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

最优停止理论研究的是选择采取特定行动的时间，以使期望报酬最大化或期望成本最小化的问题。反射倒向随机微分方程可以看作是最优停时问题的推广，当回报泛函也依赖于解时。这样的问题可以在统计学、经济学和数理金融学的许多领域中找到（例如美式期权的定价问题）。原始和对偶的方法已经在文献中产生的Monte Carlo算法高维停止问题。通常，这些算法会导致一些问题的功能凸优化，其中的原始目标泛函估计的Monte Carlo。尽管具有凸性，但这些优化算法的性能将随着底层状态空间维数的增加而急剧恶化，除非存在最优值函数的良好低维近似。该项目的目的是开发几种基于相应经验目标泛函惩罚的新型方法，这些方法能够恢复状态空间中最重要的分量，或者识别给定函数类中值函数的稀疏表示。

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.