An adaptive splitting method for the Cox-Ingersoll-Ross process

We propose a new splitting method for strong numerical solution of the Cox-Ingersoll-Ross model. For this method, applied over both deterministic and adaptive random meshes, we prove a uniform moment bound and strong error results of order $1/4$ in $L_1$ and $L_2$ for the parameter regime $\kappa\theta>\sigma^2$. Our scheme does not fall into the class analyzed in Hefter & Herzwurm (2018) where convergence of maximum order $1/4$ of a novel class of Milstein-based methods over the full range of parameter values is shown. Hence we present a separate convergence analysis before we extend the new method to cover all parameter values by introducing a 'soft zero' region (where the deterministic flow determines the approximation) giving a hybrid type method to deal with the reflecting boundary. From numerical simulations we observe a rate of order $1$ when $\kappa\theta>\sigma^2$ rather than $1/4$. Asymptotically, for large noise, we observe that the rates of convergence decrease similarly to those of other schemes but that the proposed method displays smaller error constants. Our results also serve as supporting numerical evidence that the conjecture of Hefter & Jentzen (2019) holds true for methods with non-uniform Wiener increments.