2. Multivariate insurance risk model

The firm operates an insurance business comprising $M\in\unicode{x2115}$ underwriting lines, each corresponding to a distinct type of insurance risk. Claims within the same business line share the same loss or severity distribution, and claims from different lines may occur simultaneously. The payout process for a typical customer in line $j\in\left\{1,\ldots.M\right\}$ follows a compound Poisson process of the form

(1) \begin{equation}S_t^{\,j}=\sum_{k=1}^{N_t^{\,j}}Y_k^{\,j}, \ \ t\geq 0,\end{equation}

where $N_t^j$ is a time-homogeneous Poisson process with intensity parameter $\lambda_j,$ and $Y_k^{j}$ are nonnegative i.i.d. random variables independent of $N_t^j$ with absolutely continuous distribution $Y_k^j\sim F_j(dy_j)$ defined on a probability space $(\Omega,\unicode{x2119},\mathcal{F})$ . As usual, the parameter $\lambda_j$ is the arrival rate of claims per unit of time, and $Y_k^j$ models the severity of claims incurred and reported by the customer. Throughout the paper, $y_j$ denotes a realization of the severity in line j, and $F_j(\mathrm{d}y_j)$ is the standard measure-theoretic notation for integration with respect to the marginal law of severity $Y_k^j$ . If the law admits a density $f_j(y_j)$ , then $F_j(\mathrm{d}y_j)=f_j(y_j)\,\mathrm dy_j.$

In order to model dependence in frequencies and severities among business lines, we employ the characterization of the dependence among components of compound Poisson processes via Lévy copulas and tail integrals of spectrally positive Lévy processes. We recall briefly the definition of tail and marginal tail integrals. For details concerning the following definitions see the Appendix A and Kallsen and Tankov (Reference Kallsen and Tankov2006). In what follows, we denote $y=(y_1,\ldots,y_M)^\top\in[0,\infty)^M$ , and $\underline{0}=(0,\ldots,0)^\top$ as the zero vector in $\unicode{x211D}^M$ .

Definition 1. Let $\nu(dy)$ denote the Lévy measure of the multivariate compound Poisson process $S=(S^1,\ldots,S^M).$ The tail integral of $\nu(dy)$ is the function $\mathcal U \; : \; [0, \infty]^{M} \rightarrow[0, \infty]$ defined as

\begin{align*}\mathcal U\left(y_{1}, \ldots, y_{M}\right)\,:\!=\, \begin{cases}0, & \text { if } y_{i}=+\infty \text { for at least one } i \in\{1, \ldots M\}, \\ \nu\left(\left[y_{1}, \infty\right) \times \cdots \times\left[y_{M}, \infty\right)\right) & \text { if }\left(y_{1}, \ldots, y_{M}\right)^{\top} \in[0, \infty)^{M} \backslash\{\underline 0\}, \\ +\infty & \text { if }\left(y_{1}, \ldots, y_{M}\right)^{\top}=\underline 0.\end{cases}\end{align*}

Similarly, the marginal tail integrals $\mathcal U_j$ of $S^{\,j}$ are defined as

\begin{align*}\mathcal U_j(y_j)=\nu_j([y_j,+\infty))\,:\!=\,\lambda_j F_j([y_j,+\infty)), \ \ y_j\gt 0.\end{align*}

Throughout, we assume the marginal claim distributions $F_j$ are absolutely continuous with respect to the Lebesgue measure on $[0,+\infty)$ and do not have atoms at zero, that is, $F_j(\{0\})=0$ for all $\,j=1,\ldots,M$ . Hence, the marginal tail integrals satisfy

\begin{align*}\mathcal U_j(y_j)=\lambda_j\bar F_j(y_j), \ \ y_j\gt 0,\end{align*}

where $\bar F_j$ is the survival function $\bar F_j(y_j)\,:\!=\,F_j((y_j,+\infty)).$ By Sklar’s Theorem for multivariate spectrally positive Lévy processes, there exists a Lévy copula $\mathfrak C \; : \; [0,+\infty]^M\to [0,+\infty]$ such that the tail integral $\mathcal U$ of $\nu(dy)$ satisfies

(2) \begin{equation} \mathcal U(y)=\mathfrak C(\Psi(y;\;\lambda)), \ \ y_j\geq 0, \ y\neq\underline 0\end{equation}

where

\begin{align*}\Psi(y;\;\lambda)\,:\!=\,(\lambda_1 \bar F_1(y_1),\ldots,\lambda_M \bar F_M(y_M)), \ \ \lambda=(\lambda_1,\ldots,\lambda_M)^\top,\end{align*}

Since we assume the marginal tail integrals $\lambda_j \bar F_j$ are absolutely continuous on $(0,+\infty)$ , then $\mathfrak C$ is unique. In what follows, we denote $\nu(dy;\;\lambda)$ to emphasize the dependence on the vector $\lambda\in(0,\infty)^M$ of the Lévy measure of S via the identity (2).

By definition, the tail integral $\mathcal U$ of a Lévy measure is finite except at zero. However, for multivariate compound Poisson processes, it is easy to verify (see Appendix A) that the limit $\mathcal U(\underline 0^+)$ exists and coincides with the arrival rate of common jumps $\lambda^{\|}\gt 0$ , which in turn, given the Lévy-copula characterization (2), satisfies

\begin{align*}\lambda^{\|}=\mathfrak{C}\left(\lambda_{1},\ldots,\lambda_{M}\right).\end{align*}

The role of Lévy copulas is twofold: they provide a complete characterization of the dependence structures for the jump part of multidimensional Lévy processes and enable the construction of a multidimensional Lévy process from a given collection of one-dimensional Lévy processes complying with particular dependence structures.

In the case of multivariate stable Lévy processes, the Lévy copula carries the same information as the spectral measure, see Eder and Klüppelberg (Reference Eder and Klüppelberg2012). Note, however, that the spectral measure restricts to stable processes, whereas the Lévy copula models the dependence for all Lévy processes.

The following two examples of Lévy copulas are presented first intentionally, as they represent extreme dependence structures.

Example 1 (Independence). For an independent multivariate compound Poisson process, the tail integral of the multivariate process is expressed as

\begin{align*}\mathcal U\left(y_1, \ldots, y_M\right)=\mathcal U_1\left(y_1\right) \boldsymbol1_{\left\{y_2=\cdots=y_M=0\right\}}+\cdots+\mathcal U_M\left(y_M\right) \boldsymbol1_{\left\{y_1=\cdots=y_{M-1}=0\right\}},\end{align*}

see Bregman and Klüppelberg (Reference Bregman and Klüppelberg2005). This means that the tail integral of a independent M-dimensional Lévy process is equal to 0 except for the cases where it is equal to the marginal tail integral. In particular, the Lévy measure $\nu$ has support only on the coordinate axes. The independence Lévy copula is given by

\begin{align*}\mathfrak C_{\bot}(u)=\sum_{j=1}^M u_j\boldsymbol{1}_{\left\{u_i=+\infty, i\neq j\right\}}, \ \ u=(u_1,\ldots,u_M)^\top\in[0,\infty]^M.\end{align*}

Note that the independence Lévy copula is entirely different from the ordinary independence copula, which is the product of its components.

Example 2 (Co-monotonic or complete dependence). Complete dependence corresponds to the co-monotonic Lévy copula

\begin{align*}\mathfrak{C}_{\|}(u)=\min \left(u_1, \ldots, u_M\right), \ \ u=(u_1,\ldots,u_M)^\top\in[0,\infty]^M,\end{align*}

see Cont and Tankov (Reference Cont and Tankov2003). This implies that the tail integral of a completely dependent M-dimensional Lévy process is given by the smallest of the marginal tail integrals.

