Unconditionally positivity-preserving explicit order-one strong approximations of financial SDEs with non-Lipschitz coefficients
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| Format: | Preprint |
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2025
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| _version_ | 1866912660638924800 |
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| author | Wu, Xiaojuan Liu, Ruishu Xu, Jiahao |
| author_facet | Wu, Xiaojuan Liu, Ruishu Xu, Jiahao |
| contents | In this paper, we are interested in positivity-preserving approximations of stochastic differential equations (SDEs) with non-Lipschitz coefficients, arising from computational finance and possessing positive solutions. By leveraging a Lamperti transformation, we develop a novel, explicit, and unconditionally positivity-preserving numerical scheme for the considered financial SDEs. More precisely, an implicit term $c_{-1}Y_{n+1}^{-1}$ is incorporated in the scheme to guarantee unconditional positivity preservation, and a corrective operator is introduced in the remaining explicit terms to address the challenges posed by non-Lipschitz (possibly singular) coefficients of the transformed SDEs. By finding a unique positive root of a quadratic equation, the proposed scheme can be explicitly solved and is shown to be strongly convergent with order $1$, when used to numerically solve several well-known financial models such as the CIR process, the Heston-3/2 volatility model, the CEV process and the Aït-Sahalia model. Numerical experiments validate the theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02703 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unconditionally positivity-preserving explicit order-one strong approximations of financial SDEs with non-Lipschitz coefficients Wu, Xiaojuan Liu, Ruishu Xu, Jiahao Numerical Analysis In this paper, we are interested in positivity-preserving approximations of stochastic differential equations (SDEs) with non-Lipschitz coefficients, arising from computational finance and possessing positive solutions. By leveraging a Lamperti transformation, we develop a novel, explicit, and unconditionally positivity-preserving numerical scheme for the considered financial SDEs. More precisely, an implicit term $c_{-1}Y_{n+1}^{-1}$ is incorporated in the scheme to guarantee unconditional positivity preservation, and a corrective operator is introduced in the remaining explicit terms to address the challenges posed by non-Lipschitz (possibly singular) coefficients of the transformed SDEs. By finding a unique positive root of a quadratic equation, the proposed scheme can be explicitly solved and is shown to be strongly convergent with order $1$, when used to numerically solve several well-known financial models such as the CIR process, the Heston-3/2 volatility model, the CEV process and the Aït-Sahalia model. Numerical experiments validate the theoretical findings. |
| title | Unconditionally positivity-preserving explicit order-one strong approximations of financial SDEs with non-Lipschitz coefficients |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2510.02703 |