Numerical Solutions of the Hattendorff Differential Equation for Multi-State Markov Insurance Models

Abstract

We use the representation of a continuous time Hattendorff differential equation and Matlab to compute 2σt(j), the solution of a backwards in time differential equation that describes the evolution of the variance of Lt(j), the loss at time t random variable for a multi-state Markovian process, given that the state at time t is j. We demonstrate this process by solving examples of several instances of a multi-state model which a practitioner can use as a guide to solve and analyze specific multi-state models. Numerical solutions to compute the variance 2σt(j) enable practitioners and academic researchers to test and simulate various state-space scenarios, with possible transitions to and from temporary disabilities, to permanent disabilities, to and from good health, and eventually to a deceased state. The solution method presented in this paper allows researchers and practitioners to easily compute the evolution of the variance of loss without having to resort to detailed programming.