Comonotonicity in a multivariate compound Poisson process implies that all jumps in one dimension are functions of the jumps in the other. However, unless the rates of jumps in the marginal compound Poisson processes are equal, unique jumps will always exist in the multivariate compound Poisson process with a comonotonic Lévy copula. This phenomenon arises from the discontinuity of the tail integral at zero.

In contrast to the independence case, the co-monotonic Lévy copula $\mathfrak{C}_\|(u_1,\ldots,u_M)=\min(u_1,\ldots,u_M)$ coincides with the upper Fréchet–Hoeffding bound for ordinary copulas. This coincidence has a natural explanation: in both the distributional and the Lévy settings, the minimum function characterizes the situation of maximal positive dependence, where all components move together as deterministic functions of a single common source of randomness. Since this extreme property is defined purely through the ordering structure of the marginals, without reference to whether one is working with distribution functions or tail integrals, the same function represents complete dependence in both frameworks.

The following example is analogous to the Archimedean copula construction. It provides a method for obtaining parametric families of Lévy copulas in arbitrary dimensions, with the number of parameters remaining independent of the dimension.

Example 3 (Archimedean-Lévy (AL) copula). Let $\psi\; : \; [0,\infty]\to[0,\infty]$ be continuous and strictly decreasing such that $\psi(0)=+\infty,$ $\psi(+\infty)=0$ and $\psi^{-1}$ is completely monotonic. Then

\begin{align*}\mathfrak C(u)=\psi^{-1}\left(\psi(u_1)+\cdots+\psi(u_M)\right), \ \ u=(u_1,\ldots,u_M)^\top\in[0,\infty]^M\end{align*}

is a Lévy copula, usually referred to as Archimedean-Lévy (AL) copula, and $\psi$ is called a Lévy generator function, see Kallsen and Tankov (Reference Kallsen and Tankov2006). Two emblematic examples of AL copula are

1. 1. Clayton-Lévy copula: $\psi(u)=u^{-\theta}, \ \theta \gt 0$ . The limit cases $\theta\to+\infty$ and $\theta\to 0^+$ correspond to complete dependence and independence, respectively (see Examples 1 and 2 above), so the Clayton–Lévy copula interpolates continuously between the two extreme dependence structures $\mathfrak{C}_\|$ and $\mathfrak{C}_\perp$ .

2. 2. Gumbel–Lévy copula: $\psi(u)=[\ln(u+1)]^{-\theta}, \ \theta\gt 0.$

Remark 2.1. In the bivariate case $M=2$ , the dependence parameter $\theta$ of the Clayton-Lévy copula has a direct interpretation in terms of the linear correlation between the marginal claim frequency processes $N_t^1$ and $N_t^2$ . As shown in Appendix C

\begin{align*}\mathrm{Corr}[N_t^1,N_t^2]= \frac{\lambda^\|}{\sqrt{\lambda_1\lambda_2}}= \frac{(\lambda_1^{-\theta}+\lambda_2^{-\theta})^{-1/\theta}} {\sqrt{\lambda_1\lambda_2}}.\end{align*}

This correlation is strictly increasing in $\theta$ , ranging from 0 as $\theta\to 0^+$ (independence, $\mathfrak{C}_\perp$ in Example 1) to $\min\{\lambda_1,\lambda_2\}/\sqrt{\lambda_1\lambda_2}$ as $\theta\to\infty$ (complete dependence, $\mathfrak{C}_\|$ in Example 2). In particular, the upper bound is strictly less than 1 whenever $\lambda_1\neq\lambda_2$ , confirming that the Clayton–Lévy copula parameter $\theta$ is also a monotone but nonlinear measure of frequency dependence.

With this multivariate dependence structure and the Lévy-copula representation of insurance claims, we are now equipped to address how premium rates can be optimally set to control the insurer’s risk exposure and profitability. The next section formalizes the dynamic premium optimization problem and explicitly incorporates the dependence structure introduced above into the insurer’s decision-making process.

3. Premium control model and optimization problem

Building on the characterization of dependence using Lévy copulas and tail integrals from the previous section, we now explore how premium rates can be used to manage the insurer’s exposure to different risk types. We initiate this analysis by assuming a fixed volume of business for each line: let $n_j$ denote the number of contracts underwritten for risk type j in the insurance portfolio. If the loss per individual policy in line j follows a compound Poisson distribution with arrival rate $\lambda_j$ and severity distribution $F_j$ , then the aggregate loss of $n_j$ policies is also compound Poisson with arrival rate $n_j\lambda_j$ and the same severity distribution $F_j$ . That is, the total payout up to time t of claims from line j has the same distribution as

(3) \begin{equation}L_t^j=\sum_{k=1}^{N_t^{j,n_j}}Y_k^{j},\end{equation}

where $N_t^{j,n_j}$ is a time-homogeneous Poisson process with intensity $n_j\lambda_j$ and $Y_k^j\sim F_j(dy_j)$ , see e.g. Proposition 3.3.4 in Mikosch (Reference Mikosch2004). In other words, the collective risk model with $n_j$ policies is distributionally equivalent to a single-policy model with claim arrival rate $n_j\lambda_j$ .

Based on the previous argument, the total payout of the insurance portfolio up to time $t\geq 0$ is given by

\begin{align*}L_t=\sum_{j=1}^M\sum_{k=1}^{N_t^{j,n_j}}Y_k^{j}.\end{align*}

Given the Lévy-copula characterization (2), we can rewrite this as

\begin{align*}L_t=\int_0^t\int_{\unicode{x211D}_+^M}y^\top\underline 1\,N(dy,dt;\; n\odot\lambda), \ \ t\geq 0,\end{align*}

where $\underline 1=(1,\ldots,1)^\top\in\unicode{x211D}^M$ and $N(dy,dt;\;n\odot\lambda)$ is the counting measure on $\unicode{x211D}_+^M\times [0,\infty)$ with compensator $\nu(dy;\;n\odot\lambda)\,dt,$ and $n\odot\lambda = (n_1\lambda_1,\ldots,n_M\lambda_N)^\top$ denotes the element-wise (Hadamard) multiplication of the vector of numbers of contracts $n=(n_1,\ldots,n_M)^\top$ with the vector of claim arrival rates $\lambda=(\lambda_1,\ldots,\lambda_M)^\top.$

The insurer receives constant-valued premiums from policyholders as payment for the insurance policies. Let $p_j$ denote the premium rate per unit of time charged to each individual policyholder for insurance risk type $j=1,\ldots,M.$ Then, the revenue per unit of time is given by $n^\top p$ , where $p=(p_1,\ldots,p_M)^\top\in[0,+\infty)^M.$ As in Christensen et al. (Reference Christensen, Parra-Alvarez and Serrano2021), we assume the insurance portfolio includes as many policies as the firm can attract in the insurance market, given the premiums offered by the firm.

