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Optimal dividend and capital injection problem in a BMAP model. (Chinese. English summary) Zbl 1363.91027

Summary: This paper studies the optimal dividend and capital injection problem of a company in a BMAP (Batch Markov Arrival Process) model. The parameters in the process of the company’s surplus are modulated by the phase process of the BMAP, which is an observable continuous-time Markov chain. The possible dividend and capital injection are restricted to some random discrete time points which are determined by the same BMAP. The company has both dividend and capital injection opportunities or only has dividend but not capital injection opportunities at some of these time points, while can do nothing at other random time points. By transforming the BMAP model to an auxiliary Markov modulated model, we study the optimal dividend and capital injection problem of the company under the assumption that the company will not bankrupt. This paper aims to maximize the difference between the total expected discounted dividend and the amount of capital and obtain the exact solution of the value functions and the optimal dividend and capital injection strategy.

MSC:

91B30 Risk theory, insurance (MSC2010)
91G10 Portfolio theory
62P05 Applications of statistics to actuarial sciences and financial mathematics
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