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August 2015 On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function
F. Avram, Z. Palmowski, M. R. Pistorius
Ann. Appl. Probab. 25(4): 1868-1935 (August 2015). DOI: 10.1214/14-AAP1038


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.


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F. Avram. Z. Palmowski. M. R. Pistorius. "On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function." Ann. Appl. Probab. 25 (4) 1868 - 1935, August 2015.


Received: 1 December 2012; Revised: 1 May 2014; Published: August 2015
First available in Project Euclid: 21 May 2015

zbMATH: 1322.60055
MathSciNet: MR3348997
Digital Object Identifier: 10.1214/14-AAP1038

Primary: 60J99 , 93E20
Secondary: 60G51

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

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.25 • No. 4 • August 2015
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