An Alternative for Time Series Models

Abstract

Two methods for construction of new stochastic processes with discrete time are presented. One of the methods employs as the defining tool �triangular (more specifically �pseudoaffine�) transformations� which are extended from the Euclidean Rnto infinite dimension space. They transform any well-known discrete time stochastic process into the constructed one. The other, more flexible, method is the �method of parameter dependence�, extended to infinite dimension. Properties of the obtained stochastic processes (by either method) indicate the possibility to apply them for financial analysis, as an alternative for the classical time series models. The advantage of the presented models over the existing ones first of all relies on expected better accuracy. This follows from the fact that the typically held assumption on Markovianity in the existing models can easily be relaxed. The defined processes may incorporate a quite long memory including, among others, the k-Markovian cases for k ? 2. Regardless the non-Markovianity of the models they still are tractable in an analytical or numerical way. The stochastic processes defined in this paper provide more flexible and more general tools than the existing time series models for modeling financial problems. Among others, they make it possible to incorporate the influence of environmental (explanatory) random variables on the underlying stochastic models� behavior. These additional features turn out to be describable by the method of parameter dependence. Some suggestions for an associated preliminary statistical analysis are included.