In this paper we consider an offshoot of the classical de Finetti’s optimal dividends problem in continuous time for which a transaction cost is incurred each time a dividend payment is made. Because of this fixed cost, it is no longer feasible to pay out dividends at a certain rate and therefore only lump sum dividend payments are possible.
Within this problem we assume that the underlying dynamics of the risk process is described by a spectrally negative Lévy process which is now widely accepted and used as a replacement for the classical Cramér–Lundberg process (cf. Albrecher et al. (2008), Avram et al. (2007), Dufresne and Gerber (1993), Dufresne et al. (1991), Furrer (1998), Huzak et al. (2004), Kyprianou and Palmowski (2007), Kyprianou et al. (in press), Loeffen, 2008 and Loeffen, 2009 and Renaud and Zhou (2007)). Recall that a Cramér–Lundberg risk process {Xt:t≥0}{Xt:t≥0} corresponds towhere x>0x>0 denotes the initial surplus, the claims C1,C2,…C1,C2,… are i.i.d. positive random variables with expected value μμ, c>0c>0 represents the premium rate and N={Nt:t≥0}N={Nt:t≥0} is an independent Poisson process with arrival rate λλ. Traditionally it is assumed in the Cramér–Lundberg model that the net profit condition c>λμc>λμ holds, or equivalently that XX drifts to infinity. In this paper XX will be a general spectrally negative Lévy process and the condition that XX drifts to infinity will not be assumed.
We will now state the control problem considered in this paper. As mentioned before, X={Xt:t≥0}X={Xt:t≥0} is a spectrally negative Lévy process which is defined on a filtered probability space (Ω,F,F={Ft:t≥0},P)(Ω,F,F={Ft:t≥0},P) satisfying the usual conditions. Within the definition of a spectrally negative Lévy process it is implicitly assumed that XX does not have monotone paths. We denote by {Px,x∈R}{Px,x∈R} the family of probability measures corresponding to a translation of XX such that X0=xX0=x, where we write P=P0P=P0. Further ExEx denotes the expectation with respect to PxPx with EE being used in the obvious way. The Lévy triplet of XX is given by (γ,σ,ν)(γ,σ,ν), where γ∈Rγ∈R, σ≥0σ≥0 and νν is a measure on (0,∞)(0,∞) satisfying Note that even though XX only has negative jumps, for convenience we choose the Lévy measure to have only mass on the positive instead of the negative half line. The Laplace exponent of XX is given by and is well defined for θ≥0θ≥0. Note that the Cramér–Lundberg process corresponds to the case that σ=0σ=0, View the MathML sourceν(dx)=λF(dx) where FF is the law of C1C1 and View the MathML sourceγ=c−∫(0,1)xν(dx). The process XX will represent the risk process/reserves of the company before dividends are deducted.
We denote a dividend or control strategy by ππ, where View the MathML sourceπ={Ltπ:t≥0} is a non-decreasing, left-continuous FF-adapted process which starts at zero. FurtherHere we mean by View the MathML sourceΔLsπ=Ls+π−Lsπ the jump of the process LπLπ at time ss.
The random variable View the MathML sourceLtπ will represent the cumulative dividends the company has paid out until time tt under the control ππ. We define the controlled (net) risk process View the MathML sourceUπ={Utπ:t≥0} by View the MathML sourceUtπ=Xt−Ltπ. Let View the MathML sourceσπ=inf{t>0:Utπ<0} be the ruin time and define the value function of a dividend strategy ππ by we assume that the process LπLπ is a pure jump process, i.e. where q>0q>0 is the discount rate and β>0β>0 is the transaction cost incurred for each dividend payment. Note that because of (2) we can write View the MathML sourcevπ(x)=Ex[∑0≤t<σπe−qt(ΔLtπ−β1{ΔLtπ>0})]. By definition vπ(x)=0vπ(x)=0 for x<0x<0. A strategy ππ is called admissible if ruin does not occur due to a lump sum dividend payment, i.e. View the MathML sourceΔLtπ≤Utπ for t<σπt<σπ. Let ΠΠ be the set of all admissible dividend policies. The control problem consists of finding the optimal value function v∗v∗ given by Since control strategies of the form (2) are known as impulse
controls, we refer to this problem as the impulse control problem.
