Functional limit theorems for additive and multiplicative schemes in the Cox--Ingersoll--Ross model

In this paper, we consider the Cox--Ingersoll--Ross (CIR) process in the regime where the process does not hit zero. We construct additive and multiplicative discrete approximation schemes for the price of asset that is modeled by the CIR process and geometric CIR process. In order to construct these schemes, we take the Euler approximations of the CIR process itself but replace the increments of the Wiener process with iid bounded vanishing symmetric random variables. We introduce a "truncated" CIR process and apply it to prove the weak convergence of asset prices. We establish the fact that this "truncated" process does not hit zero under the same condition considered for the original nontruncated process.