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 
(e.g., Cadenillas et al. [
(e.g., Bensoussan et al. [
and a call for funds as an upside impulse to increase the reserve with cost
). It should be noted that the reserve process 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. [
A pure two-sided impulse control problem (i.e., without a classical rate control) was investigated by Constantinides [
with
. 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 
with
, 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 
is relatively large in comparison to a finite threshold
, then the optimal band control is reduced to a downside only (i.e., a dividend payment only) control in the form of 
with
. 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
. In Section 4, we prove the verification theorem and verify the optimal control. Finally, we make concluding remarks in Section 5.
is the amount of the surplus available at time
, quantity 
represents the premium rate, 
is the Poisson process of incoming claims and 
is the size of the ith claim. This surplus process can be approximated by a diffusion process with drift 
and diffusion coefficient
, where 
is the intensity of the Poisson process
. We assume that the insurer always sets 
(i.e.
). Thus, with no control, the reserve process 
is described by
is a standard Brownian motion.
, that is endowed with information filtration 
and a standard Brownian motion 
on 
adapted to
. 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.
is a predictable process with respect to
, the random variables 
constitute an increasing sequence of stopping times with respect to
, and 
is a sequence of 
-measurable random variables,
.
represents the fraction (reinsurance share) of the premium and risk (loss) incurred by the mutual insurance at time
. Suppose that 
is chosen at time
. Then, in the diffusion approximation (2.1), drift 
and diffusion coefficient 
are reduced by factor 
(see Cadenillas et al. [
is adapted to information filtration means that any decision has to be made on the basis of past rather than the future information. The stopping times 
represent the times when the ith intervention to change the reserve level is made. If
, then the decision is to raise cash by calling the members/ clients. If
, then the decision is to make a refund. The fact that 
is a stopping time and 
is 
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.
is called admissible for initial position 
if, for 
for any
,
.
, simply by making an infinitely large refund. The second condition of admissibility is a rather natural technical condition of integrability.
is defined as
and 
denote the positive and negative components of
, that is, 
and 
. 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.
, such that
, define the infinitesimal generator
. For any twice continuously differentiable function 
be the inf-convolution operator, defined as
type, where the four parameters used to construct the optimal control must be computed as a part of a solution to the problem (see Cadenillas and Zapatero [
and 
represent the levels at which the intervention (application of impulses) must be made, whereas 
and 
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 
and 
understood as the two bands that determine the nature of the optimal control. The interval 
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
.
, where the level 
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.
satisfies all of the QVI conditions: (2.12), (2.13) and (2.14). First note that the function 
is a decreasing function of
, and thus
. 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.
. Also
.
minimizes the function 
in foregoing equation. If 
then
. (Note that if
, then (3.16) cannot be satisfied and we exclude 
from consideration).
and 
are free constants to be determined later, and
. From (3.17), we obtain the expression for 
(provided that 
which will be verified later):
such that 
when
. As
, by virtue of the equation (3.21), we obtain the following expression for
:
,
; and the corresponding differential equation becomes
. Standard arguments show that
(see e.g., Cadenillas et al. [
are the points at which the impulse control (intervention) is initiated then the boundary conditions at these points become
(see Cadenillas and Zapatero [
such that
an 
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 
and b,
is defined by (3.22).
, 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
). We begin by constructing such a function. The main technique is not to consider the function 
itself, but rather first to construct
.
is shown in 
and (3.17), we have
. By the continuity on 
and 
at
, and by (3.23), we have
. From this relation and (3.24), we have
. Then, 
, and we can write
, and from the continuity of 
at
, we obtain the expression for
:
. (Obviously, 
since
.) Now, we can write 
in terms of 
and
:
and
. Once these constants are found, we have 
and
, and thus
. Let
.
and
, then it is easy to show that for
, 
,
and
,
. Therefore, 
is decreasing on 
and 
is concave on
. In the remainder of this section, we find 
and 
and complete the construction of the function
. We do this in an implicit manner by adopting an auxiliary problem in which no contingent calls are allowed and by using the optimal value function of that problem to construct the function
.
for which 
on the right-hand side of (2.2) are negative allowed. This problem is similar to that considered in Cadenillas et al. [
be the optimal value function for this problem. As was shown in [
satisfies the same HJB equation, except for boundary conditions (3.28) and (3.29). These conditions are replaced by
.
should be sought as a solution to (3.39)-(3.44) below.
.
, where 
is the same as in (3.20) and 
is a free constant, and a general solution to (3.40) is 
, where 
and 
are the same as in (3.25), (3.26).
, which is defined as follows.
and 
are chosen in such a way that 
and
. That is, the functions 
and 
are continuous at
. (Note that 
is a derivative of 
on 
and 
is a derivative of 
on
.) We next examine the family of functions
, where
. We seek 
such that 
becomes the derivative of the optimal value function
.
and 
such that 
and
. Note that 
is a concave function, which is easily checked by differentiation. Let
, the point at which the maximum of 
is achieved (it is easy to see that
, whereas
, which shows that
exists; in view of the fact that
, it is unique). It is obvious by virtue of the concavity of 
that, for any
, 
and 
exist.
