Generalized stochastic integrals and equations

Abstract

1. Introduction. In his fundamental memoir [7] K. Ito introduced an important class of stochastic differential equations which are now known as Ito equations. These equations are based on his definitions of stochastic integrals with respect to Brownian motion and random measures with independent values. The importance of these equations is due to the fact that a large class of Markov processes in Rn can be represented as solutions of such equations. A thorough account of Ito's work and various extensions of it has been given by A. V. Skorohod [18]. It is also possible to represent certain other Markov processes of finite order as solutions of vector Ito equations. In this paper we consider an extension of Ito integrals to integrals with respect to generalized processes with independent values in the sense of I. M. Gel'fand and N. Ja. Vilenkin [5]. We do not consider the most general case which would involve integration with respect to generalized processes with independent values defined in Rn, n a 1 (cf. K. Ito [7] and A. V. Skorohod [18] for random measures) but restrict ourselves to the case n=X. The relation between the generalized Ito integral and the multiplication of (Schwartz) distributions is discussed. This involves a study of /?-transformations of Q)', the space of distributions, where p. is a measure on 3>'. Following the example of K. Ito we then discuss stochastic differential equations defined in terms of these stochastic integrals and prove several existence theorems. It is shown that the solutions of generalized Ito equations with local coefficients have local splitting a-fields and are weakly Markov of finite order. Generalized Ito equations may also be considered in connection with P. Levy's problem of the representation of stochastic processes in terms of differential innovation processes