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A time-continuous Markov chain interest model with applications to insurance. (English) Zbl 1067.91509

In classical insurance mathematics only the claims processes were modelled as stochastic. One mainstream in present-day insurance mathematics undertakes to incorporate financial risk in the form of stochastic interest. A general motivation and discussion of the issue were given by H. Bühlmann [Insur. Math. Econ. 11, No. 2, 113–127 (1992; Zbl 0763.62055)] and P. Devolder [Finance stochastique, Éditions de l’Université de Bruxelles (1993)], where lists of key references are also provided. In this paper the force of interest is modelled by a homogeneous time- continuous Markov chain with finite state space. For expected values of various functionals of this process the author derives the ordinary differential equations, in particular, for moments of present values of payments streams. The homogeneity of the interest process invites studies of stationary (time- independent) functionals, e.g., moments perpetuities, and gives rise to explicit formulae for expected value of other stationary functionals. Applications are made to some standard forms of insurance.

MSC:

91B30 Risk theory, insurance (MSC2010)
91G30 Interest rates, asset pricing, etc. (stochastic models)
62P05 Applications of statistics to actuarial sciences and financial mathematics
34A30 Linear ordinary differential equations and systems
60J27 Continuous-time Markov processes on discrete state spaces

Citations:

Zbl 0763.62055
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References:

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