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

Simulation of Non-Lipschitz Stochastic Differential Equations Driven by $\alpha$-Stable Noise: A Method Based on Deterministic Homogenization

“…We show in Figure 7 into the region Z < 0. We remark that one may use positivity-preserving schemes to mitigate against this leakage (see for example [49]). However, our approach does not require knowing in advance the existence or location of a natural boundary and such information might not be readily available.…”

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. We stress that although the derived scheme in (3) might appear similar to the one given in Alfonsi [1] and Li and Taguchi [26], it was initially not clear how such a scheme can be obtained.…”

“…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. We stress that although the derived scheme in (3) might appear similar to the one given in Alfonsi [1] and Li and Taguchi [26], it was initially not clear how such a scheme can be obtained. Due to the presence of infinite activity jumps, jump-adapted schemes devised through combining the Lamperti transform and the backward Euler scheme (see [31]) are not feasible.…”

“…Numerically speaking, the advantage of the scheme proposed in (3) is similar to that in the cases of the CIR process in Alfonsi [1] and the alpha-CIR process in Li and Taguchi [26], that the scheme is obtained by solving, at each step, a quadratic equation. Therefore, even in the diffusion case (σ2 = 0), we do not need to solve for the positive root of a non-linear equation as done in [2,11,28], where the scheme was obtained through combining the Lamperti transform and the backward Euler scheme.…”

“…Another technical difficulty, which we dealt with in Lemma 1.7, is to show the existence of the β-moment for the numerical scheme for β ∈ [1, α). This result is fundamental in removing the boundedness assumption on the jump coefficient h in [18,19,26,27] and thus the removal of the truncation step and the restriction that α > √ 2 in [26]. Even in the case of the Euler-Maruyama scheme, this integrability issue was only studied recently in Frikha and Li [15], where a mean-field extension of the equation from Li and Mytnik [25] together with the corresponding propagation of chaos property and Euler-Maruyama scheme were studied.…”