Mixed Band Control of Mutual Proportional Reinsurance *

In this paper, we investigate the optimization of mutual proportional reinsurance—a mutual reserve system that is intended for the collective reinsurance needs of homogeneous mutual members, such as P&I Clubs in marine mutual insurance and reserve banks in the US Federal Reserve, where a mutual member is both an insurer and an insured. Compared to general (non-mutual) insurance models, which involve one-sided impulse control (i.e., either downside or upside impulse) of the underlying insurance reserve process that is required to be positive, a mutual insurance differs in allowing two-sided impulse control (i.e., both downside and upside impulse), coupled with the classical proportional control of reinsurance. We prove that a special band-type impulse control   , , , a A B b a with and a A coupled with a proportional reinsurance policy (classical control), is optimal when the objective is to minimize the total maintenance cost. That is, when the reserve position reaches a lower boundary of 0 a  B b    , 0  , the reserve should immediately be raised to level A; when the reserve reaches an upper boundary of b, it should immediately be reduced to a level B. An interesting finding produced by the study reported in this paper is that there exists a situation such that if the upside fixed cost is relatively large in comparison to a finite threshold, then the optimal band control is reduced to a downside only (i.e., dividend payment only) control in the form of   , B b 0,0; with 0 a A   . In this case, it is optimal for the mutual insurance firm to go bankrupt as soon as its reserve level reaches zero, rather than to jump restart by calling for additional contingent funds. This finding partially explains why many mutual insurance companies, that were once quite popular in the financial markets, are either disappeared or converted to non-mutual ones.


