Markov Processes, Delay Stochastic Differential Equation

This paper deals with nonlinear filtering problems with delay, i.e. we consider a system (X,Y), which can be represented by means of a sys-tem (X, Y ̂), in the sense that Yt = Ŷa(t), where a(t) is a delayed time transformation. We start with X being a Markov process, and then study Markovian systems, not necessarily diffusive, with correlated noises. The interest is focused on existence of explicit representations of the corresponding filters as functionals depending on the observed trajectory. Different assumptions on the function a(t) are considered