Convergence and non-negativity preserving of the solution of balanced method for the delay CIR model with jump

Abstract

In this work, we propose the balanced implicit method (BIM) to approximate the solution of the delay Cox-Ingersoll-Ross (CIR) model with jump which often gives rise to model an asset price and stochastic volatility dependent on past data. We show that this method preserves non-negativity property of the solution of this model with appropriate control functions. We prove the strong convergence and investigate the pth moment boundedness of the solution of BIM. Finally, we illustrate those results in the last section.

A positivity preserving numerical scheme for the mean-reverting alpha-CEV process

“…In the current literature, numerical schemes for jump-extended CEV and jump-extended CIR models have started to receive increasing attention, we refer to Yang and Wang [32], Fatemion Aghdas [14] and Stamatiou [29]. However, to the best of our knowledge, for the jump-extended CEV process and the jumpextended CIR process, the existing results have all focused on the case of Poisson jumps (finite activity jumps) and results on positivity preserving strong approximation schemes in the case of infinite activity jumps have only appeared in our previous work, Li and Taguchi [26], in the case of the alpha-CIR process.…”

On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case

“…For the classic diffusion case we mention the works of Alfonsi [3][4], Berkaoui et al [5], Brigo and Alfonsi [7], Dereich et al [8], Neuenkirch and Szpruch [23] and the references within. The jump-extended case (with and without delays) has recently received increasing attention, we refer to Yang and Wang [28], and the recent working papers of Fatemion Aghdas [12] and Stamatiou [24]. To the best of our knowledge, for jump-extended CIR/CEV models, the existing results have all focused on the case of finite activity jumps (the jumps are governed by a Poisson process) and results on positivity preserving strong approximation schemes in the case of infinite activity jumps have yet to be obtained.…”