Thiele's differential equation with stochastic interest of diffusion type

The classical Thiele's differential equation for the prospective reserve of an insurance policy has been generalized to models with counting process driven payments and deterministic interest. Here the result is extended to situations with diffusion driven stochastic interest. The technique of proof consists in identifying the null part of the martingale associated with the initial present value of the payments. The presentation centers on life insurance, but the theory can be adapted to more general stochastic payment streams.