ar X iv : 0 70 5 . 23 02 v 1 [ m at h . PR ] 1 6 M ay 2 00 7 ON RANDOMIZED STOPPING

It is known that optimal stopping problems for controlled diffusion processes can be transformed into optimal control problems by using the method of randomized stopping (see [10]). Since only a few optimal stopping problems can be solved analytically (see [16]), one has to resort to numerical approximations of the solution. In such case one would like to know the rate of convergence of these approximations. Embedding optimal stopping problems into the class of stochastic control problems allows one to apply numerical methods developed for stochastic control [5]. The price one pays for it is the unboundedness of the reward function, as a function of the control parameter. Recently a major break-through has been made in estimating the rate of convergence of finite difference approximations for the payoff functions of stochastic control problems in [11], followed by [12] and [13]. Applying Krylov’s methods, new rate of convergence estimates can be found in [6], [7], [2], [9], [1] and [8]. New estimates applicable to numerical approximations of normalized Bellman equations appear in [14]. Our main result, Theorem 2.1, formulates the method of randomized stopping in a general setting. Applying it to optimal stopping problems of controlled diffusion processes we easily get, see Theorem 3.2, that under general conditions the payoff function of optimal stopping problem of controlled diffusions equals the payoff function of the control problem obtained by randomized stopping. This result is known from [10] in the case when the coefficients of the controlled diffusions are bounded in the control parameter (see Section 4 of Chapter 3 in [10]). In Theorem 3.2 the coefficients of the diffusions and the functions, defining the payoff may be unbounded functions of the control