Multivariate higher order moments in multi-state life insurance

It is well-known that combining life annuities and death benefits introduce opposite effects in payments with respect to the unsystematic mortality risk on lifetime of insured. In a general multi-state framework multiple product types, such joint are less trivial. this paper, we consider multivariate payment process insurance, where components defined terms same Markovian state process. The present value future introduced, derive differential equations integral representations its conditional moments moment generating function. Special attention given pair-wise covariances between two values, results closely connected Hattendorff type for variance derived. Numerical examples shown disability model spouse illustrate applicability insurance single as well dependent lives.