The Annals of Applied Probability

On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function

F. Avram, Z. Palmowski, and M. R. Pistorius

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This paper concerns an optimal dividend distribution problem for an insurance company whose risk process evolves as a spectrally negative Lévy 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.

Article information

Ann. Appl. Probab. Volume 25, Number 4 (2015), 1868-1935.

Received: December 2012
Revised: May 2014
First available in Project Euclid: 21 May 2015

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Mathematical Reviews number (MathSciNet)

Primary: 60J99: None of the above, but in this section 93E20: Optimal stochastic control
Secondary: 60G51: Processes with independent increments; Lévy processes

Stochastic control singular control impulse control state-constraint problem stochastic solution integro-differential HJB equation Lévy process De Finetti model barrier/band strategy Gerber–Shiu function


Avram, F.; Palmowski, Z.; Pistorius, M. R. On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function. Ann. Appl. Probab. 25 (2015), no. 4, 1868--1935. doi:10.1214/14-AAP1038.

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