On Strict-Sense Forms of the Hida-Cramer Representation
This paper is an attempt to delineate more clearly than before the role of the standard Brownian motion and Poisson processes in generating general stochastic processes (Ω δt,Xt, P). In discrete time, the analog would be to use a sequence of independent Bernoulli random walks. This is a very different setting, and one about which we have nothing to contribute. Evidently such a sequence does not go far toward generating a general discrete parameter process, at least in the sense we have in mind here. The situation in continuous time, however, is antithetical. One can obtain representation theorems of considerable generality, which to some extent crystalize an important aspect of all continuous time processes. We consider this as the aspect which involves randomness of time without randomness of place. In some sense, there is a natural dichotomy of the two kinds of randomness, and our aim is to isolate and study the case in which the role of randomness of place can be eliminated. A basic tool in our investigation is the “prediction process” Zt of , but it is no more than that. Theoretically, one could attempt to define respresentations of Zt itself, analogous to those obtained below but valid for all P. In the present paper, however, each P is treated separately.
KeywordsBrownian Motion Poisson Process Wiener Process Gaussian Case Continuous Component
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