Namely, we assume the number of contracts in line j depends on the premium vector p via a demand law function $n_j(p).$ The arrival rate $\lambda_j(p)$ is also assumed to depend on p to allow for adverse selection effects associated with a shift in coverage toward higher pooled risks as the premium increases. Hence, the quantities $p_j$ are used by the company to control its exposure to each type of insurance risk. For a given premium vector $p=(p_1,\ldots,p_M)^\top\in[0,+\infty)^M$ , the cumulative gains/losses to the insurance portfolio are given by the surplus process

\begin{align*}I_t^p\,:\!=\,n(p)^\top pt-\int_0^t\int_{\unicode{x211D}_+^M}y^\top\underline 1\,\mathcal N(dy,dt;\;p), \ \ t\geq 0,\end{align*}

where $\mathcal N(dy,dt;\;p)$ is the counting measure on $\unicode{x211D}_+^M\times [0,\infty)$ with compensator $\pi(dy;\;p)\,dt$ , where $\pi(dy;\;p)$ is the Lévy measure controlled by p via $n\odot\lambda$

\begin{align*}\pi(dy;\;p)=\nu(dy;\;[n\odot\lambda](p)).\end{align*}

In particular, in this setting, if the Lévy-copula $\mathfrak C$ associated with the Lévy measure $\nu(dy;\;\lambda)$ is continuous, then the (aggregate) joint claim arrival rate is also a function of the premium vector $p\in[0,+\infty)^M$

\begin{align*}\lambda^{\|}(p)=\mathfrak{C}\left([n\odot\lambda](p)\right).\end{align*}

We assume that the firm backs the insurance liabilities with the collected premiums and the returns from investing in a risk-free money-market account with continuously compounded interest rate $R\gt 0$ . Let $V_t$ denote the value of the reserve process at time $t\geq 0$ . Then, the gains/losses $G_t$ of investing the reserve in the risk-free money-market account satisfy $dG_t=RV_t\,dt.$

Given an initial endowment $x\in\mathbb{R}$ , the reserve process $V_t$ is funded from two sources: investment returns $G_t$ from the risk-free money-market account, and the cumulative insurance surplus $I_t^p$ . Any excess $x+G_t+I_t^p-V_t\geq 0$ is available for consumption outflow such as dividend payments to the owner-manager or administrative expenses. We allow for negative initial wealth $x\lt 0$ , corresponding to an insurer that has borrowed against future premium income, which is a standard assumption in the surplus process literature; see e.g. Asmussen et al. (Reference Asmussen, Christensen and Taksar2013).

Given a premium vector $p\in[0,\infty)^M$ and an instantaneous consumption rate process $c=(c_t)_{t\geq 0}$ , the policy pair (p, c) is said to be self-financed if the reserve process satisfies the budget constraint

(4) \begin{equation} dV_t = (RV_t - c_t)\,dt + dI_t^p, \qquad V_0 = x\in\mathbb{R},\end{equation}

with p and c as control variables. We assume that the firm can borrow from the risk-free money-market account at the same rate R, so that the budget constraint (4) remains self-financed for any initial wealth $x\in\mathbb{R}$ . Note that the premium vector p affects the wealth process through two interacting channels. On the revenue side, higher premiums increase the unit price but reduce demand, so the net effect on total revenue $n(p)^\top p$ is ambiguous and depends on demand elasticity. On the cost side, lower demand reduces the effective claim exposure. The insurer therefore faces a trade-off between risk exposure and revenue generation.

The goal of the firm is to find the strategy (c, p) that maximizes the expected discounted utility from consumption

\begin{align*}\mathcal{J}(p,c)\,:\!=\, \mathbb{E}\!\left[\int_0^\infty e^{-\delta t}U(c_t)\,dt \,\Bigl|\,V_0=x\Bigr.\right],\end{align*}

for the discount rate $\delta\gt 0$ and the exponential utility function $U(x)=-\frac{1}{\eta}e^{-\eta x}$ with constant absolute risk aversion coefficient $\eta\gt 0$ .

The policy (p, c) is admissible if $\mathcal{J}(p,c)$ is strictly negative for all initial values $x\in\mathbb{R}$ . Since $U(x)=-\frac{1}{\eta}e^{-\eta x}\lt 0$ for all x, the value function $\mathcal{J}(p,c)$ is always nonpositive, and the admissibility condition $\mathcal{J}(p,c)\lt 0$ excludes degenerate policies under which the discounted utility integral converges to zero. This occurs when consumption grows so rapidly that the exponential dis-utility $e^{-\eta c_t}$ vanishes in the limit, making the integral trivially zero rather than reflecting a genuinely optimal trade-off. Admissibility therefore imposes a regularity condition on the growth rate of $c_t$ rather than a solvency constraint. In what follows, we denote the set of admissible policies by $\mathcal{A}$ .

Exponential utility functions play a prominent role in insurance mathematics and actuarial practice, since they are the only utility functions under which the principle of “zero utility” gives a fair premium that is independent of the level of reserves of an insurance company, see, e.g., Section II.6. This optimization criterion has also the following interesting interpretation: instead of considering the expected utility from consumption one can equivalently maximize the (time-integral) of $\frac{1}{\eta}\ln \unicode{x1D53C}[e^{-\eta c_t}].$ This is the entropic risk measure of consumption. For small values of the risk-aversion coefficient, this is approximately equal to the expected consumption penalized by its variance. This is a risk-sensitive criterion, but can also be seen as the Lagrange function of a mean-variance problem, see, e.g., Chapter 4 of Föllmer and Schied (Reference Föllmer and Schied2011).

Another interesting feature of this criterion is that it has a dual representation related to relative entropy or Kullback–Leibler distance between the probability measure $\unicode{x2119}$ and measures $\unicode{x211A}\ll\unicode{x2119}.$ Here $\unicode{x2119}$ is the reference probability measure under which the wealth dynamics (4) are specified, and $\unicode{x211A}\ll\unicode{x2119}$ denotes an absolutely continuous alternative measure representing a perturbation of the baseline model. From this representation, it is easy to see that the case $\eta\to+\infty$ corresponds to the case of a robust optimization problem or worst-case optimization problem in which the insurer maximizes the expected consumption if nature chooses the least favorable measure for the model, see, e.g., Bäuerle and Leimcke (Reference Bäuerle and Leimcke2021). For $\eta\gt 0$ , this means that potentially a whole range of beliefs about $\unicode{x2119}$ is considered but deviations from the baseline model $\unicode{x2119}$ are penalized. This discussion illustrates the versatility of the exponential utility function: by adjusting the risk aversion coefficient, the insurer can shift between a risk sensitive and a robust approach, all while preserving high tractability, as explored in subsequent sections.

4. Hamilton–Jacobi–Bellman equation and main result

The HJB equation characterizes the optimal consumption and premium strategies by balancing immediate utility from consumption with the expected future utility, accounting for changes in the insurer’s wealth due to investment returns, premium income, and stochastic claim outflows.

As usual, if there exists an admissible optimal pair $(\hat c,\hat p)\in\mathcal A$ , and the optimal value function

\begin{align*}\vartheta(x)=\sup_{(c,p)\in \mathcal A}\unicode{x1D53C}\left[\int_0^\infty e^{-\delta t}U(c_t)\,dt\,\Bigl|\,V_0=x \Bigr.\right],\end{align*}

is sufficiently differentiable and satisfies the transversality condition

(5) \begin{equation}\lim_{T\to\infty}\unicode{x1D53C} \left[e^{-\delta T}\vartheta(V^{\hat c,\hat p}_T)\right]=0,\end{equation}

then $\vartheta(x)$ solves the non-linear integral-differential equation of the HJB type

\begin{align*}-\delta\vartheta(x)+\sup_{(c,p)\in\unicode{x211D}\times[0\infty)^M}\left \{U(c)+[\mathcal{L}^{c,p}\vartheta](x)\right\}=0, \ \ x\in\unicode{x211D},\end{align*}

where $\mathcal{L}^{c,p}$ is the operator

\begin{align*} [\mathcal{L}^{c,p}\vartheta](x) =&\bigl[Rx+n(p)^\top p-c\bigr]\vartheta'(x)+\int_{[0,+\infty)^M\setminus\left\{\underline 0\right\}}\Bigl[\vartheta(x-y^\top\underline 1)-\vartheta(x)\Bigr]\,\pi(dy;\;p),\end{align*}

see, e.g., Section III.7 of Fleming and Soner (Reference Fleming and Soner2006). Conversely, the so-called verification theorem links the solution of the above HJB equation with sufficient conditions for existence of optimal strategies. In order to find a smooth solution, we propose the following educated guess

\begin{align*}\vartheta(x)=-\frac{g(x)}{R\eta}, \ \ g(x)=e^{-R\eta(x+\mathcal B)},\end{align*}

with $\mathcal B\in\unicode{x211D}$ to be determined below. Inserting this guess in the HJB equation and dividing through by g(x), we end up with the following optimization problems: maximize

(6) \begin{equation} U(c)-ce^{-R\eta(x+\mathcal B)} \end{equation}

over $c\in\unicode{x211D},$ and

(7) \begin{equation} Q(p)\,:\!=\,n(p)^\top p-\frac{1}{R\eta}\int_{\unicode{x211D}_+^M}\bigl[e^{R\eta y^\top\underline 1}-1\bigr]\,\pi(dy;\;p),\end{equation}

over $p\in[0,+\infty)^M.$ By inserting our candidate exponential solution for the value function into the HJB equation, we separate the problem into two independent optimization problems. Intuitively, (7) captures the dynamic trade-off faced by the insurer: higher premiums increase revenue but might reduce demand, while lower premiums boost sales but increase exposure to claim risks.

