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Optimal dividends and capital injections in the dual model with a random time horizon. (English) Zbl 1341.49021
Summary: This paper investigates an optimal dividend and capital injection problem in the dual model with a random horizon. Both fixed and proportional costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends and the penalized discounted capital injections during the horizon, which is described by the minimum of the time of ruin and an exponential random variable. By the fluctuation theory of Lévy processes, the optimal dividend and capital injection strategy is obtained. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Besides, numerical examples are studied to illustrate our results.

MSC:
49J55 Existence of optimal solutions to problems involving randomness
60G51 Processes with independent increments; Lévy processes
93E20 Optimal stochastic control
91G80 Financial applications of other theories
91G60 Numerical methods (including Monte Carlo methods)
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