Stability estimating in optimal stopping problem

We consider the optimal stopping problem for a discrete-time Markov process on a Borel state space X. It is supposed that an unknown transition probability p(.vertical bar x), x is an element of X, is approximated by the transition probability (p) over tilde (.vertical bar x), x is an element of X, and the stopping rule (tau) over tilde*, optimal for (p) over tilde, is applied to the process governed by p. We found an upper bound for the difference between the total expected cost, resulting when applying (tau) over tilde*, and the minimal total expected cost. The bound given is a constant times sup(x is an element of X) parallel to p(.vertical bar x) - (p) over tilde(.vertical bar x)parallel to where parallel to . parallel to is the total variation norm.