The integral term represents the expected exponential utility cost arising from claims, adjusted for risk aversion and dependence among insurance lines captured by the Lévy copula. The resulting structure highlights the insurer’s need to strategically balance revenue generation and risk management through premium setting. We have the following verification result

Proposition 4.1. Suppose there exists $\hat p$ that maximizes $Q(p).$ Let $\hat c(x)$ be the Markov feedback consumption policy

(8) \begin{equation} \hat c(x)=Rx+Q(\hat p)+\frac{1}{\eta}\Bigl[\frac{\delta}{R}-1\Bigr]. \end{equation}

Then, the strategy $(\hat c(x),\hat p)$ is optimal.

Proof. The optimal policy that maximizes (6) is given by $\hat c(x)=R(x+\mathcal B).$ Using this in the terms with consumption in the HJB equation, we get

\begin{align*}U(\hat c(x))+[Rx-\hat c(x)]\vartheta'(x)&=-\frac{1}{\eta}e^{-R\eta(x+\mathcal B)}-R\mathcal B e^{-R\eta(x+\mathcal B)}\\&=-\bigg[\frac{1}{\eta}+R\mathcal B\bigg]g(x).\end{align*}

Note the first factor in the right-hand side does not depend on $x\in\unicode{x211D}.$ Inserting the candidates for optimal policies in the HJB equation and dividing through by g(x), we get

\begin{align*}\frac{1}{R\eta}(\delta -R)-R\mathcal B+Q(\hat p)=0,\end{align*}

so that (8) follows. It only remains to prove that the transversality (5) holds. Using (8), we see that the wealth process $\hat V=V^{\hat c,\hat p}$ satisfies

\begin{align*} d\hat V_t=\Bigl\{-\frac{1}{R\eta}(\delta -R)+\frac{1}{R\eta}\int_{\unicode{x211D}_+^M}\bigl[e^{R\eta y^\top\underline 1}-1\bigr]\,\pi(dy;\;\hat p)\Bigr\}\,dt-\int_{\unicode{x211D}_+^M}y^\top\underline 1\,\mathcal N(dy,dt;\;\hat p).\end{align*}

Applying Itô’s lemma for the jump process $\hat V$ with the value function $\vartheta(x)=-g(x)/(R\eta),$ and using $\vartheta'(x)=g(x),$ we get

\begin{align*}d\vartheta(\hat V_{t})&=\vartheta'(\hat V_{t-})\,d\hat V_t+d\,\left\{\sum_{0\lt s\le t}\left[\vartheta(\hat V_s)-\vartheta(\hat V_{s-})-\vartheta'(\hat V_{s-})\Delta \hat V_{s}\right]\right\} \\&=g(\hat V_{t-})\,d\hat V^{\textbf{c}}_t+d\,\left\{\sum_{0 \lt s\le t}\left[\vartheta(\hat V_s)-\vartheta(\hat V_{s-})\right]\right\}.\end{align*}

Here, $d[\!\cdot\!]$ denotes the differential of a finite-variation process, $\hat{V}_{t-}=\lim_{s\nearrow t}\hat{V}_s$ denotes the left limit of the wealth process at time t, and $\Delta\hat{V}_s=\hat{V}_s-\hat{V}_{s-}$ is the jump of $\hat{V}$ at time $s\le t$ . The sum $\sum_{0 \lt s\leq t}$ aggregates these individual jump contributions over all jump times in the interval (0, t]. The process $\hat V^{\textbf{c}}$ is the continuous part of $\hat V$ with differential

\begin{align*}d\hat V^{\textbf{c}}_t=\Bigl[-\frac{1}{R\eta}(\delta -R)+\frac{1}{R\eta}\int_{\unicode{x211D}_+^M}\bigl[e^{R\eta y^\top\underline 1}-1\bigr]\,\pi(dy;\;\hat p)\Bigr]\,dt.\end{align*}

We can write the sum of the increments $\vartheta(\hat V_s)-\vartheta(\hat V_{s-})$ as an integral with respect to the random counting measure $\mathcal N(\mathrm{d} y,\mathrm{d} t;\;\hat p)$ as follows:

\begin{align*}\sum_{0 \lt s\le t}\left[\vartheta(\hat V_{s})-\vartheta(\hat V_{s-})\right]&=\sum_{0 \lt s\le t}\left[\vartheta(\hat V_{s-}+\Delta \hat V_{s})-\vartheta(\hat V_{s-})\right]\\&=\int_0^t\int_{\unicode{x211D}^M_+}\vartheta(\hat V_{s-})[e^{R\eta y^\top\underline 1}-1]\,\mathcal N(\mathrm{d} y,\mathrm{d} s;\;\hat p).\end{align*}

Compensating these jump integrals and taking expected values, we get

\begin{align*}\unicode{x1D53C}\left\{\sum_{0 \lt s\le t}\bigl[\vartheta(\hat V_{s})-\vartheta(\hat V_{s-})\bigr]\right\}=\unicode{x1D53C}\left\{\int_0^t\int_{\unicode{x211D}^M_+}\vartheta(\hat V_{s-})\bigl[e^{R\eta y^\top\underline 1}-1\bigr]\,\pi(\mathrm{d} y;\;\hat p)\,\mathrm{d} s\right\},\end{align*}

see, e.g., Proposition 14.19 of Pascucci (Reference Pascucci2011). Using again $\vartheta(x)=-g(x)/(R\eta)$ and the definition (7) of Q(p), we see that the function $h(t)\,:\!=\,\unicode{x1D53C}[\vartheta(\hat V_{t-})|\hat V_{0}=x_0]$ satisfies

\begin{align*}h'(t)=\frac{d}{dt}\unicode{x1D53C}[\vartheta(\hat V_{t-})|\hat V_{0}=x_0]=(\delta-R)h(t),\end{align*}

with $h(0)=\vartheta(x_0).$ Therefore, the transversality condition

\begin{equation*} e^{-\delta T}\unicode{x1D53C}[\vartheta(\hat V_{T-})]=e^{-R T}\vartheta(x_0)\to 0, \ \mbox{ as } T\to\infty,\end{equation*}

holds. Thus, the conditions for the verification theorem linking the solution to the HJB equation with sufficient conditions for the existence of optimal strategies are satisfied, and the desired result follows, see, e.g., Theorem 9.1 in Section III.9 of Fleming and Soner (Reference Fleming and Soner2006).

Then, Q(p) is continuous on $[0,\infty)^M$ and attains a maximum on some compact cube $[0,P]^M$ .

Note that uniqueness generally fails without additional concavity structure. However, our results do not require uniqueness.

