Reconstructing stochastic processes from partial observations: stochastic filtering theory and stability

Stochastic filtering theory is concerned with the reconstruction of an instance of a stochastic process, called signal process, knowing some partial observations on it, given by the so-called observation process. Following the work done by Ramon van Handel in his PhD thesis, we first introduce the subject and develop some basic facts on the theory following the reference probability method. After that, we focus on the stability issue, that is we investigate how changes in the initial measure of the signal process affect the reconstructing procedure, namely the conditional law of the signal. More precisely, first we show that exponential stability holds for the case of continuous time Markov signals with finite state space under the mixing condition. Lastly, we show that exponential stability also holds, under suitable assumptions, in the case of diffusion signals with "linear + gradient" drift type and linear observation function. In doing this, we use tools from stochastic optimal control theory and coupling techniques.