标题： Heston模型及相关过程的逼近和正则性结果 显示英文标题

标题： Approximation and regularity results for the Heston model and related processes

摘要： 本博士论文通过三篇相互关联的研究，探讨了 Heston 随机波动率模型的近似值和规律性。 第一篇研究重点开发了 Cox-Ingersoll-Ross (CIR) 过程的高阶弱近似，这对于金融建模至关重要，但由于平方根扩散项阻碍了标准方法的应用，因此颇具挑战性。 通过采用基于 Alfonsi (2010) 二阶格式的随机网格技术 (Alfonsi & Bally, 2021)，该研究证明了对于光滑测试函数，可以实现任意阶的弱近似。这在比著名的 Feller 公式限制更少的条件下成立。数值结果证实了 CIR 和 Heston 模型的收敛性，并显示出显著的计算时间改进。 第二篇研究将随机网格技术扩展到对数 Heston 过程。 介绍了两种二阶格式（一种采用精确波动率模拟，另一种采用与上文相同限制条件下的二宫-维克托尔分裂）。严格证明了格式可以收敛到任意期望阶。数值实验验证了格式对欧式和亚式期权定价的有效性，并表明其对多因子/粗糙赫斯顿模型具有潜在的适用性。 第三项工作研究了与对数-赫斯顿模型相关的偏微分方程 (PDE)。它扩展了经典解的结果，并在不依赖 Feller 条件的情况下证明了粘性解的存在性和唯一性。即使对于某些不连续的初始数据，也证明了其唯一性，这与数字期权等定价工具相关。 此外，在初始数据的松弛正则性（连续性）下，证明了混合数值格式与粘性解的收敛性。 附录包含 CIR 过程的补充结果。 显示英文摘要

摘要： This Ph.D. thesis explores approximations and regularity for the Heston stochastic volatility model through three interconnected works. The first work focuses on developing high-order weak approximations for the Cox-Ingersoll-Ross (CIR) process, essential for financial modelling but challenging due to the square root diffusion term preventing standard methods. By employing the random grid technique (Alfonsi & Bally, 2021) built upon Alfonsi's (2010) second-order scheme, the work proves that weak approximations of any order can be achieved for smooth test functions. This holds under a condition that is less restrictive than the famous Feller's one. Numerical results confirm convergence for both CIR and Heston models and show significant computational time improvements. The second work extends the random grid technique to the log-Heston process. Two second-order schemes are introduced (one using exact volatility simulation, another using Ninomiya-Victoir splitting under a the same restriction used above). Convergence to any desired order is rigorously proven. Numerical experiments validate the schemes' effectiveness for pricing European and Asian options and suggest potential applicability to multifactor/rough Heston models. The third work investigates the partial differential equation (PDE) associated with the log-Heston model. It extends classical solution results and establishes the existence and uniqueness of viscosity solutions without relying on the Feller condition. Uniqueness is proven even for certain discontinuous initial data, relevant for pricing instruments like digital options. Furthermore, the convergence of a hybrid numerical scheme to the viscosity solution is shown under relaxed regularity (continuity) for the initial data. An appendix includes supplementary results for the CIR process.