High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids

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.