. Informally, 
is the area under the graph of 
and above the horizontal line
. It is obvious that 
is a continuous function of
. For
, we have
; Let
obtained in the previous subsection to construct the derivative of the optimal val-
. The main idea is to consider 
and try to find 
such that
. The optimal value function 
will then be sought in the form of
. To this end, we need the following proposition.
satisfies (3.39) (satisfies (3.40)); then, for any S, the function 
satisfies the same equation on the interval shifted by S to the left.
has a singularity at 0 with
. The concavity of 
on
implies that 
is increasing on 
and decreasing on 
(recall that 
is constant on
). Therefore, there exists unique 
such that
. Define
decreases to 
at 0 at the order of 
(see (3.45)); therefore, 
is integrable at 0 and, as a result,
.
and
; hence, we divide our analysis into two cases.
, 
and the graph of the function
. Obviously, 
is a continuous function of
. Because
and
, there exists an 
(see
is the derivative of the solution 
to the QVI, inequalities (2.12)-(2.14).
satisfies (3.18) on
, as well as (3.23) on
. In addition, from (3.48), we can see that
, we can also see that
and similarly
, so 
satisfies all the conditions (3.28)-(3.31).
given by (3.56) is a solution to the QVI (2.12)-(2.14).
satisfies (3.16) on
.
on
. Consequently,
. As 
satisfies (3.18), it also satisfies (3.16) because these two equations are equivalent whenever (3.16) holds.
, it is sufficient to show that for each
,
satisfies (3.23) (that is,
). By subtracting (3.58) from the left hand side of (3.23), we can see that (3.58) is equivalent to
. In view of the continuity of the first and second derivatives, and in view of the fact that (3.16) holds on
, we know that (3.59) is true for
. As both 
and 
on the left-hand side of (3.59) are decreasing, therefore is nonpositive for all
.
, where
, we have
, or if
, then
. Supposethat
. The function 
is continuously differentiable. By construction, 
is increasing on 
and decreasing on 
with 
for
. Therefore, the point 
is the only point 
such that
. Because
, we can see that
. Therefore, for
,
for all 
and also shows that
for
, we know that for any 
the function 
is an increasing function of
. Therefore, the minimum in the expression for 
is attained for
; as a result,
.
for
. Because 
for 
and 
for
, we can see that 
has a unique minimum at
. Therefore,
. Thus, 
if
, we note, that
, and hence
, then (2.12) holds.
and any
,
. As
, we obtain
.
, we have
we cannot find any S* such that (3.48) is satisfied. In this case, we set 
(that is, we have
, which corresponds to
).
, then 
is a solution to the QVI (2.12)-(2.14).
, whereas that of Proposition 3.3 requires a slight modification.
,
, then
for all 
and that 
for 
is the same as that in Proposition 3.3.
, then 
is equivalent to
, then 
still holds, due to the same argument as that in Proposition 3.3. 
we have 
only for
, whereas
in contrast to the case when
. Also, when
, we have
, whereas 
if
. Equivalently, in the case that the fixed cost to call for additional funds is relatively large (i.e.,
) , the optimal band control is reduced to 
with
. That is, as soon as the reserve reaches zero, it becomes optimal for the mutual insurance firm to go bankrupt, rather than to be restarted by calling for additional funds.
,
. In this case,
. Let 
be any admissible control defined by (2.2) and process 
be the corresponding surplus process (2.3), with
. Let 
be its ruin time given by (2.4). Let 
and
. Then,
. In view of (2.13), we have
the second sum in (4.68) is bounded by the same integrable random variable independent of
.
, process 
is continuous, and we can apply Ito’s formula to get
is also bounded by the same integrable random variable. Similar arguments show that
, and employing (4.72) and the monotone convergence theorem on, we get
, on
, the inequality (4.75) implies (4.67).
to be bounded. This is the case when
. When
, the function 
has a singularity at 0. We can, however, apply the same technique, first replacing 
by 
and then passing to a limit as
. This will yield inequality (4.74), which is all we need for the proof of Theorem 4.1 Let
, that is
, with
, 
, and 
and 
are defined below sequentially.
(that is, no 
exists, such that (3.48) holds), then
(that is, there exists an
, such that (3.48) holds), then
, then process 
is a continuous diffusion process with a drift and diffusion coefficient of, 
and 
, respectively, until the times of intervenetion. The times of intervention in this case are the times at which this diffusion process hits the level
, which are associated with the refunds of the constant amount of
. The time when 0 is hit is the ruin time.
, the process 
is a continuous diffusion process 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
. At point 0, the control is set to displace the process to point
, which corresponds to raising cash (making a call to shareholders) in the amount of
. Reaching the level 
results in the displacement of the process to the point 
which, corresponds to making a refund in the amount of
.
be the control described by (4.76) and (4.79)-(4.80). Then,
and if 
then
. Thus, we can repeat the arguments in the proof of Theorem 3.2 and see that, for
, all of the inequalities are tight. As a result, we obtain
, we have 
and, when
, the function V satisfies
, the first term on the left-hand side of (4.84) vanishes, and we get (4.82).
, which determines the threshold for set-up cost
, 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
, it can be determined in an algorithmic manner prior to solving the optimal control problem for the mutual insurance company.
. As our analysis shows, in this case, 
and
, 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 
and one for which reaching 0 corresponds to ruin and for which
. This is a rather unique feature of this particular problem that has not been observed previously.
. As can be seen from our analysiswe find a solution to this problem for the case of 
. 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 [