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A multi--state life insurance model is naturally described in terms of theintensity matrix of an underlying (time--inhomogeneous) Markov process whichdescribes the dynamics for the states of an insured person. Between and attransitions, benefits and premiums are paid, defining a payment process, andthe technical reserve is defined as the present value of all future payments ofthe contract. Classical methods for finding the reserve and higher ordermoments involve the solution of certain differential equations (Thiele andHattendorf, respectively). In this paper we present an alternativematrix--oriented approach based on general reward considerations for Markovjump processes. The matrix approach provides a general framework foreffortlessly setting up general and even complex multi--state models, wheremoments of all orders are then expressed explicitly in terms of so--calledproduct integrals (matrix--exponentials) of certain matrices. As Thiele andHattendorf type of theorems can be retrieved immediately from the matrixformulae, this methods also provides a quick and transparent approach toproving these classical results. Methods for obtaining distributions andrelated properties of interest (e.g. quantiles or survival functions) of thefuture payments are presented from both a theoretical and practical point ofview (via Laplace transforms and methods involving orthogonal polynomials).

Matrix calculations for inhomogeneous Markov reward processes, with applications to life insurance and point processes