Introduction
This study is motivated by the long lasting while still strong-going success of marine mutual insurance (e.g., a P&I club of ship owners), the longest in the insurance business with a history of over 200 years; and by the still unanswered question why the once popular mutual insurance firms almost disappeared after 1990's, either went bankrupt or converted to commercial (non-mutual) insurance business.A client of a mutual insurance is both insurer and insured, under a unique contingent regulation scheme of two-way impulse control (termed band control) which we give a brief description as follows.A mutual insurance firm is allowed: 1) to make calls for contingent injection from its clients once the reserve level is considered too low, as opposed to a commercial (non-mutual) insurance firm which would go bankrupt once its reserve becomes depleted; 2) or to make refunds (pay dividends) to its clients if the reserve is considered too high.We shall note that it is also permissible to adjust premium rate and liability policy in the continuous time horizon in conjunction of the band control as just mentioned, which in this case is referred to as a mixed band control of insurance reserves (i.e., a band type impulse control mixed with time-continuous classical control of a reserve process).Although in practice, mutual insurance engages mixed band control, most studies in the current literature are focused on the band control only, without classical control (e.g., with given constant premium rate, see [1,2] for reference).In this paper we consider mixed band control of mutual reinsurance which provides insurance to other mutual insurance firms, such as P&I clubs, of which each insures a number of ship-owners.
Reinsurance has been long investigated as an intrinsic part of commercial insurance, of which the mainstream modeling framework is profit maximization with the one-sided impulse control of an underlying reserve process.There are two types of one-sided impulse control: downside-only impulse control (such as a dividend payment) with a fixed cost K  (e.g., Cadenillas et al. [3], Hojgaard and Taksar [4]) and upside-only impulse control (such as inventory ordering) with a fixed cost K  (e.g., Bensoussan et al. [2], Eisenberg and Schmidli [5], Sulem [6]).In this paper, we examine mutual proportional reinsurance-a mutual reserve system that is intended for the collective reinsurance needs of homogeneous mutual members, such as the P&I Clubs in marine mutual insurance (e.g., Yuan [7]) and the reserve banks in the US Federal Reserve (e.g., Dawande et al. [8]).A mutual insurance differs from a general (non-mutual) insurance in two key dimensions: 1) a mutual system is not for profit, and 2) a mutual reserve involves two-sided impulse control (i.e., both a dividend refund as a downside impulse to decrease the reserve with cost K  and a call for funds as an upside impulse to increase the reserve with cost K  ).It should be noted that the reserve proc- ess for a general insurance must always be positive (above zero), and the insurance firm is considered bankrupt as soon as its reserve falls to zero.
The mutual proportional reinsurance model developed in this paper is a generalization of the proportional reinsurance models (e.g., Cadenillas et al. [3], Hojgaard and Taksar [4], Eisenberg and Schmidli [5], Løkka and Zervos [9]) and is modified with the two differing characteristics noted above.More specifically, the proportional reinsurance rate can be adjusted in continuous time, and the underlying mutual reserve process is regulated by a two-sided impulse control in terms of a contingent dividend payment (i.e., a downside impulse control to decrease the mutual reserve level) and contingent call for contributions (i.e., an upside impulse control to increase the mutual reserve level).The corresponding mathematical problem for mutual proportional reinsurance becomes a two-sided impulse control system combined with a classical rate control in continuous time, a problem yet to be posed in insurance research.A problem that involves a mix of impulse control and classical control is termed a hybrid control problem in control theory, of which the difficulty has been well noted (e.g., Bensoussanand Menaldi [10], Branicky, Borkarand Mitter [11], Abate et al. [12]).
A pure two-sided impulse control problem (i.e., without a classical rate control) was investigated by Constan-tinides [13] in the form of cash management.Constantinides and Richard [1] showed an optimal two-sided impulse control policy to exist in the form of a band control, denoted with four parameters as   , ; , a A B b with a A B b    .In other words, when the reserve position reaches a lower boundary a, then the reserve should immediately be raised to level A; when the reserve reaches upper boundary b, it should immediately be reduced to level B. For our mutual proportional reinsurance problem, we specify the corresponding Hamilton-Jacobi-Bellman (HJB) equation and the associated quasi-variational inequalities (QVI), from which we analytically solve the optimal value function.We then prove that a special band-type impulse control   0, , , A B b with 0 a  , combined with a proportional reinsurance policy (classical control), is optimal when the objective is to minimize the total maintenance cost.An interesting finding reported here is that there exists a situation such that if the upside fixed cost K  is relatively large in comparison to a finite threshold K  , then the optimal band control is reduced to a downside only (i.e., a dividend payment only) control in the form of   0, 0; B,b wi 0 th a A   .In this case, it is optimal for the mutual insurance to go bankrupt as soon as its reserve level falls to zero, rather than to restart by calling for additional contingent funds.This finding partially explains why many mutual insurance companies, that were once quite popular in the financial markets, are either disappeared or converted to non-mutual ones.
The remainder of the paper is organized as follows.In Section 2, we formulate the mathematical model and specify the HJB equation and the QVI of the corresponding stochastic control problem.We solve the QVI for the optimal value function in Section 3. In Section 3.2, we characterize and analyze the threshold K  .In Section 4, we prove the verification theorem and verify the optimal control.Finally, we make concluding remarks in Section 5.

Feasible Control
The classical Cramer-Lundberg model of an insurance reserve (surplus) is described via a compound Poisson process: where is a standard Brownian motion. t We start with a probability space


, that is endowed with information filtration t and a standard Brownian motion t on adapted to t  .Two types of controls are used in this model.The first is related to the ability to directly control its reserve by raising cash from or making refunds to members at any particular time.The second is related to the mutual insurance firm's ability to delegate all or part of its risk to a reinsurance company, simultaneously reducing the incoming premium (all or part of which is in this case channeled to the reinsurance company).In this model, we consider a proportional reinsurance scheme.This type of scheme corresponds to the original insurer paying u fraction of the original claim.The premium rate coming to the original insurer is simultaneously reduced by the same fraction.The reinsurance rate can be chosen dynamically depending on the situation.

