Stochastic integral with respect to a semi-Markov process of diffusion type

We consider a multidimensional semi-Markov process of diffusion type. A stochastic integral with respect to the semi-Markov process is defined in terms of asymptotics related to the first exit time from a small neighborhood of the starting point of the process, and, in particular, in terms of its characteristic operator. This integral is equal to the sum of two other integrals: the first one is a curvilinear integral with respect to an additive functional defined in terms of the expected first exit time from a small neighborhood, and the second one is a stochastic integral with respect to a martingale of special kind. To prove the existence and to derive the properties of the integral, both the method of deducing sequences and that of inscribed ellipsoids are used. For Markov processes of diffusion type, the new definition of the stochastic integral is reduced to the standard one. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 251–276.