一种求解金融中Hamilton-Jacobi-Bellman (HJB) 方程的惩罚方法
A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance
AI总结：
提出一种简单易实现的惩罚方法，用于求解一类广泛的HJB方程，通过将非线性离散系统近似为罚参数的一阶近似，并利用迭代方案在有限步内求解，在金融数学中给出数值例子。
摘要：
我们提出了一种简单易实现的方法，用于求解一类相当广泛的Hamilton-Jacobi-Bellman (HJB) 方程的数值解。在许多情况下，所考虑的问题只有粘性解，幸运的是，许多直观的（例如基于有限差分的）离散化方法可以证明收敛到该解。然而，特别是在使用具有理想稳定性性质的全隐式时间步进格式时，仍然面临着求解所得非线性离散系统的艰巨任务。在本文中，我们介绍了一种惩罚方法，该方法将非线性离散系统近似为罚参数的一阶近似，并且我们证明可以使用迭代方案在有限步内求解惩罚后的离散问题。我们包含了来自数学金融的几个例子，其中所描述的方法产生了严格的数值格式，并给出了数值结果。
英文摘要：
We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to first order in the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.
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