In this paper we consider an optimal dividend  problem for an insurance company which risk process evolves as a spectrally negative Levy process (in the absence of dividend payments). We assume that the management of the company controls timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividends received until the moment of ruin and a penalty payment at the moment of ruin which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. We explicitly solve the corresponding optimal control problem. The solution rests on the characterization of the value-function as (i) the unique stochastic solution of the associated HJB equation and as (ii) the pointwise smallest stochastic supersolution . We show that the optimal value process admits a dividend-penalty decomposition as sum of a martingale (associated to the penalty payment at ruin) and a potential (associated to the dividend payments). We find also an explicit necessary and sufficient condition  for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. We analyze a number of concrete examples  .
This paper concerns an optimal dividend distribution problem for an insurance company which risk process evolves as a spectrally negative L\'{e process (in the absence of dividend payments). The management of the company is assumed to control timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividend payments received until the moment of ruin and a penalty payment at the moment of ruin which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. A complete solution is presented to the corresponding stochastic control problem. It is established that the value-function is the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated HJB equation. Furthermore, a necessary and sufficient condition is identified for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. A number of concrete examples are analyzed .
R
In this paper we consider
This paper concerns
A
distribution
R
Levy
L\'{e
R
We assume that the
The
R
controls
is assumed to control
R
dividends
dividend payments
R
We explicitly solve the corresponding optimal
A complete solution is presented to the corresponding stochastic
R
The solution rests on the characterization of the
It is established that the
R
as (i)
is
R
of the associated HJB equation and as (ii)
and
R
. We show that the optimal value process admits a dividend-penalty decomposition as sum of a martingale (associated to the penalty payment at ruin) and a potential (associated to the dividend payments). We find also an explicit
of the associated HJB equation. Furthermore, a
A
is identified
R
We analyze a
A
A
are analyzed