When approximating the expectations of a functional of a solution to a stochastic differential equation, the numerical performance of deterministic quadrature methods, such as sparse grid quadrature and quasi-Monte Carlo (QMC) methods, may critically depend on the regularity of the integrand. To overcome this issue and improve the regularity structure of the problem, we consider cases in which analytic smoothing (bias-free mollification) cannot be performed and introduce a novel numerical smoothing approach by combining a root-finding method with a one-dimensional numerical integration with respect to a single well-chosen variable. We prove that, under appropriate conditions, the resulting function of the remaining variables is highly smooth, potentially affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (ie., the Brownian bridge and Richardson extrapolation on the weak error). This approach facilitates the effective treatment of high dimensionality. Our study is motivated by option pricing problems, focusing on dynamics where the discretization of the asset price is necessary. Based on our analysis and numerical experiments, we demonstrate the advantages of combining numerical smoothing with the ASGQ and QMC methods over these methods without smoothing and the Monte Carlo approach. Finally, our approach is generic and can be applied to solve a broad class of problems, particularly approximating distribution functions, computing financial Greeks, and estimating risk quantities.
翻译:当人们对一种解决悬殊差异方程式的功能期望接近于一种功能的功能时,确定性二次方程方法,例如稀疏的网格象形和半蒙特卡罗(QMC)方法的数值性能,可能关键取决于零星方法的规律性。要克服这一问题,改善问题的正常性结构,我们考虑一些情况,在这些情况下,分析性平滑(无偏差的软体化)无法实现,并且通过将根调查方法与单维数集成的单维数集成结合起来,引入新的数字平滑方法。我们证明,在适当条件下,其余变量的功能非常平稳,有可能使适应性分散式网形象(ASGQ)和QMC方法的效率得到提高,特别是当与等级转变(即布朗桥桥和理查德对差错的猜疑)相结合时,我们的研究可以促进有效解决高维度问题。我们研究的动机是选择性定价问题,重点是资产价格的离散化是需要的动态。我们所剩变量的功能的功能是高度平稳的,因此有可能提供适应适应性分散性网格的网格和通用方法。最后将我们的数字分析和数字方法与货币方法结合起来。