基于加权随机网格算法的最优停时问题的半易处理性
Semi-tractability of optimal stopping problems via a weighted stochastic mesh algorithm
摘要：
在这篇文章中，我们提出了一个加权随机网格（WSM）算法的离散和连续时间的最优停时问题的近似值。我们证明，在离散的情况下，WSM算法导致半易处理的相应的最优问题的意义上，它的复杂性是有界的顺序由$\varepsilon^{-4}\log^{d+2}(1/\varepsilon)$与$d$是底层马尔可夫链的维度。此外，我们研究了连续时间最优停时问题的WSM方法，并推导出相应的复杂性界。虽然我们不能证明在这种情况下，半易处理，我们的界限原来是已知的现有算法在文献中的界限中最紧的。我们通过一个数值例子来说明我们的理论研究结果。
英文摘要：
In this article we propose a Weighted Stochastic Mesh (WSM) Algorithm for approximating the value of a discrete and continuous time optimal stopping problem. We prove that in the discrete case the WSM algorithm leads to semi-tractability of the corresponding optimal problems in the sense that its complexity is bounded in order by $\varepsilon^{-4}\log^{d+2}(1/\varepsilon)$ with $d$ being the dimension of the underlying Markov chain. Furthermore we study the WSM approach in the context of continuous time optimal stopping problems and derive the corresponding complexity bounds. Although we can not prove semi-tractability in this case, our bounds turn out to be the tightest ones among the bounds known for the existing algorithms in the literature. We illustrate our theoretical findings by a numerical example.
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