Explicit modified Euler approximations of the Aït-Sahalia type model with Poisson jumps
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913919547736064 |
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| author | Jiang, Yingsong Liu, Ruishu Xu, Minhong |
| author_facet | Jiang, Yingsong Liu, Ruishu Xu, Minhong |
| contents | This paper focuses on mean-square approximations of a generalized Aït-Sahalia interest rate model with Poisson jumps. The main challenge in the construction and analysis of time-discrete numerical schemes is caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. Due to the presence of the Poisson jumps, additional difficulties arise in recovering the exact order $1/2$ of convergence for the time-stepping schemes. By incorporating implicitness in the term $α_{-1}x^{-1} $ and introducing the modifications functions $f_h$ and $g_h$ in the recursion, a novel explicit Euler-type scheme is proposed, which is easy to implement and preserves the positivity of the original model unconditionally, i.e., for any time step-size $h>0$. A mean-square convergence rate of order $1/2$ is established for the proposed scheme in both the non-critical and general critical cases. Finally, numerical experiments are provided to confirm the theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23947 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit modified Euler approximations of the Aït-Sahalia type model with Poisson jumps Jiang, Yingsong Liu, Ruishu Xu, Minhong Numerical Analysis 60H35, 60H15, 65C30 This paper focuses on mean-square approximations of a generalized Aït-Sahalia interest rate model with Poisson jumps. The main challenge in the construction and analysis of time-discrete numerical schemes is caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. Due to the presence of the Poisson jumps, additional difficulties arise in recovering the exact order $1/2$ of convergence for the time-stepping schemes. By incorporating implicitness in the term $α_{-1}x^{-1} $ and introducing the modifications functions $f_h$ and $g_h$ in the recursion, a novel explicit Euler-type scheme is proposed, which is easy to implement and preserves the positivity of the original model unconditionally, i.e., for any time step-size $h>0$. A mean-square convergence rate of order $1/2$ is established for the proposed scheme in both the non-critical and general critical cases. Finally, numerical experiments are provided to confirm the theoretical findings. |
| title | Explicit modified Euler approximations of the Aït-Sahalia type model with Poisson jumps |
| topic | Numerical Analysis 60H35, 60H15, 65C30 |
| url | https://arxiv.org/abs/2506.23947 |