Optimal dividend payments for a two-dimensional insurance risk process

In collective risk theory the surplus process X of an insurance company is modeled aswhere \(x>0\) denotes the initial surplus,is a compound Poisson. We assume that \(U_{i}, (i=1,2,\ldots )\) are i.i.d. distributed claims (with the distribution function G).The arrival process is a homogeneous Poisson process \(N_{t}\) with intensity \(\lambda\). The premium income is modeled by a constant premium density c and often assumed net profit condition \(c>\lambda E[U_{1}]\) gives the unrealistic property that process X(t) converges to infinity. In answer to this objection De Finetti [21] introduced the dividend barrier model for one-dimensional model (1), in which all surpluses above a given level are transferred to a beneficiary. Further, usually the payment of dividends should be made in such a way as to maximize the expected discounted sum of dividends paid up to ruin time. In 1969, Gerber [23] showed that if the free surplus of an insurance portfolio is modelled by a compound Poisson risk model, it is always optimal to pay dividends according to a so-called band strategy, which collapses to a barrier strategy for exponentially distributed claim amounts. Later, lots of works in the mathematical finance and actuarial literature concern the dividend barrier models and the problem of finding an optimal policy for paying out dividends. Gerber and Shiu [22], Grandits et al. [26] and Jeanblanc and Shiryaev [30] considered the optimal dividend problem in a Brownian setting. Irbäck [28] and Zhou [41], Zajic [40], Avram et al. [4], Kyprianou and Palmowski [31], Loeffen [32] studied the constant barrier model for a classical and spectrally negative Lévy risk process. Azcue and Muler [8] follow a viscosity approach to investigate optimal reinsurance and dividend policies in the Cramér–Lundberg model. The most general criteria currently available for barrier strategies to be optimal can be found in Loeffen and Renaud [33].

A detailed overviews on this subject from different perspectives are given in Azcue and Muler [10], Schmidli [36], Albrecher and Thonhauser [2] and Avanzi [5].

All these control problems have been formulated and studied in the one-dimensional framework. However, in recent years there has been an increased interest in risk theory in considering the multidimensional surplus model where X(t), x, c and S(t) are vectors, with possible dependence between the components. Indeed, the assumption of independence of risks may easily fail, for example in the case of reinsurance, when incoming claims have an impact on both insuring companies at the same time. In general, one can also consider situations where each claim event might induce more than one type of claim in an umbrella policy (see Sundt [39]). For some recent papers considering dependent risks, see Dhaene and Goovaerts [19, 20], Goovaerts and Dhaene [25], Müller [34, 35], Denuit et al. [17], Ambagaspitiya [3], Dhaene and Denuit [18], Hu and Wu [27] and Chan et al. [14]. Ruin probability expressions for a two-dimensional risk process were also studied in Avram et al. [6, 7] for simultaneous claim arrivals and proportional claim sizes and recently in Badila et al. [13] and Ivanovs and Boxma [29] in a more general framework.

In this paper we analyze the dividend problem for the two-dimensional risk model in which two branches of a company with initial capiotals \(x_1\) and \(x_2\) split the amount they pay out of each claim in fixed proportions \(b_{1}\) and \(b_{2}\) (\(b_{1}+b_{2}=1\)), and receive premiums at rates \(c_{1}\) and \(c_{2}\), respectively. Moreover, these two branches have the same shareholders and the objective is to maximize the total dividends received by these shareholders. That is, the main goal of this paper is identification of the value function maximizing the weighted sum of expected discounted dividend payments until ruin of at least one branch. This will lead to a fully two-dimensional stochastic control problem and we answer the question what the analogues of the optimal univariate barrier strategies are in two dimensions.

In Azcue and Muler [9], the problem of optimally transferring capital between two portfolios in the presence of transaction costs was considered, see also Badescu et al. [12]. Albrecher et al. [1] considered the problem of optimizing dividends for two collaborating insurance companies whose surpluses are modelled with independent compound Poisson processes.

