Additive Functionals Of Semi-Dynamic System And Kac’s Moment Formula For Markov Processes With Single Jump

The systems which Markov process and semi-dynamic system de-scribe have the same property:if the initial state is determinate, the future state just depend on the present state and is independent with its history. In other words, a Markov process without randomness is a semi-dynamic system, a semi-dynamic system with a sequence of stochastic jumping times is a Markov process. So Markov process have a close connection with semi-dynamic system. The study of ad-ditive functional of semi-dynamic system have a great significance on Markov process. In this paper, we mainly focus on the mathematical analysis of additive functional of semi-dynamic system.Kac(1949,1951), and Darling and Kac(1957) introduced a method for calculating the distribution of the integral Ah=∫(0,T]h(Xt)dt and the nth moment of Ah under the case when X is a Brownian motion. In this paper, we review Kac’s method and discuss the nth moment of the integral ∫0T f(Xs)dA(x, s) with the aid of the mathematical analysis of additive functionals of semi-dynamic system, where X is amarkov process with a single jump and A(x,t) is a additive function.