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We consider a class of stochastic optimal control problems with partial observation, and study their approximation by discrete-time control problems. We establish a convergence result by using the weak convergence technique of Kushner and Dupuis [Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York], together with the notion of relaxed control rule introduced by El Karoui, Huú Nguyen and Jeanblanc-Picqué [SIAM J. Control Optim., 26 (1988), pp. 1025--1061]. In particular, with a well chosen discrete-time control system, we obtain a first implementable numerical algorithm (with convergence) for the partially observed control problem. Moreover, our discrete-time approximation result would open the door to study convergence of more general numerical approximation methods, such as machine learning based methods. Finally, we illustrate our convergence result by numerical experiments on a partially observed control problem in a linear quadratic setting. [ABSTRACT FROM AUTHOR]

In the context of mean field games, with possible control of the diffusion coefficient, we consider a path-dependent version of the planning problem introduced by P. L. Lions: given a pair of marginal distributions (μ0, μ1), find a specification of the game problem starting from the initial distribution μ0 and inducing the target distribution μ1 at the mean field game equilibrium. Our main result reduces the path-dependent planning problem to an embedding problem, that is, constructing a McKean-Vlasov dynamics with given marginals (μ0, μ1). Some sufficient conditions on (μ0, μ1) are provided to guarantee the existence of solutions. We also characterize, up to integrability, the minimum entropy solution of the planning problem. In particular, as uniqueness does not hold anymore in our path-dependent setting, one can naturally introduce an optimal planning problem which would be reduced to an optimal transport problem along controlled McKean-Vlasov dynamics. [ABSTRACT FROM AUTHOR]

We consider a continuous time stochastic optimal control problem under both equality and inequality constraints on the expectation of some functionals of the controlled process. Under a qualification condition, we show that the problem is in duality with an optimization problem involving the Lagrange multiplier associated with the constraints. Then by convex analysis techniques, we provide a general existence result and some a priori estimation of the dual optimizers. We further provide a necessary and sufficient optimality condition for the initial constrained control problem. The same results are also obtained for a discrete time constrained control problem. Moreover, under additional regularity conditions, it is proved that the discrete time control problem converges to the continuous time problem, possibly with a convergence rate. This convergence result can be used to obtain numerical algorithms to approximate the continuous time control problem, which we illustrate by two simple numerical examples. [ABSTRACT FROM AUTHOR]

This paper examines the Root solution of the Skorokhod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on the finitely many marginals Root solution of Cox, Ob\loj, and Touzi [Probab. Theory Related Fields, 173 (2019), pp. 211--259]. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE. [ABSTRACT FROM AUTHOR]

We propose a new numerical scheme for Backward Stochastic Differential Equations (BSDEs) based on branching processes. We approximate an arbitrary (Lipschitz) driver by local polynomials and then use a Picard iteration scheme. Each step of the Picard iteration can be solved by using a representation in terms of branching diffusion systems, thus avoiding the need for a fine time discretization. In contrast to the previous literature on the numerical resolution of BSDEs based on branching processes, we prove the convergence of our numerical scheme without limitation on the time horizon. Numerical simulations are provided to illustrate the performance of the algorithm. [ABSTRACT FROM AUTHOR]

This is a continuation of our accompanying paper [SIAM J. Control Optim., 54 (2016), pp. 2174-2201] We provide an alternative proof of the monotonicity principle for the optimal Skorokhod embedding problem established in [Beiglböck, Cox, and Huesmann, Optimal Transport and Skorokhod Embedding, preprint, 2013]. Our proof is based on the adaptation of the Monge-Kantorovich duality in our context, a delicate application of the optional cross-section theorem, and a clever conditioning argument introduced in [Beiglböck, Cox, and Huesmann, Optimal Transport and Skorokhod Embedding, preprint, 2013]. [ABSTRACT FROM AUTHOR]

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the weak formulation of the optimal Skorokhod embedding problem in Beiglbock, Cox, and Huesmann [Optimal Transport and Skorokhod Embedding, Preprint, 2013] to the case of finitely many marginal constraints. Using the classic al convex duality approach together with the optimal stopping theory, we establish some duality results under more general conditions than Beiglböck, Cox, and Huesmann. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints. [ABSTRACT FROM AUTHOR]