{r{\plus}1}} $$ (for i=0, 1) and $$\{ A_{{1,1,l,i,j}}^{{\asterisk}} \} _{{j=1}}^{r} $$ (for l, i=0, 1) via (3.39)–(3.42) and the linear system (3.43)–(3.45), so that $\phi _{{\delta _{{123}} ,1,1}} (u;b)$ is finally evaluated with (3.37) and (3.38).□

Remark 9 Following the proof of Theorem 4 in the Appendix, it is noted that the exact value of the limit ${{\rm lim}}_{u\to\infty} \phi _{{\delta _{{12}} ,1}} (u;b)$ is indeed not required for deriving the full solution to $\phi _{{\delta _{{12}} ,1}} (u;b)$ . We only need the finiteness of the limit to conclude that the coefficient of the exponential term $$e^{{\kappa _{{1,0,r{\plus}1}} u}} $$ is 0 as $\kappa _{{1,0,r{\plus}1}}\, \gt \,0$ , and then from (3.14) the limit ${{\rm lim}}_{u\to\infty} \,\phi _{{\delta _{{12}} ,1}} (u;b)$ must equal 0 as $\kappa _{{1,0,j}} $ has negative real part for j=1, 2, …, r. This is also true for the proof of Theorem 5, except that the final term in (3.20) (when n=1) obtainable via equating the constant term in (A.17) is non-zero (and it automatically satisfies the limiting value ${{\rm lim}}_{u\to\infty} \, \varphi _{{\delta _{2} ,1}} (u;b)$ suggested by Lemma 2). The same comments are also applicable to Theorems 6–11 (although the proofs of Theorems 7–11 have been omitted).□

4. Numerical Examples

In this section, the results in section 3 are applied to compute the correlations involving the total discounted dividends until ruin $D_{{\delta _{2} }} (\tau _{b} )$ , the aggregate discounted claim amount until ruin $\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} {e^{{{\minus} \delta _{3} T_{k} }} Y_{k} } $ , and the time of ruin $$\tau _{b} $$ (which is considered only when ruin occurs). Hence, we assume a cost function of f(y)=y (i.e., $$Z_{{\delta _{3} }} (\tau _{b} )=\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} e^{{{\minus} \delta _{3} T_{k} }} Y_{k} $$ ) and a penalty of w≡1 throughout this section. With initial surplus U _b (0)=u and threshold level b, for notational convenience we denote the unconditional expectation of a random variable X by E[X∣u;b], and we shall use E _r [X∣u;b] (E _s [X∣u;b], respectively) to denote the expectation of X conditional on the event $\{ \tau _{b} \,\lt \, \infty\} $ ( $$\{ \tau _{b} =\infty\} $$ , respectively). The subscripts “r” and “s” correspond to “ruin” and “survival”, respectively. Clearly, one has

$$E_{r} [X \!\! \mid \! \! u;b]={{E[X1_{{\{ \tau _{b} \,\lt \, \infty\} }} \! \! \mid \! \! u;b]} \over {\psi (u;b)}}$$

and

$$E_{s} [X \! \! \mid \! \! u;b]={{E[X1_{{\{ \tau _{b} =\infty\} }} \! \! \mid \! \! u;b]} \over {\varphi (u;b)}}$$

where $\psi (u;b)$ and $$\varphi (u;b)$$ are the ruin probability and survival probability, respectively. Then the correlation of the random variables X ₁ and X ₂ is given by

$${\rm Corr}_{{\bullet}} (X_{1} ,X_{2} \! \! \mid \! \! u;b)={{{\rm Cov}_{{\bullet}} (X_{1} ,X_{2} \! \! \mid \! \! u;b)} \over {\sqrt {{\rm Var}_{{\bullet}} (X_{1} \! \! \mid \! \! u;b){\rm Var}_{{\bullet}} (X_{2} \! \! \mid \! \! u;b)} }}$$

where

$${\rm Cov}_{{\bullet}} (X_{1} ,X_{2} \! \! \mid \! \! u;b)=E_{{\bullet}} [X_{1} X_{2} \! \! \mid \! \! u;b]\,{\minus}\,E_{{\bullet}} [X_{1} \! \! \mid \! \! u;b]E_{{\bullet}} [X_{2} \! \! \mid \! \! u;b]$$

