Gaussian Solutions of Stochastic Equations

Abstract

Gaussian processes play an important role in the theory of linear filtering to be discussed in the next chapter. In the general stochastic filtering model it has been seen that the observation process and the innovation (Wiener) process are connected by an equation of the kind studied in Chapter 8. When the observation process is Gaussian, we have an example of the equation which will now be considered. The theory of stochastic equations whose solutions are Gaussian processes is an instructive special case of the general theory of functional stochastic differential equations because it is subsumed in the theory of nonanticipative representations of equivalent Gaussian measures and is identical with the latter if one of the measures is Wiener measure. We shall therefore present it in more detail than is strictly necessary for the purpose of solving linear filtering problems.