Hattendorff’s theorem for non-smooth continuous-time Markov models I: Theory

Extending previous work of Ramlau-Hansen (Ramlau-Hansen, H., 1998a. Scand. Actuarial J., 143–156) for smooth Markov models, Hattendorff’s theorem on the decomposition of the variance of the overall loss created by an insurance contract is generalized to policy developments given by an inhomogeneous continuous-time Markov jump process with a possibly non-smooth transition matrix. Due to the lack of smoothness assumptions, our result covers classical discrete versions of Hattendorff’s theorem as well, and it is also applicable to “mixed” situations in which some transitions have smooth transition probabilities, whereas others can only take place at discrete times. Following Ramlau-Hansen (op. cit.), we use martingale techniques for the multivariate counting process underlying the policy development, the difference being that here its compensator may have jumps. This difference leads to a discontinuity correction in the explicit variance formula for the loss.