is the covariance of X ₁ and X ₂, and

$${\rm Var}_{{\bullet}} (X \! \! \mid \! \! u;b)=E_{{\bullet}} [X^{2} \! \! \mid \! \! u;b]\,{\minus}\,(E_{{\bullet}} [X \! \! \mid \! \! u;b])^{2} $$

is the variance of X. Here the expectation E∙ can be the unconditional expectation E or the conditional expectations E _r or E _s , and this applies to other moment-based quantities as well. All intermediate quantities required in our computation involving $\tau _{b} $ , $D_{{\delta _{2} }} (\tau _{b} )$ and $Z_{{\delta _{3} }} (\tau _{b} )$ are obtainable from the Gerber–Shiu function $\phi _{{\delta _{1} ,\delta _{2} ,\delta _{3} ,n,m}} (u;b)=\phi _{{\delta _{{123}} ,n,m}} (u;b)$ in (1.5) and the joint moment $$\varphi _{{\delta _{2} ,\delta _{3} ,n,m}} (u;b)=\varphi _{{\delta _{{23}} ,n,m}} (u;b)$$ in (1.6). For example, the first joint moment $E_{r} [D_{{\delta _{2} }} (\tau _{b} )Z_{{\delta _{3} }} (\tau _{b} ) \! \! \mid \! u;b]$ conditional on ruin is the ratio of $E[D_{{\delta _{2} }} (\tau _{b} )Z_{{\delta _{3} }} (\tau _{b} )1_{{\{ \tau _{b} \,\lt \, \infty\} }} \! \! \mid \! \! u;b]=\phi _{{\delta _{{123}} ,1,1}} (u;b) \! \! \mid _{{\delta _{1} \,=\,0}} $ to $\psi (u;b)=\phi _{{\delta _{1} }} (u;b) \! \! \mid _{{\delta _{1} \,=\,0}} $ , which can be evaluated using Theorem 10 and Lemma 4. Similarly, the first joint moment $E_{s} [D_{{\delta _{2} }} (\tau _{b} )Z_{{\delta _{3} }} (\tau _{b} ) \! \! \mid \! u;b]$ conditional on survival follows from $E[D_{{\delta _{2} }} (\tau _{b} )Z_{{\delta _{3} }} (\tau _{b} )1_{{\{ \tau _{b}\, =\,\infty\} }} \! \! \mid \!\! u;b]=\varphi _{{\delta _{{23}} ,1,1}} (u;b)$ (that is available in Theorem 11) and $\varphi (u;b)=1\,{\minus}\,\psi (u;b)$ . The unconditional first joint moment $E[D_{{\delta _{2} }} (\tau _{b} )Z_{{\delta _{3} }} (\tau _{b} ) \! \! \mid \! u;b]$ is given by (1.7) when n=m=1. It is instructive to note that correlations in relation to $\tau _{b} $ only exist when $\tau _{b} \,\lt \, \infty$ , and the (joint) moments involving $\tau _{b} $ can be obtained from $\phi _{{\delta _{{123}} ,n,m}} (u;b)$ . For example, one has

$$E_{r} [\tau _{b} Z_{{\delta _{3} }} (\tau _{b} ) \! \! \mid \! u;b]={{E[\tau _{b} Z_{{\delta _{3} }} (\tau _{b} )1_{{\{ \tau _{b} \,\lt \, \infty\} }} \! \! \mid \! u;b]} \over {\psi (u;b)}}=\,{\minus}\,{{{\partial \over {\partial \delta _{1} }}\phi _{{\delta _{{123}} \!,0,1}} (u;b) \! \! \mid _{{\delta _{1} \,=\,0}} } \over {\psi (u;b)}}$$

Before discussing specific examples, we first note that in general the relationship

(4.1) $$u{\plus}c\tau _{b} {\plus} \! \! \mid \! \! U_{b} (\tau _{b} ) \! \! \mid \! \; =D(\tau _{b} ){\plus}S(\tau _{b} )\,{\rm on}\,{\rm the}\,{\rm set}\,\{ \tau _{b} \,\lt \, \infty\} $$

among the random variables is valid for sample paths leading to ruin, where $D(\tau _{b} )$ is the total dividends paid until ruin and $S(\tau _{b} )=\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{b} )} {Y_{k} } $ is the aggregate claims until ruin (both without discounting). For these sample paths, it is clear that

