## A theory of bonus life insurance.(English)Zbl 0939.62108

The paper deals with a sketch of the traditional approach to bonus theory in the framework of a time-continuous Markov chain model for a multi-state insurance policy. In this setup intensities of transitions $$\mu_{jk}(t)$$ and intensity of interest $$\delta(t)$$ are non-stochastic for the time being. Further, this model is extended by letting the second order experience basis (mortality, interest, etc.) be stochastic. More precise, the author extends the second order model by placing a probabilistic distribution on $$\mu_{jk}(t)$$ and $$\delta(t)$$ and taking the traditional model as a conditional one given these elements.
Special attention is given to a candidate model in which the second order basis is governed by a time-continuous Markov chain $$y(t)$$, where $$\delta(t)$$ and $$\mu_{jk}(t)$$ depend on the current $$y$$-state. A novel definition of the technical surplus of an insurance contract is proposed and main principles for its repayment as bonus are presented. Possibilities of model-based prognoses of future bonuses within such an approach are discussed. Numerical examples are provided.

### MSC:

 62P05 Applications of statistics to actuarial sciences and financial mathematics 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 91B30 Risk theory, insurance (MSC2010)
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