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The impact of negative interest rates on optimal capital injections. (English) Zbl 1416.91172
Summary: In the present paper, we investigate the optimal capital injection behaviour of an insurance company if the interest rate is allowed to become negative. The surplus process of the considered insurance entity is assumed to follow a Brownian motion with drift. The changes in the interest rate are described via a Markov-switching process. It turns out that in times with a positive rate, it is optimal to inject capital only if the company becomes insolvent. However, if the rate is negative it might be optimal to hold a strictly positive reserve. We establish an algorithm for finding the value function and the optimal strategy, which is proved to be of barrier type. Using the iteration argument, we show that the value function solves the Hamilton-Jacobi-Bellman equation, corresponding to the problem.
MSC:
 91B30 Risk theory, insurance (MSC2010) 93E20 Optimal stochastic control 49L20 Dynamic programming in optimal control and differential games 91G30 Interest rates, asset pricing, etc. (stochastic models)
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