Differential equations for moments of present values in life insurance

Ordinary differential equations are obtained for first and higher order conditional moments of present values of payments in respect of a life insurance policy described as a time-continuous Markov chain. Those for the first moments are the well-known Thieleʹs differential equations for the reserve. It is shown how the differential equations can be used to construct untraditional insurance products. Numerical computations of moments are performed for some forms of insurance common in practice. Applications to problems in pure probability theory are demonstrated by examples.