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Reflected forward-backward SDEs and obstacle problems with boundary conditions

机译：反映前向后SDE和边界条件的障碍物问题

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摘要

In this paper we study a class of forward-backward stochastic differential equations with reflecting boundary conditions (FBSDER for short). More precisely, we consider the case in which the forward component of the FBSDER is restricted to a fixed, convex region, and the backward component will stay, at each fixed time, in a convex region that may depend on time and is possibly random. The solvability of such FBSDER is studied in a fairly general way. We also prove that if the coefficients are all deterministic and the backward equation is one-dimensional, then the adapted solution of such FBSDER will give the viscosity solution of a quasilinear variational inequality (obstacle problem) with a Neumann boundary condition. As an application, we study how the solvability of FBSDERs is related to the solvability of anAmerican game option.
机译：在本文中，我们研究了一类具有边界条件的前向-后向随机微分方程（简称FBSDER）。更准确地说，我们考虑这样一种情况，其中FBSDER的前向分量被限制在固定的凸区域中，而后向分量将在每个固定的时间停留在可能取决于时间并且可能是随机的凸区域中。以相当普遍的方式研究了这种FBSDER的可溶性。我们还证明，如果系数都是确定性的，并且后向方程是一维的，则这种FBSDER的自适应解将给出具有Neumann边界条件的拟线性变分不等式（障碍问题）的粘度解。作为一项应用，我们研究了FBSDER的可解决性与美国游戏选项的可解决性之间的关系。

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