Czarna and Palmowski [16] studied the same dividend problem as in this paper but for a very particular dividend strategy of reflecting two-dimensional risk process from the line. Solving certain partial differential equations they managed to identify the value function for the exponential claim sizes. We will show in this paper though that this strategy is not optimal.

In this paper we prove that the value function is a viscosity solution of the respective Hamilton–Jacobi–Bellman equation (abbreviated lately by HJB) and it can be characterized as the smallest viscosity supersolution. Using this result we manage to identify the optimal strategy. In particular, under assumption that the second branch receives less premium per amount paid out, for initial surpluses \((x_{1},x_{2})\) satisfying \(x_{1}/b_{1} >x_{2}/b_{2}\) the best strategy is to pay \(\left( x_{1}-\left( b_{1} /b_{2}\right) x_{2}\right)\) by the first branch immediately as dividends.

For the symmetric case \(c_{2}/c_{1}=b_{2}/b_{1}\) and for \(x_{1}/b_{1} \le x_{2}/b_{2}\), by transferring the optimization problem into one-dimensional optimization problem, we managed to describe the optimal strategy explicitly. For \(x_{1}/b_{1} <x_{2}/b_{2}\) the optimal strategy is that the second branch immediately pays \(x_{2}-\left( b_{2}/b_{1}\right) x_{1}\) as dividends and afterwards the controlled process remains in the half line \(\mathcal {M}\) in \(\mathbf {R}_{+}^{2}\) that contains the origin with slope \(b_{2}/b_{1}\) until ruin time. In \(\mathcal {M}\), either both branches pay the incoming premium as dividends until next claim, or the second branch pays a positive amount m of money and the first branch pays \(\left( b_{1}/b_{2}\right) m,\) where m is the minimal amount that brings the surplus process back to the previous case, or, finally, both branches pay no dividends.

For general, non-symmetric case \(c_{2}/c_{1}<b_{2}/b_{1}\) we use a convergent numerical scheme to find the optimal strategy for initial surpluses \((x_{1},x_{2})\) satisfying \(x_{1}/b_{1}\le x_{2}/b_{2}\). More formally, given any \(\delta >0\), we denote \(\Delta x_{1}=c_{1}\delta\) and \(\Delta x_{2}=c_{2}\delta\) and define the grid domain \(\mathcal {G}^{\delta }\) in \(\mathbf {R}_{+}^{2}\) as \(\mathcal {G}^{\delta }:=\left\{ (n\Delta x_{1},m\Delta x_{2}):n,m\ge 0\right\} .\) We look, at each point of the grid \(\mathcal {G}^{\delta }\), for the best local strategy among the ones suggested by the operators of the HJB equation. These possible local strategies are: the first branch pays a lump sum as dividends, the second one pays a lump sum as dividends, or none of the branches pay dividends. These local strategies are modified in such a way that the controlled surplus after applying these local strategies lies in the grid. We believe that this techniques used in this paper could be applicable to other risk-sharing mechanisms.

The rest of the paper is organized as follows. In Sect 2 we present the model we deal with and state formally the problem that we want to solve. In Sect. 3 we prove that the value function is a viscosity solution of HJB. Using this result we describe the optimal strategy in some cases in Sect. 4, in Sect. 5 we describe a numerical scheme to approximate the optimal value function and the optimal strategy and in Sect. 6 we present some numerical examples. We conclude the paper with the number of comments in Sect. 7.

In this paper we consider two branches of a company that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim. This model corresponds to proportional reinsurance dependence. The total amount of claims up to time t is modeled as a compound Poisson process. We can write this two-dimensional risk model asAbove \(x_{1}\) and \(x_{2}\) are the corresponding initial surplus levels and \(c_{1}\) and \(c_{2}\) are the corresponding premium rates. The sizes of the claims \(U_{i}\)are non-negative i.i.d. random variables with absolutely continuous distribution G. The claims arrival process \(N_{t}\)is a Poisson processes with intensity \(\lambda\). Finally, the constants \(b_{1}\) and \(b_{2}\) are the proportions of the claim that each branch pays, so \(b_{1}+b_{2}=1\), \(b_{1}>0\) and \(b_{2}>0\). We assume here that the process \(N_{t}\) and the random variables \(U_{i}\) are independent of each other. Without loss of generality, we can assume that the second branch receives less premium per amount paid out, that isWe denote by \(\{\mathcal {F}_{t}\}_{\{t\ge 0\}}\) the right-continuous natural filtration of \(\overline{X}(t)\) satisfying usual conditions. Throughout this paper all stopping times and martingales are taken with respect of this filtration.

