On Anticipatory and Nonanticipatory Properties in Stochastic Differential Systems

摘要

One of the aims of this paper is to point out the existence of strong computing anticipatory property and nonanticipatory property in stochastic differential systems, which are characterized by stochastic differential equations (SDEs). Since an SDE is merely a matter of form, its definition sorely depends on the stochastic integrals, not on differentials. Among others Ito and Stratonovich-Fisk integrals are widely used, and the latter has anticipatory property in its own definition, and hence the systems, which are described by equations including the Stratonovich-Fisk integrals, are considered to be strong computing anticipatory systems in stochastic type. On the other hand, Ito differential systems are nonanticipatory systems. In computing current states some of numeriacl algorithms for solving the SDE are taking into account predicted states. Hence, such systems are weakly computing anticipatory systems.