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标准布朗运动驱动的 CIR 模型梯形数值方法
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Abstract:
本文针对标准布朗运动驱动的Cox–Ingersoll–Ross (CIR)模型探讨了梯形数值方法的强收敛性。通过Lamperti变换, 将CIR模型转换为具有局部Lipschitz条件的漂移项和具有全局Lipschitz条件的扩散项的新方程. 在适当的条件下，证明了新方程梯形数值方法的保正性和强收敛阶，并通过 Lamperti 逆变换得到了CIR模型数值解的强收敛阶。最后，利用数值模拟结果验证了理论分析。
This paper investigates the strong convergence of the trapezoidal numerical method for the Cox–Ingersoll–Ross (CIR) model driven by standard Brownian motion. Through the Lamperti transformation, the CIR model is transformed into a new equation with a drift term satisfying a local Lipschitz condition and a di？usion term satisfying a global Lipschitz condition. Under suitable conditions, the positivity preservation and strong convergence order of the trapezoidal numerical method for the new equation are proven. Furthermore, the strong convergence order of the numerical solution for the CIR model is obtained through the Lamperti inverse transformation. Finally, the theoretical analysis is validated through numerical simulation results.
| [1] | Cox, J.C., Ingersoll Jr., J.E. and Ross, S.A. (1985) A Theory of the Term Structure of Interest Rates. Econometrica, 53, 385-407. https://doi.org/10.2307/1911242 |
| [2] | Higham, D.J. and Mao, X. (2005) Convergence of Monte Carlo Simulations Involving the Mean-Reverting Square Root Process. Journal of Computational Finance, 8, 35-61. https://doi.org/10.21314/JCF.2005.136 |
| [3] | Dereich, S., Neuenkirch, A. and Szpruch, L. (2012) An Euler-Type Method for the Strong Approximation of the Cox-Ingersoll-Ross Process. Proceedings of the Royal Society A: Mathe- matical, Physical and Engineering Sciences, 468, 1105-1115. https://doi.org/10.1098/rspa.2011.0505 |
| [4] | Milstein, G.N. (1994) Numerical Integration of Stochastic Di？erential Equations. Vol. 313, Springer Science Business Media, Berlin. https://doi.org/10.1007/978-94-015-8455-5 |
| [5] | Fischer, M. and Nappo, G. (2009) On the Moments of the Modulus of Continuity of It？ Processes. Stochastic Analysis and Applications, 28, 103-122. https://doi.org/10.1080/07362990903415825 |
