We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in mathematical finance, particle systems, and elastic theory. By leveraging insights from the successful monotone discretization of the fractional Laplacian, we establish uniform boundedness, solution existence, and uniqueness for the numerical solutions of the fractional obstacle problem. We employ a policy iteration approach for efficient solution of discrete nonlinear problems and prove its finite convergence. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the method's efficacy.

翻译：我们提出了一种新颖的单调离散化方法，用于处理涉及整数阶分数拉普拉斯算子的障碍问题，该问题在具有齐次狄利克雷边界条件的有界利普希茨域上定义。此类问题广泛应用于数学金融、粒子系统和弹性理论中。通过借鉴分数拉普拉斯算子成功单调离散化的见解，我们建立了分数障碍问题数值解的一致有界性、解的存在性和唯一性。我们采用策略迭代方法高效求解离散非线性问题，并证明了其有限收敛性。改进后的策略迭代方法根据解的正则性进行调整，通过在不同区域修改离散化方案展现出优越性能。数值例子验证了该方法的有效性。