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We consider stochastic control models with Borel spaces and universallymeasurable policies. For such models the standard policy iteration is known tohave difficult measurability issues and cannot be carried out in general. Wepresent a mixed value and policy iteration method that circumvents thisdifficulty. The method allows the use of stationary policies in computing theoptimal cost function, in a manner that resembles policy iteration. It can alsobe used to address similar difficulties of policy iteration in the context ofupper and lower semicontinuous models. We analyze the convergence of the methodin infinite horizon total cost problems, for the discounted case where theone-stage costs are bounded, and for the undiscounted case where the one-stagecosts are nonpositive or nonnegative.  For undiscounted total cost problems with nonnegative one-stage costs, wealso give a new convergence theorem for value iteration, which shows that valueiteration converges whenever it is initialized with a function that is abovethe optimal cost function and yet bounded by a multiple of the optimal costfunction. This condition resembles Whittle's bridging condition and is partlymotivated by it. The theorem is also partly motivated by a result of Maitra andSudderth, which showed that value iteration, when initialized with the constantfunction zero, could require a transfinite number of iterations to converge. Weuse the new convergence theorem for value iteration to establish theconvergence of our mixed value and policy iteration method for the nonnegativecost case.

A Mixed Value and Policy Iteration Method for Stochastic Control with Universally Measurable Policies