Stochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton-Jacobi-Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ($ n\geq 3$). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme. We show that matrices resulting from spatial discretization and temporal discretization are M--matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided.

中文翻译：

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一种新的随机最优控制问题的高维拟合方案

随机最优原理导致偏微分方程 (PDE) 的求解，即 Hamilton-Jacobi-Bellman (HJB) 方程。通常，该方程无法解析求解，因此数值算法是提供准确近似值的唯一工具。本文的目的是介绍一种新的拟合有限体积方法，以解决高维（$ n\geq 3 $）随机最优控制问题的高维退化 HJB 方程。这里的挑战是由于我们的 HJB 方程的性质，它是一个退化的二阶偏微分方程加上优化问题。对于此类问题，有限差分法等标准方案失去了其单调性，因此可能无法保证向粘度解收敛。我们使用拟合有限体积方法对 HJB 方程进行离散化，众所周知，该方法可以解决退化的偏微分方程，而时间离散化是使用隐式欧拉方案执行的。我们表明由空间离散化和时间离散化产生的矩阵是 M-矩阵。提供了金融中的数值结果，证明了所提出的数值方法与标准有限差分方法相比的准确性。