In this paper, we introduce a novel, explicit, wide-stencil, two-dimensional (2D) tree–grid method for solving stochastic control problems (SCPs) with two space dimensions and one time dimension, or, equivalently, the corresponding Hamilton– Jacobi–Bellman equation. This new method can be seen as a generalization of the tree–grid method for SCPs with one space dimension that was recently developed by the authors. The method is unconditionally stable and no 2D interpolation is needed in the stencil construction. We prove the convergence of the method and exemplify it in our application to a two-factor uncertain volatility model.

中文翻译：

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二维树-网格法

在本文中，我们介绍了一种新颖的、显式的、宽模板的、二维 (2D) 树网格方法，用于解决具有两个空间维度和一个时间维度的随机控制问题 (SCP)，或者等效地，相应的 Hamilton–雅可比-贝尔曼方程。这种新方法可以看作是作者最近开发的具有一个空间维度的 SCP 的树-网格方法的推广。该方法是无条件稳定的，并且在模板构造中不需要 2D 插值。我们证明了该方法的收敛性，并在我们对双因素不确定波动率模型的应用中举例说明。