To extend the approach outlined in Sect.
3, we consider a vector of market shares
\(p=(p_1,p_2) \in [0,1]^2\), where
\(p_i\) denotes the proportion of individuals in the market holding a policy for risk
i. Let
\(p^{(1,0)}\) be the proportion of individuals in the market holding a policy for risk 1 and no policy for risk 2. Similarly,
\(p^{(0,1)} \) and
\(p^{(1,1)} \) denote the proportion of individuals holding a policy only for risk 2 and for both risks, respectively. Obviously,
\(p_1=p^{(1,1)}+p^{(1,0)}\) and
\(p_2=p^{(1,1)}+p^{(0,1)}\). If the acquisition of polices for different risks is independent, then
$$\begin{aligned} p^{(1,1)} = p_1 p_2, \qquad p^{(1,0)} = p_1 (1-p_2 ), \qquad p^{(0,1)} = p_2 (1-p_1 ). \end{aligned}$$
Under this model, the company’s surplus process is
$$\begin{aligned} \tilde{X}_t = u^{(1)} + u^{(2)} + \big ( c^{(1)} + c^{(2)} \big )t - \sum _{i = 0}^{\tilde{N}_t^{1\perp }} \tilde{Y}_i^{1\perp } - \sum _{i = 0}^{\tilde{N}_t^{2\perp }} \tilde{Y}_i^{2\perp } - \sum _{i = 0}^{\tilde{N}_t^{\parallel }} \Big (Y_i^{1\parallel } + Y_i^{2\parallel } \Big ), \end{aligned}$$
where
\(\tilde{N}_t^{1\perp }\),
\(\tilde{N}_t^{2\perp }\), and
\(\tilde{N}_t^{\parallel }\) count the number of claims received by the company, concerning only risk 1, only risk 2, and both risks, respectively. Their intensities are, respectively,
$$\begin{aligned} \begin{aligned} \tilde{\lambda }_1^\perp&= p^{(1,0)} \big ( \lambda _1^\perp + \lambda ^\parallel \big ) + p^{(1,1)} \lambda _1^\perp = p_1 \lambda _1^\perp + p^{(1,0)} \lambda ^\parallel , \\ \tilde{\lambda }_2^\perp&= p_2 \lambda _2^\perp + p^{(0,1)} \lambda ^\parallel , \\ \tilde{\lambda }^\parallel&= p^{(1,1)} \lambda ^\parallel . \end{aligned} \end{aligned}$$
The distribution of the single risk claim amounts
\(\tilde{Y}^{1\perp }\) (resp.,
\(\tilde{Y}^{2\perp }\)) is a mixture of the distributions
\(Y^{1\perp }\) and
\(Y^{1\parallel }\) (resp.,
\(Y^{2\perp }\) and
\(Y^{2\parallel }\))
$$\begin{aligned} \begin{aligned}&F_{\tilde{Y}^{1\perp }} = \frac{p_1\lambda _1^\perp }{p_1 \lambda _1^\perp + p^{(1,0)} \lambda ^\parallel } F_{Y^{1\perp }} + \frac{p^{(1,0)} \lambda ^\parallel }{p_1 \lambda _1^\perp + p^{(1,0)} \lambda ^\parallel } F_{Y^{1\parallel }} \\&F_{\tilde{Y}^{2\perp }} = \frac{p_2 \lambda _2^\perp }{p_2 \lambda _2^\perp + p^{(0,1)} \lambda ^\parallel } F_{Y^{2\perp }} + \frac{p^{(0,1)} \lambda ^\parallel }{p_2 \lambda _2^\perp + p^{(0,1)} \lambda ^\parallel } F_{Y^{2\parallel }} \end{aligned} \end{aligned}$$
This is because some customers insure risk 1, but not risk 2 and vice-versa. Therefore, the aggregate process for the insurer is
$$\begin{aligned} \tilde{X}_t = u^{(1)} + u^{(2)} + \big ( c^{(1)} + c^{(2)} \big )t - \sum _{i = 0}^{\tilde{N}_t} \tilde{Y}_i, \end{aligned}$$
where
\(\tilde{N}_t\) is a Poisson process with intensity
$$\begin{aligned} \tilde{\lambda } = p_1 \lambda _1^\perp + p_2 \lambda _2^\perp + \big ( p^{(1,0)} + p^{(0,1)}+ p^{(1,1)} \big )\lambda ^\parallel = p_1 \lambda _1 + p_2 \lambda _2 - p^{(1,1)}\lambda ^\parallel , \end{aligned}$$
(4.1)
and
\(\tilde{Y}_i\),
\(i \in {\mathbb {N}}\) are i.i.d random variables with distribution
$$\begin{aligned} \begin{aligned} F_{\tilde{Y}}&= \frac{p_1 \lambda _1^{\perp } }{\tilde{\lambda }} F_{Y^{1 \perp }} + \frac{p_2 \lambda _2^{\perp } }{\tilde{\lambda }} F_{Y^{2 \perp }} + \frac{p^{(1,0)} \lambda ^{\parallel } }{\tilde{\lambda }} F_{Y^{1 \parallel }} + \frac{p^{(0,1)} \lambda ^{\parallel } }{\tilde{\lambda }} F_{Y^{2 \parallel }} + \frac{p^{(1,1)} \lambda ^{\parallel } }{\tilde{\lambda }} F_{Y^{1 \parallel }+ Y^{2 \parallel }} \\&=\frac{1}{p_1 \lambda _1 + p_2 \lambda _2 - p^{(1,1)} \lambda ^\parallel } \left( p_1 \lambda _1 F_{Y^{1}} + p_2 \lambda _2 F_{Y^{2}} + p^{(1,1)}\lambda ^\parallel \left( F_{Y^{1\parallel } +
Y^{2\parallel }} - F_{Y^{1\parallel }} - F_{Y^{2\parallel }} \right) \right) . \end{aligned} \end{aligned}$$
(4.2)
Thus, if the risks in the market are independent (i.e. if
\(\lambda ^\parallel = 0\)), then the risk in the company’s portfolio is just a sum of the risks
\(S^{(1)}\) and
\(S^{(2)}\), weighted by the respective market shares,
\(p_1\) and
\(p_2\), irrespective of any dependency between sales of policies for different risks. However, if the risks in the market are dependent (
\(\lambda ^\parallel \ne 0\)), then the company’s risk is not, in general, a weighted sum of
\(S^{(1)}\) and
\(S^{(2)}\). Further, this effect persists even in the case where sales of different policies are independent (i.e.,
\(p^{(1,1)} = p_1 p_2\)). On the other hand, equalities (
4.1) and (
4.2) show that in the (unlikely) situation where clients always buy insurance for only one risk, the risk exposure of the insurer is accurately computed using only the marginal distributions of each risk (i.e. assuming that the risks are independent). This is due to homogeneity of our mode, i.e., the intensities
\(\lambda _1^\perp \),
\(\lambda _2^\perp \) and
\(\lambda ^\parallel \) are constants and therefore the model does not account for events that increase temporarily the frequency of claims, like weather phenomena, natural catastrophes, or collective behaviour of customers. If
\(\lambda _1\) and
\(\lambda _2\) are stochastic processes, like in multidimensional Cox [
16] or Hawkes [
28] processes, then some (possibly very important) dependencies remain even if
\(\lambda ^\parallel \) is identically zero. This is work in progress.