In this paper, we present a finite difference scheme for a linear complementarity problem with a mixed boundary condition arising from pricing a Russian option with a finite time horizon. An implicit Euler method for the temporal discretization and second-order difference schemes on a piecewise uniform mesh for the spatial discretization are used to solve the linear complementarity problem with a mixed boundary condition. It is shown that the transformed discrete operator satisfies a maximum principle, which is used to derive the error estimate. It is proved that the scheme is first- and second-order convergent with respect to the temporal and spatial variables, respectively. Numerical experiments verify the validity of the theoretical results.

在本文中，我们针对具有混合边界条件的线性互补问题提出了一种有限差分格式，该问题由有限时间范围内的俄罗斯期权定价产生。用于时间离散化的隐式欧拉方法和用于空间离散化的分段均匀网格上的二阶差分格式用于解决具有混合边界条件的线性互补问题。结果表明，变换后的离散算子满足最大原理，用于推导误差估计。证明了该方案对于时间和空间变量分别是一阶和二阶收敛的。数值实验验证了理论结果的有效性。