# Stopping Problems for General Partially Observed Models

Lukasz Stettner^{*}

In many situations for partially observed models the observation is not corrupted by an additive noise which is required to use standard reference probability method approach. When the observation process takes at most countable number of values we can get a nice approximation of filtering processes, which in particular in the Markov case enables us to prove Markov property and other regularity properties of such general filtering processes. Our purpose is to study optimal stopping and impulse control problems, consisting of a sequence stopping times and shifts of the state process chosen at this stopping times, for processes with general discrete observation or with an approximation of the observation using discrete observation (i.e. observation taking at most countable number of values). There are many examples which come from practical implemetation. One of then is a control of solutions to stochastic differential equations driven by fractional Brownian motion, which form Markov processes, when we fully observe noise variable (which is not the case). Besides of the studies on properties of filtering processes we going to show the form of optimal or nearly optimal stopping and impulse strategies for a wide family of partially observed problems.

Mathematics Subject Classification: 93E20 93E11 60G40

Keywords: optimal stopping; impulse control; partial observation; Bellman equations;

Minisymposion: Stochastic Models, Control and Applications