(4.2) $$D(\tau _{b} )\leq \alpha \tau _{b} \,{\rm on}\,{\rm the}\,{\rm set}\,\{ \tau _{b} \,\lt \, \infty\} $$

and therefore

(4.3) $$S(\tau _{b} )\, \geq \, u{\plus}c_{2} \tau _{b} {\plus} \! \! \mid \! \! U_{b} (\tau _{b} ) \! \! \mid \! \! \,{\rm on}\,{\rm the}\,{\rm set}\,\{ \tau _{b} \,\lt \, \infty\} $$

On the other hand, concerning the sample paths for which the process survives, one has that

(4.4) $$D(t){\plus}S(t)\leq u{\plus}ct\,{\rm for}\,{\rm all}\,t\, \geq \, 0\,{\rm on}\,{\rm the}\,{\rm set}\,\{ \tau _{b} =\infty\} $$

The parameter values that are used for all numerical illustrations are summarised in Table 1. In each subsequent figure, the quantity of interest is plotted against the initial surplus level u under three different claim size distributions, namely

1. i. a sum of two exponentials (“Sum Exp”) with density p(y)=3e ^−(3/2)y −3e ^−3y ;

2. ii. an exponential distribution (“Exp”) with density p(y)=e ^−y ; and

3. iii. a mixture of two exponentials (“Mixed Exp”) with density p(y)=(1/6)e ^−(1/2)y +(4/3)e ^−2y .

Table 1 Parameters used in all numerical examples.

All these distributions belong to the class of combinations of exponentials (see (3.1)) and have the same mean of 1 (and the loading condition $c_{2} \gt \lambda E[Y_{1} ]$ holds true). However, they have different amount of variability as evident in their variances of 0.56, 1, and 2, respectively. The curves corresponding to the above claim distributions are marked in solid, dashed, and dotted lines, respectively.

Conditional on ruin occurring, Figures 1–3 show how the pairwise correlations of $\tau _{{10}} $ and $Z_{{0.01}} (\tau _{{10}} )$ and $D_{{0.01}} (\tau _{{10}} )$ vary with u for 0≤u≤200. (For simplicity, we shall write $\tau $ , Z, and D instead of $\tau _{{10}} $ , $Z_{{0.01}} (\tau _{{10}} )$ , and $D_{{0.01}} (\tau _{{10}} )$ , respectively, in the y-axis of the plots.) From Figure 1, it is observed that ${\rm Corr}_{r} (\tau _{{10}} ,Z_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ for all three claim distributions is of the same shape. Specifically, ${\rm Corr}_{r} (\tau _{{10}} ,Z_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ starts with a positive value of over 0.9, and it decreases as u increases and then becomes negative when u reaches approximately 95. This complements figures 1 and 2 in Cheung & Woo (Reference Cheung and Feng2016), which demonstrated a sign change of the covariance of the ruin time and the aggregate discounted claims until ruin in a dependent Sparre Andersen risk model without dividends as u increases. Some interpretations therein are indeed applicable: for fixed u, two opposing effects are in place when one analyses sample paths for which $\tau _{b} $ is large. Intuitively, the aggregate (non-discounted) claim amount $S(\tau _{{10}} )=\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{{10}} )} {Y_{k} } $ tends to be large because more claims arise as the process survives longer (see also (4.3)). But these claims occur over a longer time horizon and a large claim does not happen early (otherwise it would have caused early ruin), meaning that the discounted amount $$Z_{{0.01}} (\tau _{{10}} )=\mathop{\sum}\nolimits_{k\,=1\,}^{N(\tau _{{10}} )} {e^{{{\minus} 0.01T_{k} }} Y_{k} } $$ possibly has a tendency to become smaller due to discounting. Figure 1 suggests that the former effect is more dominant until the correlation changes sign at around u=95. As u increases further from 95, the effect of discounting starts to dominate because the discounting on the nominal amount u appearing on the right-hand side of (4.3) is getting significant.

Figure 1 Correlation of ruin time and aggregate discounted claims conditional on ruin.