An important type of strategy for the impulse control problem is
the one we call in this paper the .c1I c2/ policy and which is similar
to the well known .s; S/ policy appearing in inventory control
models, see e.g. Bather (1966) and Sulem (1986). The .c1I c2/ policy
is the strategy where each time the reserves are above a certain
level c2, a dividend payment is made which brings the reserves
down to another level c1 and where no dividends are paid out
when the reserves are below c2. In case X is a Brownian motion
plus drift, Jeanblanc-Picqué and Shiryaev (1995) showed that an
optimal strategy for the impulse control problem is formed by
a .c1I c2/ policy. Paulsen (2007) considered the case when X is
modeled by a diffusion process and showed that under certain
conditions a .c1I c2/ policy is optimal. Note that in Paulsen (2007)
this type of strategy is referred to as a lump sum dividend barrier
strategy. Further, Alvarez and Rakkolainen (2009) study the case
where the driving process is a spectrally negative Lévy diffusion
with a jump component of geometric form. In this paper we
will investigate when an optimal strategy for our impulse control
problem is formed by a .c1I c2/ policy.
When the assumption (2) is dropped and the transaction cost
is taken to be equal to zero, then the impulse control problem
transforms into the classical de Finetti optimal dividends problem.
The latter optimal dividends problem will be referred to as the
de Finetti problem in the remainder of the paper. This particular
problem was introduced by de Finetti (1957) in a discrete time setting for the case that the risk process evolves as a simple random
walk. Thereafter the de Finetti problem has been studied in a
continuous time setting for the case that X is a CramérLundberg
risk process (Gerber, 1969; Azcue and Muler, 2005) and for the
case that the risk process is a general spectrally negative Lévy
process (Avram et al., 2007; Kyprianou et al., in press; Loeffen,
2008). For this problem an important strategy is the so-called
barrier strategy. The barrier strategy at level a is the strategy where
initially (in case the starting value of the reserves are above a) a
lump sum dividend payment is made to bring the reserves back
to level a and thereafter each time the reserves reach the level a,
non-lump sum dividend payments are made in such a way that
the reserves do not exceed the level a, but where no dividends are
paid out when the reserves are strictly below a. Mathematically
this corresponds to reflecting the risk process X at a. The barrier
strategy at level a may be seen (at least intuitively) as a limit of
.c1I c2/ policies where c1 and c2 converge to the barrier a.
Gerber (1969) proved that an optimal strategy for the de Finetti
problem is formed by a barrier strategy in the case where X is
a CramérLundberg risk process with exponentially distributed
claims. Building on the work of Avram et al. (2007), Loeffen (2008)
showed that optimality of the barrier strategy for the de Finetti
problem depends on the shape of the so-called scale function of
a spectrally negative Lévy process. To be more specific, the q-scale
function of X, W.q/ V R ! T0;1/ where q 0, is the unique
function such thatW.q/.x/ D 0 for x < 0 and on T0;1/ is a strictly
increasing and continuous function characterized by its Laplace transform which is given by Theorem 2 of Loeffen (2008) then says that if W.q/ is sufficiently
smooth and ifW.q/0 is increasing on .a;1/ where a is the largest
point where W.q/0 attains its global minimum, then the barrier
strategy at a is optimal for the de Finetti problem. HereW.q/ being
sufficiently smooth means that W.q/ is once/twice continuously
differentiable when X is of bounded/unbounded variation. It was
then shown in Loeffen (2008) that when X has a Lévy measure
which has a completely monotone density, these conditions on the
scale function are satisfied and in particular that W.q/0 is strictly
convex on .0;1/. (Note that it was shown in Loeffen (2009) that
W.q/0 is actually strictly log-convex.) Shortly thereafter, Kyprianou
et al. (in press) proved that W.q/0 is strictly convex on .a;1/
under the weaker condition that the Lévy measure has a density
which is log-convex and then used Theorem 2 from Loeffen (2008)
mentioned above, to conclude that the barrier strategy at a is
optimal (though they needed to relax the sufficiently smoothness
assumption). It is important to note that without a condition on
the Lévy measure the barrier strategy is not optimal in general.
Indeed Azcue and Muler (2005) have given an example for which
no barrier strategy is optimal.
In this paper we will show that the results for the de Finetti
problem mentioned in the previous paragraph have their counterparts
for the impulse control problem, whereby the role of the barrier
strategy is now played by the .c1I c2/ policy. In particular we
will give a theorem similar to Theorem 2 in Loeffen (2008) and then
use this theorem to show that a certain .c1I c2/ policy is optimal if
the Lévy measure has a log-convex density. Moreover we give an
example for which no .c1I c2/ policy is optimal.
The outline of this paper is as follows. In the next section we
review some properties concerning scale functions and in Section 3
we give sufficient conditions under which the .c1I c2/ policy is
optimal.Wetreat the case when the Lévy measure has a log-convex
density in Section 4 and show that the optimal strategy is formed
by a unique .c1I c2/ policy. Further we show how to numerically
find the optimal values of c1 and c2. In the last section we treat two
explicit examples including one for which we show that no .c1I c2/
policy is optimal.
