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Risk-sensitive dividend problems. (English) Zbl 1341.91087

Summary: We consider a discrete time version of the popular optimal dividend payout problem in risk theory. The novel aspect of our approach is that we allow for a risk averse insurer, i.e., instead of maximising the expected discounted dividends until ruin we maximise the expected utility of discounted dividends until ruin. This task has been proposed as an open problem in [H. U. Gerber and E. S. W. Shiu, N. Am. Actuar. J. 8, No. 1, 1–20 (2004; Zbl 1085.62122)]. The model in a continuous-time Brownian motion setting with the exponential utility function has been analysed in [P. Grandits et al., Scand. Actuar. J. 2007, No. 2, 73–107 (2007; Zbl 1164.62080)]. Nevertheless, a complete solution has not been provided. In this work, instead we solve the problem in discrete time setup for the exponential and the power utility functions and give the structure of optimal history-dependent dividend policies. We make use of certain ideas studied earlier in [N. Bäuerle and U. Rieder, Markov decision processes with applications to finance. Berlin: Springer (2011; Zbl 1236.90004)], where Markov decision processes with general utility functions were treated. Our analysis, however, includes new aspects, since the reward functions in this case are not bounded.

MSC:

91B30 Risk theory, insurance (MSC2010)
60J05 Discrete-time Markov processes on general state spaces
90C40 Markov and semi-Markov decision processes
91B16 Utility theory
91G80 Financial applications of other theories
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