Markov models and Thieleʹs integral equations for the prospective reserve

Extending previous work of Hoem (1968, 1969) and Norberg (1990, 1991), a mathematical framework for the insurance of persons is proposed, which jointly comprises the “discrete method” and the “continuous method” of insurance mathematics as well as intermediate cases. Our main tool for modelling the policy development is the theory of Markov jump processes based on cumulative transition intensities, as developed by Jacobsen (1972) and Gill and Johansen (1990), which is reviewed here to some extent. Within this set-up, generalizations of Hoemʹs and Norbergʹs version of Thieleʹs differential equations for the prospective reserve are derived. These integral equations turn out to be equivalent to the backward integral equations connecting the transition matrix to the cumulative transition intensity matrix of a Markov jump process. The uniqueness of their solution is established. Applications given include general recursion formulae for the prospective reserve, a generalization of Cantelliʹs theorem and premium calculation in pension insurance with pension age chosen by the insured.