We present new high order approximations schemes for the Cox–Ingersoll–Ross (CIR) process that are obtained by using a recent technique developed by Alfonsi and Bally (2021, A generic construction for high order approximation schemes of semigroups using random grids. Numer. Math., 148, 743–793) for the approximation of semigroups. The idea consists in using a suitable combination of discretization schemes calculated on different random grids to increase the order of convergence. This technique coupled with the second order scheme proposed by Alfonsi (2010, High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comp., 79, 209–237) for the CIR leads to weak approximations of order $2k$, for all $k\in{{\mathbb{N}}}^{\ast }$. Despite the singularity of the square-root volatility coefficient, we show rigorously this order of convergence under some restrictions on the volatility parameters. We illustrate numerically the convergence of these approximations for the CIR process and for the Heston stochastic volatility model and show the computational time gain they give.

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使用随机网格的 Cox-Ingersoll-Ross 过程半群的高阶近似

我们提出了 Cox-Ingersoll-Ross (CIR) 过程的新高阶近似方案，该方案是通过使用 Alfonsi 和 Bally 开发的最新技术获得的（2021，使用随机网格的半群高阶近似方案的通用构造。Numer。 Math., 148, 743–793) 用于半群的近似。该想法在于使用在不同随机网格上计算的离散化方案的适当组合来增加收敛阶数。该技术与 Alfonsi 提出的二阶方案（2010，CIR 过程的高阶离散方案：仿射项结构和 Heston 模型的应用。Math. Comp., 79, 209–237）针对 CIR 提出了弱近似$2k$ 阶，对于所有 $k\in{{\mathbb{N}}}^{\ast }$。尽管平方根波动系数存在奇点，我们在波动性参数的一些限制下严格地展示了这种收敛顺序。我们以数值方式说明了 CIR 过程和 Heston 随机波动率模型的这些近似值的收敛性，并显示了它们给出的计算时间增益。