尤拉方程的两个自由边界问题的相容性

The Compatibility of Two Free Boundary Problem of Euler Equation

摘要: 本文将永久美式期权的自由边界问题归结为在半无界区域具有多个（或单个）奇异点的边值问题来研究，引入广义特征函数法获得了多个奇异点的数学模型的精确解。只有一个奇异点的情形，所得到的解函数在奇异点处取最大值；并得到了左、右自由边界问题同时有一致解的相容性条件。证明了在相容性条件下，左、右自由边界点与奇异点三点合一，从而左、右自由边界点与奇异点都是永久美式期权最佳实施边界点。具有多个奇异点的情形，获得了判断左、右自由边界点成为最佳或较佳实施边界点的条件。

Abstract: In this paper, we model the free boundary problem of the Perpetual American Option as boundary value problem with multiple (or single) singular points in the semi infinite domain, and introduce the generalized characteristic function method to be able to obtain the exact solution of the mathematical model of multiple singular point. In the single singular point case, our solution function takes the maximum value at the singular point. We deduce the consistency condition of the left and right free boundary problem. Under the compatibility condition, the three points, the left and right free boundary points and singular point are the same, so that they all are the optimal implementation point of the Perpetual American Option. In the case of multiple singular points, the conditional judgment of the left and right free boundary points to be the optimal or nearly optimal implementation point is obtained.