Figure 2 Correlation of ruin time and discounted dividends conditional on ruin.

Figure 3 Correlation of aggregate discounted claims and discounted dividends conditional on ruin.

Next, when we look at Figure 2, which depicts the behaviour of ${\rm Corr}_{r} (\tau _{{10}} ,D_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ , it is noted that the correlation is always positive. This is unlike ${\rm Corr}_{r} (\tau _{{10}} ,Z_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ in Figure 1 where there is a change in sign as u increases. A possible explanation is that u does not appear on the right-hand side of (4.2) (as dividend is paid from part of the premium income but not the initial surplus), and thus the effect of discounting on u is absent in this case. As a result, the positive correlation between $\tau _{{10}} $ and $D_{{0.01}} (\tau _{{10}} )$ is simply attributed to the fact that, for each fixed u, the surplus process is more likely to stay above the threshold more often when the ruin time is large, resulting in more dividends.

In Figure 3, the correlation ${\rm Corr}_{r} (Z_{{0.01}} (\tau _{{10}} ),D_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ conditional on ruin takes on positive values when u increases to about 100 and then it becomes negative. From the above discussions, we argue that the aggregate non-discounted values of the claim amount $S(\tau _{{10}} )$ and dividends $D(\tau _{{10}} )$ both tend to increase with the ruin time $\tau _{{10}} $ . However, $S(\tau _{{10}} )$ and $D(\tau _{{10}} )$ may also move in opposite directions because only part of the claims are paid from the premium, while all dividend payments come from the premium (see also (4.1) for the constraint on the sum $D(\tau _{{10}} ){\plus}S(\tau _{{10}} )$ ). Under discounting, Figure 3 suggests that the former effect dominates for u less than 100, while the latter becomes dominant when u exceeds 100. Interestingly, we observe that ${\rm Corr}_{r} (Z_{{0.01}} (\tau _{{10}} ),D_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ in Figure 3 changes sign at roughly the same place as ${\rm Corr}_{r} (\tau _{{10}} ,Z_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ does in Figure 1. Note also that the curves in Figure 3 are ordered according to the variance of the individual claim size distribution.

Now, we turn to Figure 4 concerning the correlation of $Z_{{0.01}} (\tau _{{10}} )$ and $D_{{0.01}} (\tau _{{10}} )$ conditional on survival. Clearly, the shape of ${\rm Corr}_{s} (Z_{{0.01}} (\tau _{{10}} ),D_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ is completely different from that of ${\rm Corr}_{r} (Z_{{0.01}} (\tau _{{10}} ),D_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ in Figure 3. In particular, ${\rm Corr}_{s} (Z_{{0.01}} (\tau _{{10}} ),D_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ in Figure 4 begins at a negative value between −0.55 and −0.60. It increases with u, stays negative and converges to 0 from below. Note also that the above pattern appears to kick in earlier when the claim size has smaller variance. The reason for negative correlation is the constraint (4.4), which makes it impossible for both $Z_{{0.01}} (\tau _{{10}} )$ and $D_{{0.01}} (\tau _{{10}} )$ to be large in the presence of discounting. The convergence of ${\rm Corr}_{s} (Z_{{0.01}} (\tau _{{10}} ),D_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ to 0 as u increases can be explained by zero covariance at the limit. Indeed, we can apply Lemma 2 three times with (n, m)=(1, 1), (1, 0), and (0, 1) to see that

$$\eqalignno{&#x0026; \mathop {{\rm lim}}\limits_{u\to\infty} E[D_{{\delta _{2} }} (\tau _{b} )Z_{{\delta _{3} }} (\tau _{b} )1_{{\{ \tau _{b} =\infty\} }} \! \! \mid \! \! u;b] \cr &#x0026; \quad=\left( {\mathop {{\rm lim}}\limits_{u\to\infty} E[D_{{\delta _{2} }} (\tau _{b} )1_{{\{ \tau _{b} =\infty\} }} \! \! \mid \! \! u;b]} \right)\left( {\mathop {{\rm lim}}\limits_{u\to\infty} E[Z_{{\delta _{3} }} (\tau _{b} )1_{{\{ \tau _{b} =\infty\} }} \! \! \mid \! \! u;b]} \right)$$

Figure 4 Correlation of aggregate discounted claims and discounted dividends conditional on survival.