Remark 4.3. Since $n(p)^\top p$ is the insurer’s revenue per unit of time, the optimization (7) is equivalent to the long-run profit maximization problem for a multiproduct monopolist with nonseparable cost function

\begin{align*} H(n,p) = \frac{1}{R\eta}\int_{\mathbb{R}_+^M} \bigl[e^{R\eta y^\top\underline{1}}-1\bigr]\, \nu\!\left(\mathrm{d}y;\;n\odot\lambda(p)\right),\end{align*}

so that $Q(p) = n(p)^\top p - H(n(p),p)$ . The nonseparability identified is precisely what distinguishes this problem from M independent single-line monopolist problems: even if $\lambda$ is independent of p and the demand functions $n_j(p_j)$ are separable (each $n_j$ depends only on its own price $p_j$ ), the first-order conditions for $\hat{p}$ are coupled across lines through the cross-derivatives of H(n(p), p).

5. Numerical example

In this section, we present a numerical example to illustrate the premium optimization framework. The time unit is years. The firm underwrites insurance for $M=2$ business lines, and marginal tail integrals $\lambda_j\bar F_j$ are assumed to be absolutely continuous, with $f_j$ denoting the density function of the distribution $F_j(dy_j).$ For simplicity, we assume constant arrival rates $\lambda_j$ for each insurance line throughout the numerical example. This means the illustration focuses exclusively on the demand effect of pricing through the logit functions $n_j(p)$ , while holding claim intensities fixed.

We follow the notation of Section 2. In the two-dimensional case, if the Lévy copula $\mathfrak C$ is sufficiently smooth, the Lévy density $f(y;\;\lambda)$ of the Lévy measure $\nu(dy;\;\lambda)$ can be obtained through differentiation

\begin{align*}f(y_1,y_2;\;\lambda_1,\lambda_2)=\mathfrak c(\lambda_1\bar F_1(y_1),\lambda_2\bar F_2(y_2))\lambda_1 \lambda_2f_1(y_1)f_2(y_2), \ \ \mathfrak c(u_1,u_2)=\frac{\partial^2\mathfrak C}{\partial u_1\partial u_2},\end{align*}

see, e.g., Theorem 5.4 in Cont and Tankov (Reference Cont and Tankov2003). Using this and the decomposition of bivariate compound Poisson processes from Appendix C, the integral with respect to the Lévy measure $\pi(dy;\;p)=\nu(dy;\;[n\odot\lambda](p))$ splits into three mutually singular components

(9) \begin{equation}\int_{\mathbb{R}_+^2}\bigl[e^{R\eta(y_1+y_2)}-1\bigr]\,\pi(dy;\;p)= I^{\|}(p) + I_1^{\perp}(p) + I_2^{\perp}(p),\end{equation}

where the joint component

(10) \begin{equation}I^{\|}(p)= \prod_{j=1}^2 \lambda_j n_j(p) \int_{\mathbb{R}_+^2} \bigl[e^{R\eta(y_1+y_2)}-1\bigr] \mathfrak{c}\bigl(\lambda_1 n_1(p)\bar{F}_1(y_1),\, \lambda_2 n_2(p)\bar{F}_2(y_2)\bigr) f_1(y_1)f_2(y_2)\,dy_1\,dy_2,\end{equation}

accounts for simultaneous claims in both lines and the idiosyncratic components

(11) \begin{align}I_1^{\perp}(p)&= \lambda_1 n_1(p)\int_{\mathbb{R}_+} \bigl[e^{R\eta y_1}-1\bigr] \biggl[1 - \frac{\partial\mathfrak{C}}{\partial u_1} \bigl(\lambda_1 n_1(p)\bar{F}_1(y_1), \lambda_2 n_2(p)\bigr)\biggr] \, f_1(y_1)\,dy_1,\end{align}

(12) \begin{align}I_2^{\perp}(p)&= \lambda_2 n_2(p)\int_{\mathbb{R}_+} \bigl[e^{R\eta y_2}-1\bigr] \biggl[1 - \frac{\partial\mathfrak{C}}{\partial u_2} \bigl(\lambda_1 n_1(p), \lambda_2 n_2(p)\bar{F}_2(y_2)\bigr)\biggr] \, f_2(y_2)\,dy_2,\end{align}

account for claims occurring in line j alone with no simultaneous event in the other line. In the line-1 integral, the line-2 threshold is $y_2=0$ , so $\bar{F}_2(0)=1$ and the second argument of $\mathfrak{C}$ reduces to the full exposure $\lambda_2 n_2(p)$ . The term $\partial\mathfrak{C}/\partial u_1(u_1\bar{F}_1(y_1),u_2)$ is therefore the conditional probability that a line-1 claim of size $y_1$ is accompanied by a simultaneous line-2 event. The same reasoning applies symmetrically to $I_2^\perp$ .

After the change of variables $x_j = \bar{F}_j(y_j)$ , the three integrals become

(13) \begin{align}I^{\|}(p) &= u_1 u_2 \int_{[0,1]^2} \bigl[e^{R\eta[F_1^{-1}(1-x_1)+F_2^{-1}(1-x_2)]}-1\bigr] \mathfrak{c}(u_1 x_1,\,u_2 x_2)\,dx_1\,dx_2, \end{align}

(14) \begin{align}I_j^{\perp}(p) &= u_j \int_0^1 \bigl[e^{R\eta F_j^{-1}(1-x_j)}-1\bigr] \Bigl[1 - \frac{\partial\mathfrak{C}}{\partial u_j} (u_1 x_1,\, u_2 x_2)\big|_{x_{-j}=1}\Bigr] dx_j, \quad j=1,2, \end{align}

where $u_j = \lambda_j n_j(p)$ and $x_{-j}=1$ means the other variable is evaluated at the upper boundary of the integration domain.

For this example, we assume dependence in frequency and severity among the two business lines is modeled using a Clayton–Lévy copula

\begin{align*}\mathfrak C_\theta(u_1,u_2)=(u_1^{-\theta}+u_2^{-\theta})^{-1/\theta},\end{align*}

where the dependence parameter $\theta\gt 0$ affects how claims and arrival rates from the two insurance lines are correlated, which in turn influences risk pooling, loss variability, and pricing strategies. In this case, we have

\begin{align*}\mathfrak c_\theta(u_1,u_2)=\frac{\partial^2\mathfrak C_\theta}{\partial u_1\partial u_2}&=(\theta+1)\,(u_1u_2)^{-\theta-1}\,(u_1^{-\theta}+u_2^{-\theta})^{-2-\frac{1}{\theta}}\\&=(\theta+1)(u_1u_2)^\theta(u_1^{\theta}+u_2^{\theta})^{-2-\frac{1}{\theta}}.\end{align*}

Note that the decomposition (9) is consistent with the marginal Lévy–Khintchine formula: as $\theta\to 0^+$ (independence), $\partial\mathfrak{C}_\theta/\partial u_j\to 0$ and $I^{\|}\to 0$ , so

(15) \begin{equation} I^{\|}(p) + I_1^\perp(p) + I_2^\perp(p) \;\xrightarrow{\theta\to 0^+}\; \sum_{j=1}^2 u_j\int_{\mathbb{R}_+} \bigl[e^{R\eta y_j}-1\bigr]\,f_j(y_j)\,dy_j,\end{equation}

recovering the sum of two independent marginal Lévy integrals, as expected. For each $j=1,2$ , the claim severity $Y^j$ is assumed to have a Gamma distribution with density function

\begin{align*}f_j(y_j)=\frac{1}{\Gamma(\alpha_j)\gamma_j^{\alpha_j}}y_j^{\alpha_j-1}e^{-y_j/\gamma_j}, \ \ y_j\geq0.\end{align*}

Here, $\alpha_j\gt 0$ is the shape parameter and $\gamma_j\gt 0$ is the scale parameter. The number of policies in each line is given by the (multiproduct) logit demand functions of the form

\begin{align*}n_j(p)=N\frac{e^{-b_jp_j}}{1+e^{-b_1p_1}+e^{-b_2p_2}}, \ \ j=1,2,\end{align*}

with coefficients $b_j\gt 0$ and market size N. Equivalently, $n_j(p)/N$ is the multinomial-logit probability that a potential customer chooses product j as a function of the vector prices p. Similar demand functions have been used recently by Gudmundarson et al. (Reference Gudmundarson, Guerra and Moura2023). These functions capture customer choice behavior and price sensitivity in insurance competitive markets. They also account for substitution effects, ensuring that demand shifts dynamically among available insurance products as premiums change. The remaining parameters are $N=1,000$ , $\eta=4,$ $R=4\%,$ and those listed in Table 1. This calibration is not empirically identifiable or grounded in observed data. It is, however, reasonable and intended purely for illustration purposes. Figure 1 contains the plot of Q(p) for $\theta=0.5$ (left) and $\theta=3$ (right).