 W Ω
Mathematically, control U takes a triple form:  [14], Hojgaard and Taksar [4]).u The fact that the process   u t is adapted to informa- tion filtration means that any decision has to be made on the basis of past rather than the future information.The stopping times i  represent the times when the ith intervention to change the reserve level is made.If 0 i   , then the decision is to raise cash by calling the members/ clients.If 0 i   , then the decision is to make a refund.
The fact that i  is a stopping time and i  is i   measurable also indicates that the decisions concerning when to make a contingent call and how much cash to raise are made on the basis of only past information.The same applies to the refund decisions.
Once control U is chosen, the dynamics of the reserve process becomes: Define the ruin time as and if We denote the set of all admissible controls by  .The meaning of admissibility is as follows.At any time the decision to make a refund is made, the refund amount cannot exceed the available reserve.As can be seen in the following, if this condition is not satisfied, then one can always achieve a cost equal to , simply by making an infinitely large refund.The second condition of admissibility is a rather natural technical condition of integrability.

Cost Structure and Value Function
The objective in this model is to minimize the operational cost and the lost opportunity to invest the money in the market.Cost function g is defined as Here,   and   denote the positive and negative components of  , that is, . The costs associated with refunds are of a different nature.A contingent call always increases the total cost, whereas a refund decreases it.However fixed set-up costs and are incurred regardless of of the size of a contingent call or a refund.In addition, when the call is made and the cash is raised, there is a proportional cost associated with the amount raised.The constant represents the amount of cash that needs to be raised in order for one dollar to be added to the reserve.If the reserve is used for a refund, then a part of it may be charged as tax.The constant   represents the amount actually received by the shareholders for each dollar taken from the reserve.
Given a discount rate r, the cost functional associated with the control U is defined as The objective is to find the value function, and optimal control , such that

Variational Inequalities for the Optimal Value Function
For each , define the infinitesimal generator .For any twice continuously differentiable function Let M be the inf-convolution operator, defined as together with the tightness condition

Solution of the QVI
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The HJB Equation in the Continuation Region
In this model, the application of the control that is related to calls and refunds results in a jump in the reserve process.This type of model is considered in the framework of the so-called impulse control.Because we also have a control whose application changes the drift and the diffusion coefficient of the controlled process, the resulting mathematical problem becomes a mixed regular-impulse control problem (e.g., Cadenillas et al. [14]).In the case of a pure impulse control, the optimal policy is of the

 
, , , a A B b type, where the four parameters used to con- struct the optimal control must be computed as a part of a solution to the problem (see Cadenillas and Zapatero [15], Constantinides and Richard [10], Harrison and Taylor [16], and Paulsen [17]).Parameters a and b represent the levels at which the intervention (application of impulses) must be made, whereas A and B stand for the positions that the controlled process must be in after the intervention is made.This is a so-called band-type policy, with   We conjecture that, in our case, the optimal intervention (impulse control) component of the problem is also of the band type.Moreover, as the following analysis implicitly shows, we can narrow our search for the optimal policy to a special band-type control   0, , , A B b , where the level a associated with the contingent calls is set to zero.Therefore, only three of the four band-type policy parameters remain unknown.After finding these parameters (and determining the optimal drift/diffusion control in the continuation region), we will see that the cost function associated with this policy satisfies the QVI.
The derivation of the value function is similar to [3] and [14].Suppose that   V x satisfies all of the QVI conditions: (2.12), (2.13) and (2.14).First note that the function   V x is a decreasing function of x , and thus 0 V   .To satisfy (2.14), for any , at least one of the two functions on the left side of the equation should be equal to zero.We conjecture that the value function has the following structure.
provided that the right-hand side of (3.17) belongs to , then (3.16) cannot be satisfied and we exclude The general solution for (3.18) is where 1 and are free constants to be determined later, and x .From (3.17), we obtain the expression for (provided that which will be verified later): Note that the solution of (3.18) coincides with the solution of (3.16)only in the region where From this expression, we conjecture that there is a switching point 0 , by virtue of the equation (3.21), we obtain the following expression for 0 x : ; and the corresponding differential equation becomes The general solution for (3.23) is where (see e.g., Cadenillas et al. [14]).The boundary conditions for the equation are rather tricky.If 0 and are the points at which the impulse control (intervention) is initiated then the boundary conditions at these points become However, if bankruptcy is allowed and no intervention is initiated when the process reaches 0, then the boundary condition at 0 becomes straightforward:   0 V 0  (see Cadenillas and Zapatero [3] and Cadenillas, et al. [14]).In our case, whether 0 is the point that corresponds to the intervention in the form of a contingent call or whether it corresponds to bankruptcy is not given a priori; rather it is part of the solution to the problem.
We seek the solution by finding a function such that To find the free constants in the expressions for 1 an 2 and to paste different pieces of the solution together we apply the principle of smooth fit by making the value and the first derivatives to be continuous at the switching points where 0 x is defined by (3.22).(It should be noted that the function , which is constructed from (3.32)-(3.34)subject to conditions (3.28)-(3.31)and (3.35)-(3.37),corresponds to the case in which the optimal policy leads to V    ).We begin by con- structing such a function.The main technique is not to consider the function itself, but rather first to construct . By the continuity on and V V  at 0 x , and by (3.23), we have From this relation and (3.24), we have , and we can write 0 ; e e We can easily get the inequalities: , and from the continuity of at V x , we obtain the expression for 1 C : .) Now, we can write in terms of and : What remains is to determine and .Once these constants are found, we have ; e e ,f o r Note that if and , then it is easy to show that for , , ; , , , V x C C  e on 0 x  .In the r der of , we find 2 C an and complete the construction of the function .We this in an implicit manner by adopting an auxi ry problem in which no contingent calls are allowed and by using the optimal value function of that problem to construct the function V .
Let's consider a lig where

