标题： 带股票借贷费用的期权定价中的HJB有限元方法 显示英文标题

标题： Finite Element Method for HJB in Option Pricing with Stock Borrowing Fees

摘要： 在数学金融学中，许多来自带有摩擦市场的衍生品可以表述为HJB框架下的最优控制问题。 解析最优控制可能会导致高度非线性的偏微分方程（PDE），这可能产生不稳定的数值结果。 准确且收敛的数值方案对于充分利用套期保值过程的优势至关重要。 在这项研究中，我们采用有限元方法结合非均匀网格，用于带股票借贷费用的期权定价任务，从而得到一个绕过解析最优控制而直接离散化PDE的HJB方程。 时间积分使用theta格式，并按照Rannacher流程进行初始修正。 在每个时间步长处，应用Newton型算法处理类似惩罚项。 进行了数值实验，结果与基准问题一致，并显示出强匹配性。 达到预期结果所需的CPU时间表明，P2-FEM优于FDM和线性P1-FEM，且P2-FEM表现出更优的收敛性。 本文为HJB问题提供了一种高效的替代框架，并通过引入基于有限元方法（FEM）的HJB应用解决方案，为数学金融领域的文献做出了贡献。 显示英文摘要

摘要： In mathematical finance, many derivatives from markets with frictions can be formulated as optimal control problems in the HJB framework. Analytical optimal control can result in highly nonlinear PDEs, which might yield unstable numerical results. Accurate and convergent numerical schemes are essential to leverage the benefits of the hedging process. In this study, we apply a finite element approach with a non-uniform mesh for the task of option pricing with stock borrowing fees, leading to an HJB equation that bypasses analytical optimal control in favor of direct PDE discretization. The time integration employs the theta-scheme, with initial modifications following Rannacher`s procedure. A Newton-type algorithm is applied to address the penalty-like term at each time step. Numerical experiments are conducted, demonstrating consistency with a benchmark problem and showing a strong match. The CPU time needed to reach the desired results favors P2-FEM over FDM and linear P1-FEM, with P2-FEM displaying superior convergence. This paper presents an efficient alternative framework for the HJB problem and contributes to the literature by introducing a finite element method (FEM)-based solution for HJB applications in mathematical finance.