Table 1.

Parameter values for lines $j=1,2.$

Table 1. Long description

A table titled Parameter values for lines with two columns and four rows. The columns are labeled j = 1 and j = 2. The rows are labeled with different parameters: alpha_j, gamma_j, lambda_j, and b_j. Row 1: alpha_j, 2.6, 2.0. Row 2: gamma_j, 1.0, 1.4. Row 3: lambda_j, 0.8, 0.5. Row 4: b_j, 0.48, 0.71.

Figure 1.

Plots of function Q(p).

Figure 2 shows level curves of Q(p) for $\eta\in\{2,4\}$ and $\theta\in\{0.2,1.0,3.0\}$ . The optimal premium vectors are listed in Table 2, computed via two-stage grid search over $[0.4,21]^2$ .

Figure 2.

Level curves of Q(p) for $\eta=2,4$ and $\theta=0.2,1,3.$

Figure 2. Long description

Panel A: A contour plot with eta = 2.0 and theta = 0.2. The x-axis is labeled p1 and ranges from 0 to 15. The y-axis is labeled p2 and ranges from 0 to 14. The color bar ranges from -945 to 270. Panel B: A contour plot with eta = 4.0 and theta = 0.2. The x-axis is labeled p1 and ranges from 0 to 15. The y-axis is labeled p2 and ranges from 0 to 14. The color bar ranges from -1200 to 150. Panel C: A contour plot with eta = 2.0 and theta = 1.0. The x-axis is labeled p1 and ranges from 0 to 15. The y-axis is labeled p2 and ranges from 0 to 14. The color bar ranges from -1050 to 300. Panel D: A contour plot with eta = 4.0 and theta = 1.0. The x-axis is labeled p1 and ranges from 0 to 15. The y-axis is labeled p2 and ranges from 0 to 14. The color bar ranges from -1440 to 180. Panel E: A contour plot with eta = 2.0 and theta = 3.0. The x-axis is labeled p1 and ranges from 0 to 15. The y-axis is labeled p2 and ranges from 0 to 14. The color bar ranges from -1050 to 300. Panel F: A contour plot with eta = 4.0 and theta = 3.0. The x-axis is labeled p1 and ranges from 0 to 15. The y-axis is labeled p2 and ranges from 0 to 14. The color bar ranges from -1600 to 200.

Table 2.

Optimal premium vector $\hat{p}=(\hat{p}_1,\hat{p}_2)$ and optimal value $\hat{Q}$ for $\eta\in\{2,4\}$ and $\theta\in\{0.2,1.0,3.0\}$ .

Table 2. Long description

A table with four rows and five columns. The columns are labeled theta, eta, p-hat-1, p-hat-2, and Q-hat. The rows present data for different combinations of theta and eta values. Row 1: theta 0.2, eta 2, p-hat-1 4.765, p-hat-2 3.354, Q-hat 341.8. Row 2: theta 0.2, eta 4, p-hat-1 5.081, p-hat-2 3.613, Q-hat 283.9. Row 3: theta 1.0, eta 2, p-hat-1 4.765, p-hat-2 3.626, Q-hat 316.3. Row 4: theta 1.0, eta 4, p-hat-1 5.030, p-hat-2 4.980, Q-hat 224.0. Row 5: theta 3.0, eta 2, p-hat-1 4.689, p-hat-2 3.904, Q-hat 305.2. Row 6: theta 3.0, eta 4, p-hat-1 4.891, p-hat-2 5.954, Q-hat 217.8.

Two patterns are immediately visible. First, for fixed $\theta$ , higher risk aversion ( $\eta=4$ , right column) produces a more concentrated profitable region and a lower optimal value $\hat{Q}$ , as the insurer assigns greater weight to the exponential cost of claims. Second, for fixed $\eta$ , increasing dependence ( $\theta$ rising across rows) shifts the optimal point rightward in $p_2$ and compresses the profitable region, with the effect substantially more pronounced at $\eta=4$ than at $\eta=2$ . At $\eta=2$ , the optimal line-2 premium $\hat{p}_2$ rises only modestly from $3.35$ to $3.90$ as $\theta$ increases from $0.2$ to $3.0$ . However, at $\eta=4$ , the same increase in $\theta$ drives $\hat{p}_2$ from $3.61$ to $5.95$ . This is a 65% increase in line-2 premium that reduces the optimal value $\hat{Q}$ by $23.3\%$ . The level curves at $\eta=4$ show a visibly more concentrated profitable region that contracts sharply across rows, making the dependence effect clearly legible in the figure.

All computations were carried out in Python 3 using the NumPy and SciPy libraries, with parallelization provided by joblib. The two-dimensional integral over $[0,1]^2$ in Q(p) was evaluated using scipy.integrate.dblquad, an adaptive Gaussian quadrature routine, with absolute and relative tolerances set to $\varepsilon_{\mathrm{abs}}=\varepsilon_{\mathrm{rel}}=10^{-4}$ . The integration domain was restricted to $[\varepsilon,1-\varepsilon]^2$ with $\varepsilon=0.01$ to avoid the boundary singularities of the Clayton–Lévy copula density $\mathfrak{c}_\theta(u_1,u_2)$ near $u_j=0$ and of the inverse survival functions $F_{Y_j}^{-1}(1-x_j)$ near $x_j\in\{0,1\}$ . The inverse CDFs $F_{Y_j}^{-1}$ for the Gamma $(\alpha_j,\gamma_j)$ marginals were precomputed on an uniform grid of 5,000 nodes over $[\varepsilon,1-\varepsilon]$ and evaluated inside the quadrature loop via linear interpolation, which avoids repeated calls to scipy.stats.gamma.ppf at each quadrature node and yields a tenfold reduction in computation time per integral evaluation. The optimal premium vector $\hat{p}(\theta)$ was located via a two-stage grid search: a coarse $60\times 60$ grid over $[0.4,\,21]^2$ identified the approximate location of the maximum, followed by a fine $80\times 80$ grid over a neighborhood of radius 3 around the coarse optimum, yielding a final grid resolution of approximately $0.08$ per dimension. Grid evaluations were parallelized across all available CPU cores using joblib.Parallel. The replication code is provided in Appendix D.

6. Sensitivity of optimal value with respect to the Lévy copula

The numerical example of the previous section illustrates how the profile Hamiltonian Q(p) and its maximizer $\hat{p}(\theta)$ depend on the severity distributions and risk-aversion coefficient. We now turn to a systematic analysis of how the dependence parameter $\theta$ of the Clayton–Lévy copula affects the insurer’s optimal premiums and long-run profitability. By leveraging stochastic ordering techniques for Lévy processes in two dimensions, we first establish the theoretical guarantee that stronger dependence reduces optimal profitability, then quantify this effect numerically and assess the cost of misspecifying $\theta$ .