A Solution to the Auxiliary Problem
First note that a general solution to (3.39), (3.41) is cx   , where  is the same as in (3.20) and c is a free constant, and a general solution to (3.40) is , where 1  and 2  are the same as in (3.25), (3.26).
To solve our auxiliary problem we apply the same technique as that used in Cadenillas et al. [14].We begin with   H x , which is defined as follows.

 
this expression, constants and are chosen in such a way that is the op value function iliary pr

The Optimal Value Function for the Original
We nction oblem timal (see Figure 2).The proof here is identical to that of a similar statement in Cadenillas et al. [14] and thus we omit it.

Problem employ the fu
H  in the pre obtained vious subsection to construct the rivative of the optimal val-de ). T re, there exis herefo ts unique 0 Note that H  decreases to at 0 at the order of  x   (see (3. ); therefore, 45) H  is integrable at 0 and, sult, as a re K    .The qualitative nature of t ution to the original pr he sol oblem depends on the relationship between K  and K  ; hence, we divide our analysis into two cases.In what follows, we show that

The Case of
is the deriv e QVI, inequalities ative of the solution V to (2.12)- (2.14). Let From the construction of the function , we se We also have on   0 0, x .As satisfies (3.18), it also satisfies (3.16) e that (3.16) holds for V because these two equations are equivalent whenever (3.16) holds.
2) To prov because, in view of Proposition 3.1, the function ).By subtracti The foregoing inequality al n, From Proposition 3.2 we know 4) is true for For any because V is a decreasing function.This completes the proof of Theor

The Case of
we cannot find an ch that (3.48) ase, we se The foregoing inequality is always true because By convention,


. In view of (2.13), we have In view of (2.6) From this equation, using (2.6) and standard but rather tedious arguments, we can deduce that the first (4.69)for all is also bounded by the same integrable random variable.Similar arguments show that (see Cadenillas et al. [7]).Note that the sec e right-hand side of (4.70) is a martingale whose ctation vanishes.However, in view of (2.12), the integrand in the first integral of (4.70) is nonne Therefore, ond integral on th expe gative.
can see that the expectation of the first sum on the right-hand side of (4.68) is nonnegative.
From (2.13), we can see that and, taking into account the dominated convergence theorem, we Substituting this inequality into (4.2) and taking expectations of both sides, we obtain Letting and employing (4.72) and the monotone convergence theorem on, we get imp Remark 4.1.For the expectation of the stochastic integral on the right hand side of (4.70) to vanish, it ficient for its integrand t be bounded.

