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Optimal control problems are inherently hard to solve as the optimizationmust be performed simultaneously with updating the underlying system. Startingfrom an initial guess, Howard's policy improvement algorithm separates the stepof updating the trajectory of the dynamical system from the optimization anditerations of this should converge to the optimal control. In the discretespace-time setting this is often the case and even rates of convergence areknown. In the continuous space-time setting of controlled diffusion thealgorithm consists of solving a linear PDE followed by maximization problem.This has been shown to converge, in some situations, however no global rate ofis known. The first main contribution of this paper is to establish global rateof convergence for the policy improvement algorithm and a variant, called herethe gradient iteration algorithm. The second main contribution is the proof ofstability of the algorithms under perturbations to both the accuracy of thelinear PDE solution and the accuracy of the maximization step. The prooftechnique is new in this context as it uses the theory of backward stochasticdifferential equations.

Exponential Convergence and stability of Howards's Policy Improvement Algorithm for Controlled Diffusions