Theorem 6.1. Let $M=2$ and let $\mathfrak C,\tilde{\mathfrak C}$ be two Lévy copulas satisfying $\mathfrak C\le\tilde{\mathfrak C}$ pointwise, i.e. $\mathfrak{C}(u_1,u_2)\leq\tilde{\mathfrak{C}}(u_1,u_2)$ for all $(u_1,u_2)\in[0,\infty)^2$ . Let $\nu$ and $\tilde\nu$ be Lévy measures with tail integrals

\begin{align*}\mathcal U(y)=\mathfrak C(\Psi(y;\;\lambda\odot n)), \ \ \tilde{\mathcal U}(y)=\tilde{\mathfrak C}(\Psi(y;\;\lambda\odot n)),\end{align*}

respectively, and the same marginal tail integrals. Let Q(p) and $\tilde Q(p)$ be the profile Hamiltonians defined as in (7) with compensators

\begin{align*}\pi(dy;\;p)=\nu(dy;\;[n\odot\lambda](p)), \ \ \tilde\pi(dy;\;p)=\tilde\nu(dy;\;[n\odot\lambda](p)),\end{align*}

respectively. Then,

\begin{align*}Q(p)\geq \tilde Q(p), \ \ \mbox{ for all }p=(p_1,p_2)\in[0,+\infty)^2.\end{align*}

Proof. Let $\mathcal N(dy,dt;\; p)$ and $\tilde{\mathcal N}(dy,dt;\; p)$ denote the counting measures on $\unicode{x211D}_+^M\times [0,\infty)$ with compensators $\pi(dy;\;p)$ and $\tilde\pi(dy;\;p)$ , and let $S_t^p$ and ${\tilde S}_t^p$ be the corresponding 2-dimensional compound Poisson processes. Since the function $y\mapsto e^{R\eta (y_1+y_2)}$ is super-modular (see Definition 3 in Appendix B) by Theorem B.1 in Appendix B, we have

\begin{align*}\unicode{x1D53C}\bigl[e^{R\eta {S_t^p}^\top\underline 1}\bigr]\le \unicode{x1D53C}\bigl[e^{R\eta\tilde{S_t^p}^\top\underline 1}\bigr]\end{align*}

for all $p\in[0,+\infty)^2$ . By the Lévy–Khintchine formula, it follows that

\begin{align*}\int_{\unicode{x211D}_+^2}\bigl[e^{R\eta (y_1+y_2)}-1\bigr]\,\pi(dy;\;p)\leq \int_{\unicode{x211D}_+^2}\bigl[e^{R\eta (y_1+y_2)}-1\bigr]\,\tilde\pi(dy;\;p).\end{align*}

Using the definition of Q(p), we obtain the desired result.

In particular, ignoring or underestimating claim dependence results in a welfare loss for the insurer, in the sense that the optimal value $\hat{Q}$ achievable under the true model exceeds that achievable under the misspecified model with weaker dependence. In the case of Lévy–Clayton copulas, we have the following result

Proof. Let $\hat p(\theta)$ be the vector of optimal prices for dependence parameter $\theta.$ Then,

\begin{align*}\hat Q_\theta=Q(\hat p(\theta);\;\theta)\geq Q(\hat p(\theta');\;\theta),\end{align*}

and by the previous Theorem, we have $Q(\hat p(\theta');\;\theta)\geq Q(\hat p(\theta');\;\theta')=\hat Q_{\theta'}.$

Figure 3 shows the optimal premium vector $\hat{p}(\theta)=(\hat{p}_1(\theta),\hat{p}_2(\theta))$ and the corresponding optimal value $\hat{Q}(\theta)$ as functions of $\theta\in[0.2,3.0]$ for $\eta=4$ .

Figure 3.

Optimal premium vector $\hat{p}(\theta)=(\hat{p}_1(\theta),\hat{p}_2(\theta))$ (left and center panels) and optimal value $\hat{Q}(\theta)$ (right panel) as functions of the Clayton–Lévy dependence parameter $\theta$ , for $\eta=4$ . Computed via two-stage grid search over $[0.4,21]^2$ .

Note that $\hat{Q}(\theta)$ is strictly decreasing in $\theta$ throughout, falling from $283.9$ at $\theta=0.2$ to $217.8$ at $\theta=3.0$ . The monotone decrease in $\hat{Q}$ coexists with a nonmonotone response in the optimal premium vector. Indeed, for low dependence $\theta\lt 1.0$ , the optimal line-2 premium $\hat{p}_2$ rises from $3.61$ to $4.98$ , reflecting the insurer’s incentive to suppress correlated line-2 demand as joint claim exposure grows. Meanwhile, $\hat{p}_1$ moves modestly, rising to $5.18$ then retreating to $5.03$ . Here, line-1 pricing is influenced by the nonseparability of H(n, p) but is not the primary instrument of adjustment.

For $\theta\geq 1.4$ , the optimal premiums move only marginally. This plateau arises because the Clayton–Lévy copula $C_\theta$ approaches its co-monotone limit as $\theta\to\infty$ . Indeed, for large $\theta$ , the copula is near-maximal, additional increases in $\theta$ change the joint distribution very little, and the insurer’s optimal strategy has already adjusted to the high-dependence environment. The total decline in $\hat{Q}$ over the plateau is only $1.9$ , which confirms that most of the dependence cost is incurred in the transition regime.

6.1. Welfare loss analysis

Recall from Proposition 4.1 that the optimal value function takes the exponential form

\begin{align*} \vartheta(x;\;p,\theta) = -\frac{1}{R\eta}\,e^{-R\eta\,[x+\mathcal{B}(p,\theta)]},\end{align*}

where

\begin{align*} \mathcal{B}(p,\theta) = \frac{1}{R}\!\left[Q(p;\;\theta)+\frac1\eta\Bigl(\frac{\delta}{R}-1\Bigr)\right].\end{align*}

Suppose $\theta$ is the true Clayton–Lévy copula dependence parameter. The welfare-equivalent loss $\mathrm{WEL}(\theta,\theta')$ is defined as the additional initial capital that makes the insurer indifferent between starting at wealth x with the optimal strategy $\hat{p}(\theta)$ and starting at wealth $x+\mathrm{WEL}$ with the misspecified strategy $\hat{p}(\theta')$

\begin{align*} \vartheta\bigl(x+\mathrm{WEL};\;\hat{p}(\theta'),\theta\bigr) = \vartheta\bigl(x;\;\hat{p}(\theta),\theta\bigr),\end{align*}

both evaluated under the true parameter $\theta$ . Substituting the exponential form and taking logarithms, the wealth level x cancels on both sides, which is a direct consequence of the constant absolute risk aversion (CARA) structure, yielding

\begin{align*} \mathrm{WEL}(\theta,\theta') = \hat{\mathcal{B}}(\theta) - \mathcal{B}(\hat{p}(\theta'),\theta) = \frac{1}{R}[\hat{Q}(\theta) - Q(\hat{p}(\theta');\theta)],\end{align*}

where $\hat{Q}(\theta)=Q(\hat{p}(\theta);\;\theta)$ is the optimal value of the profile Hamiltonian. Table 3 reports the welfare-equivalent loss $\mathrm{WEL}(\theta,\theta')$ and the profit loss percentage $(\hat{Q}(\theta)-Q(\hat{p}(\theta');\;\theta))/\hat{Q}(\theta)$ for each combination of true parameter $\theta$ and misspecified parameter $\theta'$ . When true dependence lies in the transition or high-dependence regime, the cost of underestimating it increases sharply with $\theta$ . If the insurer assumes near-independence ( $\theta'=0.01$ ) when the true dependence is $\theta=1.0$ , the resulting profit loss is $8.42\%$ , with a welfare-equivalent loss of $\mathrm{WEL}=471.2$ . These figures rise to $19.32\%$ and $1{,}054.3$ at $\theta=2.0$ , and to $23.17\%$ and $1{,}261.6$ at $\theta=3.0$ . The mechanism is apparent from the misspecified premium vectors: near-independence leads to $\hat p_2(\theta')\approx 3.63$ , well below the true optimum of $5.83$ – $5.95$ , thereby attracting an excessive volume of correlated line-2 risk.

