Continuous-time Martingale Optimal Transport and Optimal Skorokhod Embedding.

Summary This thesis presents three main research topics, the first two being independent and the last one indicating the relation of the first two problems in a concrete case.In the first part we focus on the martingale optimal transport problem in Skorokhod space, whose first goal is to study systematically the tension of martingale transport schemes. We first focus on the upper semicontinuity of the primal problem with respect to the marginal distributions. Using the S-topology introduced by Jakubowski, we derive the upper semicontinuity and show the first duality. We also give two dual problems concerning the robust overcoverage of an exotic option, and we establish the corresponding dualities, by adapting the principle of dynamic programming and the discretization argument initiated by Dolinsky and Soner.The second part of this thesis deals with the optimal Skorokhod folding problem. We first formulate this optimization problem in terms of probability measures on an extended space and its dual problems. Using the classical duality. convex approach and the optimal stopping theory, we obtain the duality results. We also relate these results to martingale optimal transport in the space of continuous functions, from which the corresponding dualities are derived for a particular class of payment functions. Next, we provide an alternative proof of the monotonicity principle established by Beiglbock, Cox and Huesmann, which allows us to characterize optimizers by their geometric support. We show at the end a stability result which contains two parts: the stability of the optimization problem with respect to the target marginals and the connection with another problem of the optimal folding.The last part concerns the application of stochastic control to the martingale optimal transport with the local time dependent payoff function, and to the Skorokhod folding. For the case of one marginal, we find the optimizers for the primal and dual problems via the Vallois solutions, and consequently show the optimality of the Vallois solutions, which includes the optimal martingale transport and the optimal Skorokhod folding. For the case of two marginals, we obtain a generalization of the Vallois solution. Finally, a special case of several marginals is studied, where the stopping times given by Vallois are well ordered.