Both branches use part of their surpluses to pay dividends. The dividend payment strategy \(\overline{L}(t)=\left( L_{1}(t){\small ,}L_{2}(t)\right)\) is the total amount of dividends paid by the two branches up to time t. We define the associated controlled process with initial surplus \((x_{1} ,x_{2})\) asThe dividend payment strategy \(\overline{L}=\left( L_{1}(t){\small ,} L_{2}(t)\right) _{t\le \tau }\) is called admissible if it is non-decreasing, càglàd (left continuous with right limits), predictable with respect to the filtration generated by the bivariate process \(\overline{X}(t)\) and satisfies \(L_{1}(t)\le X_{1}(t)\), \(L_{2}(t)\le X_{2}(t)\). The first time when the two-dimensional risk process first leaves the positive quadrant will be our ruin time:That is, the ruin time is the first time at which at least one branch get ruined. We denote by \(\mathbf {R}_{+}=[0,\infty )\) and by \(\mathbf {R}_{+} ^{2}=(0,\infty )^{2}\) the first quadrant. Let \(\Pi _{x_{_{1}},x_{2}}\) be the set of admissible dividend strategies with the initial surplus \(\overline{x}=(x_{1},x_{2})\in \mathbf {R}_{+}^{2}\). Given an admissible dividend strategy \(\overline{L}\in \Pi _{x_{_{1}},x_{2}},\) the expected discounted dividends paid by the two branches until \(\widehat{\tau }\) iswhere \(q>0\) is a constant discount factor.

The main goal of this paper is identification of the optimal value function defined by:for \(\overline{x}\in \mathbf {R}_{+}^{2}.\)

In the next section, we will see that V is well defined. Another crucial problem that we treated in this paper is the existence of an optimal strategy defined as follows. Given a family of admissible strategieswe define the value function\(V_{\pi }:\mathbf {R}_{+}^{2} \rightarrow \mathbf {R}_{+}\) as \(V_{\pi }(\overline{x})=V_{\overline{L}_{\overline{x}}}(\overline{x})\). We say that \(\pi ^{*}\) is the optimal strategy if \(V_{\pi ^{*}}=V\).

In this section we first state some basic results concerning regularity and growth at infinity of the optimal value function V defined in (6). We then deduce the Hamilton–Jacobi–Bellman equation associated to this optimization problem and see that V is a viscosity solution of the HJB equation. Moreover, we can characterize V as the smallest viscosity supersolution of this equation with a suitable growth condition. Finally, we obtain a verification result: any viscosity supersolution which is a value function of a family of admissible strategies is the optimal value function.

The following two lemmas are the two-dimensional counterparts of Lemmas 2.1 and 2.2 of [8].

In order to obtain the Hamilton–Jacobi–Bellman (HJB) equation associated to the optimization problem (6), we need to state the so-called Dynamic Programming Principle (DPP). Since \(\overline{X}(t)\) is a Markov process the proof follows the same arguments as the ones given in Lemma 1.2 of Azcue and Muler [10]. They use only the fact that V is increasing and continuous in \(\mathbf {R}_{+}^{2}\). Obviously we need also to extend the definition of V to \(\mathbf {R}^{2}\) defining V as zero outside the first quadrant.

For any continuously differentiable function u defined in \(\mathbf {R} _{+}^{2}\), we define the infinitesimal generator \(\widetilde{\mathcal {G}}\) of the controlled process \(\overline{X}^{\overline{L}}(t\wedge \hat{\tau })\) bysee [10, Sec. 1.4] for details.

