Abstract
The present work is devoted to strong approximations of a generalized Aït-Sahalia model arising from mathematical finance. The numerical study of the considered model faces essential difficulties caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. In this paper, a novel explicit Euler-type scheme is proposed, which is easily implementable and able to preserve positivity of the original model unconditionally, i.e., for any time step-size \(h >0\). A mean-square convergence rate of order 0.5 is also obtained for the proposed scheme in both non-critical and general critical cases. Our work is motivated by the need to justify the multi-level Monte Carlo (MLMC) simulations for the underlying model, where the rate of mean-square convergence is required and the preservation of positivity is desirable particularly for large discretization time steps. Numerical experiments are finally provided to confirm the theoretical findings.
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Acknowledgements
This work was supported by the Natural Science Foundation of China (12071488, 12371417, 11971488) and the Natural Science Foundation of Hunan Province (2020JJ2040).
Funding
This work is supported by the National Natural Science Foundation of China (Grant Nos. 12071488, 12371417, 11971488) and the Natural Science Foundation of Hunan Province (Grant No.2020JJ2040).
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Liu, R., Cao, Y. & Wang, X. Unconditionally positivity-preserving explicit Euler-type schemes for a generalized Aït-Sahalia model. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01810-2
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DOI: https://doi.org/10.1007/s11075-024-01810-2