On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency.

*(English)*Zbl 1291.91088Summary: We consider the dual model, which is appropriate for modeling the surplus of companies with deterministic expenses and stochastic gains, such as pharmaceutical, petroleum or commission-based companies. Dividend strategies for this model that can be found in the literature include the barrier strategy (e.g., [the first author et al., ibid. 41, No. 1, 111–123 (2007; Zbl 1131.91026)]) and the threshold strategy (e.g., [the second author, “‘Recursive calculation of the dividend moments in a multi-threshold risk model’, Andrei Badescu and David Landriault, January 2008”, N. Am. Actuar. J. 12, No. 3, 336–340 (2008; doi:10.1080/10920277.2008.10597525)]), where dividend decisions are made continuously. While in practice the financial position of a company is typically monitored frequently, dividend decisions are only made periodically along with the publication of its books. In this paper, we introduce a dividend barrier strategy whereby dividend decisions are made only periodically, but still allow ruin to occur at any time (as soon as the surplus is exhausted). This is in contrast to H. Albrecher et al. [Astin Bull. 41, No. 2, 645–672 (2011; Zbl 1239.91072)], who introduced periodic dividend payments in the CramĂ©r-Lundberg surplus model, albeit with periodic ruin opportunities as well.

Under the assumption that the time intervals between dividend decisions are Erlang(\(n\)) distributed, we derive integro-differential equations for the Laplace transform of the time to ruin and the expected present value of dividends until ruin. These are then solved with the help of probabilistic arguments. We also provide a recursive algorithm to compute these quantities. Finally, some numerical studies are presented, which aim at illustrating how our assumptions about dividend payments and ruin occurrence compare with those of the classical barrier strategy.

Under the assumption that the time intervals between dividend decisions are Erlang(\(n\)) distributed, we derive integro-differential equations for the Laplace transform of the time to ruin and the expected present value of dividends until ruin. These are then solved with the help of probabilistic arguments. We also provide a recursive algorithm to compute these quantities. Finally, some numerical studies are presented, which aim at illustrating how our assumptions about dividend payments and ruin occurrence compare with those of the classical barrier strategy.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

91G50 | Corporate finance (dividends, real options, etc.) |