We consider the optimal stopping problem for a stochastic process ϑ which satisfies the stochastic differential equation   when the reward criterion is   It is assined that the process ϑ cannot be observed itself, and that only some information about it can be obtained by observing a stochastic process, satisfying   where ε is a small positive number and W a BROWNian motion which is independent from W.

In this situation the optimal meau reward is   where   is the class of stopping rules with respect to the family of σ-fields generated by the process ξ It is proved that εeconverges to the optimal mean reward of the analogous stopping problem in the case of complete information   if s⁰<∞ holds   is the class of stopping rules with respect to the family of σ-fields generated by the process ϑ It is shown that this convergence is of an order not less than   In the case s⁰=∞ it is proved that also ,s^ε=∞ holds. Finally, an analogous result for the case of a reward criterion of a more general structure is given.