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Dividend moments in the dual risk model exact and approximate approaches. (English) Zbl 1256.91026
Summary: In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in [B. Avanzi et al., Insur. Math. Econ. 41, No. 1, 111–123 (2007; Zbl 1131.91026)]. Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

91B30 Risk theory, insurance (MSC2010)
91G80 Financial applications of other theories
Full Text: DOI
[1] DOI: 10.2143/AST.21.2.2005364 · doi:10.2143/AST.21.2.2005364
[2] Insurance Risk and Ruin (2005) · Zbl 1060.91078
[3] Insurance: Mathematics and Economics 7 pp 1– (1988)
[4] Insurance: Mathematics and Economics 41 pp 111– (2007)
[5] Scandinavian Actuarial Journal pp 6– (2005)
[6] DOI: 10.1017/S0515036100013878 · doi:10.1017/S0515036100013878
[7] Stochastic Theory of a Risk Business (1969) · Zbl 0196.23501
[8] Loss Models: From Data to Decisions (2004)
[9] DOI: 10.1017/S0515036100014392 · doi:10.1017/S0515036100014392
[10] DOI: 10.2143/AST.36.2.2017931 · Zbl 1162.91374 · doi:10.2143/AST.36.2.2017931
[11] Insurance: Mathematics and Economics 35 pp 205– (2004)
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