We will consider now the admissible strategy \(\overline{L}\) in which both branches pay dividends with constant rates \(l_{1}\ge 0\) and \(l_{2}\ge 0\) respectively until ruin time \(\hat{\tau }\). Then using [10, Rem. 1.8] we havewhereand \(\tau _{1}\) denotes the first claim arrival.

Assume now that V is continuously differentiable. Note that from Lemma 3.3 we have:Thus by (8) dividing above inequality by t and taking \(t\downarrow 0\) give:whereTaking \(l_{1}=l_{2}=0\), \(l_{1}\rightarrow \infty\) with \(l_{2}=0\); and \(l_{2}\rightarrow \infty\) with \(l_{1}=0\), we obtainWe now associate to our problem the following HJB equation:Since the optimal value function could be not differentiable, we have to use the notion of viscosity solution and see that V is a viscosity solution of the associated HJB equation. Let us define this notion (see for instance Crandall and Lions [15] and Soner [37]).

We now show that the optimal value function V can be characterized as the smallest viscosity supersolution of (12) with a suitable growth condition. We say that the function \(u:\mathbf {R}_{+}^{2}\rightarrow \mathbf {R}\) satisfies the growth condition, if there exists \(K>0\) such thatWe have the following result.

The proof of this proposition is a straightforward extension to the two-dimensional case of the corresponding proof of Proposition 4.4 in [10], taking again the zero-expectation martingale defined in (13) and using Lemmas 3.1 and 3.2.

From the last proposition and Theorem 3.5, we conclude the following corollary.

First in this section, we introduce some special families of admissible strategies which depends only on the current surplus levels.

Assuming that V is differentiable in \(\mathbf {R}_{+}^{2}\), the Eq. (12) suggests how the dividends are paid depending on the current surplus \(\overline{x}=(x_{1},x_{2})\in \mathbf {R}_{+}^{2}\):If V is not differentiable at some points, still these sets could be defined in a viscosity sense.

Let us define \(\mathcal {M}\) as the half line in \(\mathbf {R}_{+}^{2}\) that contains the origin with slope \(b_{2}/b_{1}\). Note that, when the current surplus is outside \(\mathcal {M}\), there exists the possibility that the ruin occur leaving one branch with positive surplus; if the current surplus is in the setthen the first branch will be the one with eventual positive surplus at ruin time and conversely, if the current surplus is in the setthe same could happen with the second branch.

In this section we present a convergent numerical scheme to approximate the optimal value function for the case \(c_{1}/b_{1}>c_{2}/b_{2}\) in \(\mathbf {R}_{+}^{2}\). In fact, by Corollary 4.2, we have that the value function satisfyand so it is enough to approximate the optimal value function in \(D^{2} \cup \mathcal {M}\). This numerical scheme can be seen as a particular case of the one described in [11] for the case that the joint multivariate distribution function of the downward jumps is given by \(F(x,y):=G(\left( x/b_{1}\right) \wedge \left( y/b_{2}\right) )\) and both the penalty and the switch-value functions are identically zero.

Given any \(\delta >0\), consider \(\Delta x_{1}=c_{1}\delta\) and \(\Delta x_{2}=c_{2}\delta\), define the grid domain \(\mathcal {G}^{\delta }\) in \(\mathbf {R}_{+}^{2}\) asWe look, at each point of the grid \(\mathcal {G}^{\delta }\), for the best local strategy among the ones suggested by the operators of the HJB equation (12); these possible local strategies are: the first branch pays a lump sum as dividends, the second one pays a lump sum as dividends, or none of the branches pay dividends. These local strategies are modified in such a way that the controlled surplus after applying these local strategies lies in the grid. The possible control actions at any point of the grid \(\mathcal {G}^{\delta }\) are defined as follows: let \((n\Delta x_{1},m\Delta x_{2})\in \mathcal {G} ^{\delta }\) be the initial surplus and \(\tau\) and U be the time and size of the first claim respectively.For technical reasons, an extra control action \(\mathbf {E}_{s}\) is considered, under this control action no more dividends are paid. The set of control actions is denoted by \(\mathcal {E}=\{\mathbf {E}_{1},\mathbf {E}_{2} ,\mathbf {E}_{0},\mathbf {E}_{s}\}\).

