Weak convergence rate for the Cox-Ingersoll-Ross process

We study the weak convergence rate of a drift-implicit discretisation scheme for the Cox-Ingersoll-Ross (CIR) process in the regime where the process remains strictly positive. Specifically, we consider scheme~(4) of Alfonsi~\cite{A}, which arises naturally from applying a drift-implicit Euler step to the SDE satisfied by the square root of the CIR process and admits a unique positive closed-form solution at each time step. Using a PDE approach combined with a continuous-time SDE representation of the discretised process, we prove that the weak convergence rate is $\mathcal{O}(1/N)$ under the Feller condition $2\alpha\geq\theta^2$ and mild polynomial growth conditions on the payoff function. The proof requires only elementary techniques and, in particular, avoids the semi-exact simulation machinery used in earlier work. The methodology is expected to extend to a broader class of diffusion processes.