A counting process approach to stochastic interest

The aim of the present paper is to propose a stochastic approach for describing the return of an investment, and study its applications in insurance. The process governing the return of the investment is assumed to have bounded variation over finite intervals and possess a jump part. Attention is restricted to cases where the process has independent increments and is subject to fluctuations given by a Markovian environment. In the first case direct calculations are obtainable for evaluating moments of present and accumulated values. In the last case we establish differential equations akin to the celebrated Thieleʹs differential equation in life insurance.