Define \(\Pi _{n\Delta x_{1},m\Delta x_{2}}^{\delta }\subset \Pi _{n\Delta x_{1},m\Delta x_{2}}\) as the set of all the admissible dividend strategies with initial surplus \((n\Delta x_{1},m\Delta x_{2})\in \mathcal {G}^{\delta }\) which can be obtained by a sequence (finite or infinite) of control actions in \(\mathcal {E}\) at each point of the grid. The \(\delta\)-optimal function\(V^{\delta }\) is defined at each point \((n\Delta x_{1},m\Delta x_{2})\in \mathcal {G}^{\delta }\) as the supremum of the value functions of admissible strategies which are combination of the control actions in \(\mathcal {E}\), that isNote that the extra control action \(\mathbf {E}_{s}\) corresponds in the numerical scheme described in [11] to switch immediately to a switch-value function identically 0 (which is always smaller than V), so it is never optimal to use this control; however -as we will see later on- it is used to find a simple admissible dividends strategy in \(\Pi _{n\Delta x_{1},m\Delta x_{2}}^{\delta }\) to start a recursive construction in \(\Pi _{n\Delta x_{1},m\Delta x_{2}}^{\delta }\) which will converge to \(V^{\delta }\).

Define \(v^{\delta }(n,m):=V^{\delta }(n\Delta x_{1},m\Delta x_{2})\). It is proved in [11], that the function \(v^{\delta }\) is a solution of the following discrete version of the HJB equationat \((n,m)\in \mathbf {N}_{0}^{2}\). Here the operators \(T_{0},\)\(T_{1}\) and \(T_{2}\) are defined asandwhere

Consider the operator T defined asUsing the results in [11], we have that:

We show three numerical examples with parameters \(b_{1}=b_{2}=0.5\), \(c_{1}=2\), \(c_{2}=1,q=0.05\) and \(\lambda =1\) and different claim-size distributions G. We use the numerical scheme introduced in Section 5 and obtain the numerical approximation \(V^{\delta }\) of the optimal value function V and the corresponding \(\delta\)-optimal strategy. We can see in general that, as we proved in Subsection 4.4, the optimal strategy for initial surplus in \(D^{1}\) consists in the first branch paying dividends immediately so the surplus moves to \(D^{2} \cup \mathcal {M}\). It was proved in Proposition 4.8 that it is not optimal for some initial surplus in \(\mathcal {M}\) (unlike the case \(c_{2}/c_{1}=b_{2}/b_{1}\)) to pay dividends so the controlled process remains in the set \(\mathcal {M}\) until ruin time. We will see in Figs. 2, 6 and 8 that for some initial surpluses in \(\mathcal {M}\) the best local strategy is to get out of \(\mathcal {M}\) with the first branch paying dividends immediately.

In this paper we have studied the two-dimensional optimal dividend problem in the context of two branches of an insurance company with compound Poisson surplus processes dividing claims and premiums in some specified proportions. We proved that the value function is the smallest viscosity supersolution of the respective Hamilton–Jacobi–Bellman equation and we identified the optimal strategy. In particular, for symmetric case we managed to describe the best strategy explicitly solving some auxiliary equivalent optimization problem. For general case we propose the convergent numerical scheme to find the optimal strategy. Using it, we analyzed some numerical examples.

We believe that this kind of study is still at its first step and several further extension can be done. First of all, one can consider more branches of one company. Moreover, one can think of more general claim arrival process, for example renewal one, with phase-type inter-arrival times. Finally, the analysis may be conducted for the two-dimensional classical risk process perturbed by bivariate correlated Brownian motion. It would be also interesting to consider impulse control and pay-out function related with Gerber-Shiu penalty functions. All of these problems are quite complex and left for future research.