Analysis of non-negativity and convergence of solution of the balanced implicit method for the delay Cox-Ingersoll-Ross model

The delay Cox-Ingersoll-Ross (CIR) model is an important model in the financial markets. It has been proved that the solution of this model is non-negative and its pth moments are bounded. However, there is no explicit solution for this model. So, proposing appropriate numerical method for solving this model which preserves non-negativity and boundedness of the model's solution is very important. In this paper, we concentrate on the balanced implicit method (BIM) for this model and show that with choosing suitable control functions the BIM provides numerical solution that preserves non-negativity of solution of the model. Moreover, we show the pth moment boundedness of the numerical solution of the method and prove the convergence of the proposed numerical method. Finally, we present some numerical examples to confirm the theoretical results, and also application of BIM to compute some financial quantities.