Solving optimal stopping problems and reflected backward stochastic differential equations by convex optimization and penalization

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The theory of optimal stopping is concerned with the problem of choosing a time to take aparticular action, in order to maximize an expected reward or minimize an expected cost. Re-flected backward stochastic differential equations can be considered as generalizations of optimalstopping problems when the reward functional may also depend on the solution. Such problemscan be found in many areas of statistics, economics, and mathematical finance (e.g. the pricingproblem of American options). Primal and dual approaches have been developed in the literature which give rise to Monte Carlo algorithms for high-dimensional stopping problems. Typically, these algorithms lead to some problems of functional convex optimization, where the original objective functionals are to be estimated by Monte Carlo. Despite of the convexity, the performance of these optimization algorithms will deteriorate sharply as the dimension of the underlying state space increases, unless there exists a good low-dimensional approximation for the optimal value function. The aim of this project is to develop several novel approaches based on the penalization of the corresponding empirical objective functionals which are able either to recover the most important components of the state space or to identify a sparse representation for the value function in a given class of functions.