o
In particula sufficient for V  to ounded.be b This is the case when , the function V has a singularity at 0. We can, however, apply the same technique, first replacing  and then passing to a limit as 0   .This will yield inequality (4.74), which is all we need for the proof of Theorem 4. 1 Let  se are the times his diff associated with the refunds of tant amount of The time when 0 is hit is the ruin time.When . Th V us guments in the proof of Theorem U U   , all of the inequalities ar obtain , we can repeat the ar-3.2 and see that, for e tight.As a r (4.84)Because we know that, when esult, we , .
, we have and, when 1 0 , the function V satisfies 0, the first term on the left-hand side of and we get (4.82)

Conclusions
The optimal policy in this model has several interesting nontrivi e fact t only when there ze ) is supported by intuition.However, the qualitative structure of the optidependence on the model parameters . □ al features.Th hat calls should be made is no possibility of waiting any longer (that is when the reserve reaches ro mal policy and its are not as obvious.
It turns out that it is always optimal to pay dividends, no matter what the costs associated with such payments are.However, raising cash may not be optimal when the initial set-up cost is too high.Quantity K  , which determines the threshold for set-up cost K  , such that if the cost is higher than this threshold then it is optimal to allow ruin, is in itself determined via an auxiliary problem with a one-sided impulse control.Although there is no closed-form expression for the quantity K  , it can be determined in an algorithmic manner prior to solving the optimal control problem for the m ual insurance company.
There is one rather curious feature of the optimal solution when ut . As our analysis shows, in this case,  0 S  and V v  , the same as is the case when . However, from the construction of the optimal policy, we can see that the two band-type policy is optimal in this case as well.In this borderline case, we thus have two optimal policies, one for which    with the lower band equal to   0, A and one for which reaching 0 onds to ruin and for which corresp    .a rather unique feature of this particular problem that has not been observed previously.
A natural question arises: what if ruin is explicitly disallowed, and we must find an optimal policy from among those for which This is   .As can seen from our analysis, we find a solution to this problem for the case of be K   K  .However, when this inequality does not hold, then of the stochastic control technique and the HJB equation used in this paper do not work.Another approach should be developed, as can be seen indirectly in the work of Eisenberg [18] a isenberg and Schmidli [5], where a nd E ed for 6. doi:10.1287/opre.26.4.620 similar (although not identical) problem is consider the case of a surplus process modeled via the classical Cramer-Lundberg model.This constitutes an interesting and challenging problem for future study, the nature of whose solution is not obvious at this time.

,
a A and   , B b understood as the two bands that determine the nature of the optimal control.The interval   , a b is called the continuation region.When the process falls inside the continuation region, no interventions/impulses are applied.When an intervention is initiated, the time when the process reaches one of the boundaries of the continuation region corresponds to one of i  .
problem in which only those controls U for which i  on the right-hand side of (2.2) are nega e allowed.T is problem is similar to that considered in Cadenillas et al.[14].Let tiv h v x be the optimal value function for this problem.As was shown in[14], the function v satisfies the same HJB equation, except for boundary conditions (3.28) and (3.29).These conditions are replaced by   0 0 v  .The same arguments as those above sh w ow that e can m ake the conjecture that the function v should be sought as a solution to (3.39)-(3.44)below.
MH x and a ove the horizontal line

Figure 2 .
Figure 2. Solution to the auxiliary problem.


Figure 3) su ch that

Figure 3 .
Figure 3. Determining the parameters for the value nction fu   V x by utilizing the solution of the auxiliary problem   H x 

( 3 .
58) from the left hand ), we can see that (3.58) is equivalent to ng side of(3.23

.
In view of the continuity of the first and seco ives, and in view of the fact that (3.16) holds on   0 0, x , we know that (3.59) is true for 0 4.76) From (3.46), (3.45), (3.49), and (3.53), we can see that ess with the same drift and diffusion coefficients as above, between the times of intervention.The intervention times are the times at which this process reaches either 0 or b .At point 0, the control is set to ce the process to point ontinuous diffusion displa proc A , which corresponds to raising cash (making a call to holders) in the amount of share A .Reaching the level b results in the dis- placement of the process to the point B which, corre- sponds to making a refund in the amount of b B  .Theorem 4.2.(The verification theorem) Let U  be the control described by (4.76) and (4.79)-(4.80).Then, In view of (4.67), it is sufficient to show that