On non-Gaussian innovations processes for observations with non-Gaussian noise

It is well-known that for observations with additive Gaussian noise, the innovations process is a Brownian motion process which, under certain conditions, has the same information as the observation. In this paper, it is shown that for observations with non-Gaussian noise, a Brownian motion process cannot be informationally equivalent to the observation in general, and that the role of the innovations process is undertaken by a non-Gaussian martingale. Also, a sufficient condition is shown for this martingale to be equivalent to the observation. The approach used in this paper is based on the fact that a semi-martingale with a non-Gaussian martingale is decomposed into the one with a Brownian motion process and the quadratic covariation process.