Table 3.

Welfare-equivalent loss $\mathrm{WEL}(\theta,\theta')$ and profit loss percentage for misspecified dependence, with $\eta=4$ and $R=0.04$ . WEL is in the same monetary units as initial capital x. Profit loss $= (\hat{Q}(\theta)-Q(\hat{p}(\theta');\;\theta))/\hat{Q}(\theta)$ . The case $\theta'=\theta$ (WEL $=0$ by definition) is omitted.

Table 3. Long description

The table presents data on welfare-equivalent loss (WEL) and profit loss percentage for various combinations of true and misspecified dependence parameters. It has 5 rows and 6 columns. The columns are labeled as follows: θ, θ’, p̂_1(θ’), p̂_2(θ’), WEL, and Profit loss (%). The table includes the following data: Row 1: θ, 0.5; θ’, 0.01; p̂_1(θ’), 5.043; p̂_2(θ’), 3.626; WEL, 80.9; Profit loss (%), 1.29. Row 2: θ, 0.5; θ’, 0.25; p̂_1(θ’), 5.119; p̂_2(θ’), 3.702; WEL, 45.0; Profit loss (%), 0.72. Row 3: θ, 0.5; θ’, 1.00; p̂_1(θ’), 5.043; p̂_2(θ’), 4.967; WEL, 209.6; Profit loss (%), 3.35. Row 4: θ, 0.5; θ’, 3.00; p̂_1(θ’), 4.891; p̂_2(θ’), 5.954; WEL, 579.8; Profit loss (%), 9.26. Row 5: θ, 0.5; θ’, 10.00; p̂_1(θ’), 4.891; p̂_2(θ’), 6.030; WEL, 606.3; Profit loss (%), 9.68. Row 6: θ, 1.0; θ’, 0.01; p̂_1(θ’), 5.043; p̂_2(θ’), 3.626; WEL, 471.2; Profit loss (%), 8.42. Row 7: θ, 1.0; θ’, 0.25; p̂_1(θ’), 5.119; p̂_2(θ’), 3.702; WEL, 390.3; Profit loss (%), 6.97. Row 8: θ, 1.0; θ’, 3.00; p̂_1(θ’), 4.891; p̂_2(θ’), 5.954; WEL, 90.1; Profit loss (%), 1.61. Row 9: θ, 1.0; θ’, 10.00; p̂_1(θ’), 4.891; p̂_2(θ’), 6.030; WEL, 101.7; Profit loss (%), 1.82. Row 10: θ, 2.0; θ’, 0.01; p̂_1(θ’), 5.043; p̂_2(θ’), 3.626; WEL, 1,054.3; Profit loss (%), 19.32. Row 11: θ, 2.0; θ’, 0.25; p̂_1(θ’), 5.119; p̂_2(θ’), 3.702; WEL, 939.9; Profit loss (%), 17.23. Row 12: θ, 2.0; θ’, 1.00; p̂_1(θ’), 5.043; p̂_2(θ’), 4.967; WEL, 97.8; Profit loss (%), 1.79. Row 13: θ, 2.0; θ’, 3.00; p̂_1(θ’), 4.891; p̂_2(θ’), 5.954; WEL, 2.3; Profit loss (%), 0.04. Row 14: θ, 2.0; θ’, 10.00; p̂_1(θ’), 4.891; p̂_2(θ’), 6.030; WEL, 5.1; Profit loss (%), 0.09. Row 15: θ, 3.0; θ’, 0.01; p̂_1(θ’), 5.043; p̂_2(θ’), 3.626; WEL, 1,261.6; Profit loss (%), 23.17. Row 16: θ, 3.0; θ’, 0.25; p̂_1(θ’), 5.119; p̂_2(θ’), 3.702; WEL, 1,135.9; Profit loss (%), 20.86. Row 17: θ, 3.0; θ’, 1.00; p̂_1(θ’), 5.043; p̂_2(θ’), 4.967; WEL, 142.2; Profit loss (%), 2.61. Row 18: θ, 3.0; θ’, 10.00; p̂_1(θ’), 4.891; p̂_2(θ’), 6.030; WEL, 0.6; Profit loss (%), 0.01.

By contrast, overestimating dependence is almost costless for $\theta\geq 1.0$ . Profit losses remain below $2.61\%$ for all $\theta'\geq \theta$ and fall to just $0.01\%$ when $\theta=3.0$ and $\theta'=10.0$ . This reflects the flatness of Q(p) with respect to $p_2$ on the high-dependence plateau: once the insurer has adjusted to a high-dependence environment, marginal oversuppression of line-2 demand entails only negligible profit loss.

The pattern reverses in the low-dependence regime. There, underestimating dependence is relatively inexpensive: profit losses are only $0.72\%$ – $1.29\%$ for $\theta'\leq 0.25$ , because the true optimal strategy already sets $\hat p_2$ at a moderate level close to that implied by a near-independence model. Overestimating dependence, however, becomes the more costly error. Choosing $\theta'=3.0$ or $\theta'=10.0$ pushes $\hat p_2$ to $5.95$ – $6.03$ , well above the true optimum of $4.06$ , excluding profitable line-2 customers and generating losses of $9.26\%$ – $9.68\%$ . The asymmetry reverses because $\theta=0.5$ lies in a low-dependence regime in which the optimal policy calls for only moderate line-2 premiums: aggressively suppressing line-2 demand at that point sacrifices substantial revenue with little corresponding reduction in risk.

Overall, the results convey a nuanced policy message. In high-dependence environments, failing to recognize dependence can be very costly, whereas modest overestimation is largely harmless. In low-dependence environments, however, excessive caution may be the more damaging distortion, pricing the insurer out of profitable business with only limited risk-management benefits.

7. Conclusion

This study explored the problem of optimal premium setting for a multiline insurer, incorporating demand-based pricing and dependence structures in claims using Lévy copulas. By embedding this dependence into a dynamic optimization framework, we analyzed how an insurer, acting as a monopolistic price-setter, can control risk exposure and profitability through premium adjustments. Our results highlight the critical role of dependence between underwriting lines in shaping optimal pricing strategies.

A key finding is that stronger dependence between insurance lines leads to increased overall risk exposure and a reduction in the insurer’s expected utility. For a two-line insurer, we formally demonstrate that a higher degree of dependence results in a lower optimal profit function, using supermodular ordering of Lévy measures.

This suggests that insurers should account for interdependencies when designing pricing strategies, particularly in environments with correlated risks.

Our numerical analysis, using a Clayton–Lévy copula, illustrates the impact of different dependence levels and risk aversion parameters on premium setting. The results indicate that when claim dependence is weak, insurers have more flexibility in pricing, while stronger dependence limits profitable pricing regions and necessitates more conservative premium strategies to mitigate potential losses. Additionally, we observed that risk aversion amplifies the effects of dependence on pricing decisions, with highly risk-averse insurers favoring more conservative premium structures.

This study contributes to the literature by integrating dependence modeling into dynamic premium optimization and providing insights into how insurers can balance profitability and risk. Future research directions include extending the model to incorporate demand-side complementarities and studying multivariate demand-based operational value at risk as in Böcker and Klüppelberg (Reference Böcker and Klüppelberg2010). This is highly relevant for solvency purposes and in risk-based capital regulatory systems such as Solvency II. Exploring competition among insurers in a multifirm setting and assessing regulatory implications of dependence-aware pricing strategies are also promising avenues for further investigation.