Truncated sequential guaranteed estimation for the Cox-Ingersoll-Ross models
Mohamed Ben Alaya, Thi-Bao Trâm Ngô, Serguei Pergamenchtchikov
TL;DR
This work develops truncated, guaranteed sequential estimators for drift parameters of the Cox–Ingersoll–Ross process observed on a fixed horizon with known diffusion. It first handles scalar cases, deriving non-asymptotic mean-square bounds and proving asymptotic minimax optimality that reduces the required observation time relative to fixed-duration MLEs. It then extends to the two-parameter setting, introducing a two-step, truncated sequential scheme with explicit non-asymptotic risk bounds and LAN-based optimality results, both locally and in minimax sense. Concentration inequalities underpin the non-asymptotic analysis, and the Appendix provides technical support for moment bounds, stochastic integrals, and LAN. Overall, the paper delivers practical, theoretically-backed procedures that guarantee estimation accuracy while requiring fewer observations than classical fixed-time methods, with explicit asymptotic rates.
Abstract
The drift sequential parameter estimation problems for the Cox-Ingersoll-Ross (CIR) processes under the limited duration of observation are studied. Truncated sequential estimation methods for both scalar and {two}-dimensional parameter cases are proposed. In the non-asymptotic setting, for the proposed truncated estimators, the properties of guaranteed mean-square estimation accuracy are established. In the asymptotic formulation, when the observation time tends to infinity, it is shown that the proposed sequential procedures are asymptotically optimal among all possible sequential and non-sequential estimates with an average estimation time less than the fixed observation duration. It also turned out that asymptotically, without degrading the estimation quality, they significantly reduce the observation duration compared to classical non-sequential maximum likelihood estimations based on a fixed observation duration.