SIAM Journal on Control and Optimization, Volume 58, Issue 6, Page 3709-3733, January 2020.
We investigate the linear programming framework for an exit-time stochastic control problem and apply the moment-sum-of-squares hierarchy to obtain tight pointwise bounds and global bounding functions for the value function. The primal linear program over suitable measures and the dual linear program over test functions are implemented numerically by semidefinite programs which target at moments and sum-of-squares polynomial representations, respectively. Numerically optimized bounds converge to the value function from below as polynomial degree increases to infinity under suitable technical conditions. We focus on the dual problem, which is particularly effective, as its single implementation yields a polynomial bounding function over the entire problem domain, and since it allows a flexible choice of objective function, one may improve the global bound on regions of interest.

中文翻译：

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可计算的原始和双重边界用于随机控制

SIAM控制与优化杂志，第58卷，第6期，第3709-3733页，2020年1月。
我们研究了退出时间随机控制问题的线性规划框架，并应用了平方和矩矩层次结构来获得紧密的逐点边界和值函数的全局边界函数。分别通过以时刻和平方和多项式表示为目标的半定程序在适当的量度上进行原始线性规划，并在测试函数上对偶线性规划进行数值计算。在适当的技术条件下，随着多项式次数增加到无穷大，数值优化的边界从下面收敛到值函数。我们将重点放在对偶问题上，该对偶问题特别有效，因为其单个实现会在整个问题域上产生多项式有界函数，并且由于它允许灵活选择目标函数，