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

最优停止理论是关于选择一个时间采取特殊行动的问题，目的是最大化预期收益或最小化预期成本。当报酬泛函也可能依赖于解时，反射倒向随机微分方程可以被看作是最优停止问题的推广。这样的问题可以在统计学、经济学和数学金融的许多领域找到(例如，美式期权的定价问题)。在文献中已经发展了原始方法和对偶方法，这导致了高维停止问题的蒙特卡罗算法。通常，这些算法会导致一些泛函凸优化问题，其中原始目标泛函需要用蒙特卡罗方法来估计。尽管存在凸性，但这些优化算法的性能将随着底层状态空间的维度的增加而急剧恶化，除非存在对最优值函数的良好的低维近似。这个项目的目的是开发几种新的方法，该方法基于相应的经验目标泛函的惩罚，能够恢复状态空间的最重要的分量，或者能够识别给定函数类中的值函数的稀疏表示。

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.