Division of each of the three above limits by the limiting survival probability $${{\rm lim}}_{u\to\infty} \varphi (u;b)=1$$ reveals that $${\rm Cov}_{s} (Z_{{\delta _{3} }} (\tau _{b} ),D_{{\delta _{2} }} (\tau _{b} ) \! \! \mid \! \! u;b)=E_{s} [Z_{{\delta _{3} }} (\tau _{b} )D_{{\delta _{2} }} (\tau _{b} ) \! \! \mid \! \! u;b]\,{\minus}\,E_{s} [Z_{{\delta _{3} }} (\tau _{b} ) \! \! \mid \! \! u;b]E_{s} [D_{{\delta _{2} }} (\tau _{b} )\! \! \mid \! \! u;b]$$ tends to 0 as $$u\to\infty$$ .

Lastly, Figure 5 plots the unconditional correlation of $Z_{{0.01}} (\tau _{{10}} )$ and $D_{{0.01}} (\tau _{{10}} )$ against u. As a function of u, the correlation ${\rm Corr}(Z_{{0.01}} (\tau _{{10}} ),D_{{0.01}} (\tau _{{10}} ) \! \! \mid \! \! u;10)$ first decreases from over 0.95 to negative values and finally converges to 0. Similar to Figure 4, the pattern prevails earlier when the individual claim size has less variability. Note that the unconditional correlation takes all sample paths into account regardless of whether the process ruins or survives. The contributions of these events are in accordance with the ruin probability $\psi (u;10)$ and the survival probability $\varphi (u;10)=1\,{\minus}\,\psi (u;10)$ , respectively. As u increases, $\psi (u;10)$ decreases and converges to 0 and therefore the impact of ruin occurrence becomes less significant while that of survival becomes stronger. The shape of Figure 5 is thus a combination of Figures 3 and 4.

Figure 5 Unconditional correlation of aggregate discounted claims and discounted dividends.

5. Concluding Remarks

This paper jointly analyses the aggregate discounted claims until ruin (payments to policyholders) and the discounted dividends until ruin (payments to shareholders) in the compound Poisson insurance risk model with a threshold dividend strategy. The method adopted here is based on the derivation of recursive IDEs satisfied by Gerber–Shiu-type functions involving the joint moments of these two random variables. When each claim amount is distributed as a combination of exponentials, these IDEs are transformed to ordinary differential equations that can be solved with the help of the continuity and limiting conditions.

An alternative approach will be to connect the risk process to an equivalent fluid flow model (e.g., Badescu et al., Reference Badescu, Breuer, Da Silva Soares, Latouche, Remiche and Stanford2005). In such a construction, a downward jump (caused by the arrival of a claim) of size y in the risk model is replaced by decreasing segment of slope −C in the fluid model over a time period of y/C for some constant C>0. It is well known that (e.g., Ramaswami, Reference Ramaswami2006; Ahn et al., Reference Ahn, Badescu and Ramaswami2007) the Laplace transforms of various first passage times can typically be expressed in terms of the Laplace transform of the busy period, whose evaluation can be done by numerical algorithms that converge quadratically fast (e.g., Ahn & Ramaswami, Reference Ahn and Ramaswami2005; Bean et al., Reference Bean, O’Reilly and Taylor2005). Then one may try to express our Gerber–Shiu-type functions in terms of these quantities pertaining to the fluid model. While the moments of discounted dividends were derived by Badescu & Landriault (Reference Badescu and Landriault2008) in this manner for a multi-threshold model, the aggregate discounted claim amount until ruin has never been analysed via fluid flow to the best of our knowledge. Nevertheless, in the absence of discounting, we note that the total dividend is (a scalar multiple of) the occupation time when the fluid level is in an increasing phase above the threshold level b, while the aggregate claim amount corresponds to (a scalar multiple of) the occupation time of the fluid in a decreasing phase. Finally, we also remark that another research problem will be to determine the exact joint distribution of the discounted dividends and the aggregate discounted claims until ruin (as opposed to joint moments in the present work), which is expected to be a very challenging task. We leave these as open questions.