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 [2] and [8]). Since only a few optimal stopping problems can be solved analytically (see [13]), one has to resort to numerical approximations of the solution. In such cases, 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 [4]. The price one pays for this is the unboundedness of the reward function, as a function of the control parameter. Recently, a major breakthrough has been made in estimating the rate of convergence of finite difference approximations for the pay-off functions of stochastic control problems (in [9], followed by [10] and [11]). Applying Krylov’s methods, new rate of convergence estimates can be found in [1, 5, 6, 7]. New estimates applicable to numerical approximations of normalized Bellman equations appear in [12]. 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 pay-off function of optimal stopping problem of controlled diffusions equals the pay-off function of the control problem obtained by randomized stopping. This result is known from [8] in the case where the coefficients of the controlled diffusions are bounded in the control parameter (see Section 4 of Chapter 3 in [8]). In Theorem 3.2, the coefficients of the diffusions and the functions defining the pay-off may be unbounded functions of the control parameter. Also,