发表机构
Utrecht University; RWTH Aachen University; King Abdullah University of Science and Technology (KAUST)（乌得勒支大学; 亚琛工业大学; 阿卜杜拉国王科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对粗糙Heston模型傅里叶定价的误差控制需求，提出单级与多级高斯-拉盖尔求积法，平衡两类误差，多级法计算成本更低且优于BL2近似。
AI 中文摘要
与经典Heston模型不同，粗糙Heston模型下的傅里叶定价需要在每个求积点求解一个分数阶Riccati方程。由于所需的分辨率随模型参数和求积点变化，单一的均匀时间离散化可能效率低下。我们开发了单级和多级高斯-拉盖尔求积法，以平衡时间离散化误差和傅里叶求积误差。两种方法均对拉盖尔权重进行缩放，以匹配傅里叶被积函数的衰减特性。单级方法将预设的容差分配给两种误差；多级方法将被积函数拆分为零级项和级差项，并在每个级别分别选择求积点。假设傅里叶被积函数离散化误差为$O(\triangle t^p)$，单次计算特征函数的成本为$O(\triangle t^{-\beta})$，代数高斯-拉盖尔求积误差为$O(N^{-s_{SL}/2})$（其中$s_{SL}$为光滑性指数）。在该估计和级差的正则性与衰减假设下，我们证明所提单级方法达到精度$\boldsymbol{\tau}$所需的计算工作量为$O(\boldsymbol{\tau}^{-(\beta/p+2/s_{SL})})$，而所提多级方法所需计算工作量为$O(\boldsymbol{\tau}^{-\beta/p})$。我们还研究了用于实际多级求积分配的根指数高斯-拉盖尔误差模型。数值实验验证了观测到的分数阶Riccati和傅里叶被积函数收敛速率以及根指数求积行为，并表明所提缩放方法可大幅降低求积成本；多级方法相比单级方法具有显著的计算优势。我们进一步将多级分数阶Riccati方法与BL2马尔可夫近似进行基准测试，结果显示在测试配置中总CPU时间更低。
英文摘要
Unlike the classical Heston model, Fourier pricing under the rough Heston model requires solving a fractional Riccati equation at every quadrature point. Since the required resolution varies with model parameters and quadrature point, a single uniform time discretization can be inefficient. We develop single- and multilevel Gauss-Laguerre quadrature methods that balance the time discretization and Fourier quadrature errors. Both methods scale the laguerre weight to the estimated Fourier integrand decay. The single-level method allocates a prescribed tolerance between the two errors. The multilevel method splits the integrand into a level-zero term and level differences, selecting quadrature points separately at each level. Suppose that the Fourier integrand discretization error is $O(Δt^p)$, that evaluating the characteristic function once costs $O(Δt^{-β})$, and that the algebraic Gauss-Laguerre quadrature error is $O(N^{-s_{SL}/2})$, where $s_{SL}$ is the smoothness index. Under this estimate and assumptions on the regularity and decay of level differences, we prove that the proposed single-level method requires $O(ε^{-(β/p+2/s_{SL})})$ computational work to achieve accuracy $ε$, whereas the proposed multilevel method requires $O(ε^{-β/p})$ computational work. We also study root-exponential Gauss-Laguerre error models for practical multilevel quadrature allocation. Numerical experiments support the observed fractional Riccati and Fourier integrand convergence rates and root-exponential quadrature behavior, and show substantial reductions in quadrature cost from the proposed scaling. The multilevel method provides clear computational savings over the single-level method. We further benchmark the multilevel fractional Riccati method against the BL2 Markovian approximation and report lower total CPU time in the tested configurations.
