Scope of the Signal Plus “White Noise” Model (II)

When one does not know that the SPWN model has a representation in the form of a stochastic differential equation, which is generally the case within the Cramér-Hida framework, it becomes important to know that such a representation exists, as it is that representation which allows an explicit form for the likelihood. In the previous chapter, it was seen that the existence of the likelihood has the consequence that the observations are represented in the form of a stochastic differential equation. The result is however an existence result which gives no hint as to the form of the resulting signal. When one is willing, or able, to assume some integrability conditions on the signal (that is typically the case when adjusting a model to data: one makes do with what is available, as long as the procedure is reasonable, and can be seriously evaluated), one then obtains a stochastic differential equation form in which the signal is a conditional expectation with respect to the observations. That latter stochastic differential equation representation is known under the appellation of “innovation representation,” Innovation representation and makes up the next topic. In the framework of the Cramér-Hida representation, assumptions on the integrability of the signal in the derived “white noise model” may be difficult, nay, impossible to justify. It is nevertheless useful to know that the stochastic differential equation representation is related to conditional expectations with respect to the observations process.