Convergence Analysis of Numerical Solution of Delayed Interest Rate Model with Poisson Jumps

doi:10.13438/j.cnki.jdzk.2025.05.003

Abstract

Abstract: An Ait-Sahalia interest rate model is constructed with delayed volatility driven by Poisson jumps,and its dynamic properties are investigated through a modified truncated Euler-Maruyama (EM) method.Under the theoretical framework of local Lipschitz conditions and Khasminskii-type conditions,it is proved that the numerical solution generated by the modified truncated EM method converges strongly to the true solution of the model.

Key words: delayed interest volatility, modified truncated Euler-Maruyama scheme, Ait-Sahalia, Poisson jumps

FENG Denghong, LI Zhimin. Convergence Analysis of Numerical Solution of Delayed Interest Rate Model with Poisson Jumps[J]. Journal of Jishou University(Natural Sciences Edition), 2025, 46(5): 10-18.

References

[1] VASICEK OLDRICH ALFONSO.An Equilibrium Characterization of the Term Structure[J].Journal of Financial Economics,1977,5(2):177-188.
[2] COX JOHN C,INGERSOLL JONATHAN E,ROSS JR STEPHEN A.A Theory of the Term Structure of Interest Rates[J].Econometrica,1985,53(2):385-407.
[3] AIT-SAHALIA YACINE.Testing Continuous-Time Models of the Spot Interest Rate[J].The Review of Financial Studies,1996,9(2):385-426.
[4] HIGHAM DESMOND J,MAO Xuerong,STUART ANDREW M.Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations[J].SIAM Journal on Numerical Analysis,2002,40(3):1041-1063.
[5] HUTZENTHALER MARTIN,JENTZEN ARNULF,KLOEDEN PETER E.Strong Convergence of an Explicit Numerical Method for SDEs with Nonglobally Lipschitz Continuous Coefficients[J].The Annals of Applied Probability,2012,22(4):1611-1641.
[6] MAO Xuerong.Convergence Rates of the Truncated Euler-Maruyama Method for Stochastic Differential Equations[J].Journal of Computational and Applied Mathematics,2016,296:362-375.
[7] MAO Xuerong.The Truncated Euler-Maruyama Method for Stochastic Differential Equations[J].Journal of Computational and Applied Mathematics,2015,290:370-384.
[8] LAN Guangqiang,XIA Fang.Strong Convergence Rates of Modified Truncated EM Method for Stochastic Differential Equations[J].Journal of Computational and Applied Mathematics,2018,334:1-17.
[9] JIANG Yu,LAN Guangqiang.Convergence and Exponential Stability of Modified Truncated Milstein Method for Stochastic Differential Equations[J].Journal of Computational and Applied Mathematics,2024,436:115288.DOI:10.1016/j.cam.2023.115288.
[10] DENG Shounian,FEI Weiyin,LIU Wei,et al.The Truncated EM Method for Stochastic Differential Equations with Poisson Jumps[J].Journal of Computational and Applied Mathematics,2019,355:232-257.
[11] TANG Yiyi,MAO Xuerong.The Logarithmic Truncated EM Method with Weaker Conditions[J].Applied Numerical Mathematics,2024,198:258-275.
[12] DENG Shounian,FEI Chen,FEI Weiyin,et al.Generalized Ait-Sahalia-Type Interest Rate Model with Poisson Jumps and Convergence of the Numerical Approximation[J].Physica A:Statistical Mechanics and Its Applications,2019,533:122057.DOI:10.1016/j.physa.2019.122057.
[13] EMMANUEL COFFIE,MAO Xuerong.Truncated EM Numerical Method for Generalised Ait-Sahalia-Type Interest Rate Model with Delay[J].Journal of Computational and Applied Mathematics,2021,383:113137.DOI:10.1016/j.cam.2020.113137.
[14] LAN Guangqiang.Asymptotic Exponential Stability of Modified Truncated EM Method for Neutral Stochastic Differential Delay Equations[J].Journal of Computational and Applied Mathematics,2018,340:334-341.
[15] LAN Guangqiang,WANG Qiushi.Strong Convergence Rates of Modified Truncated EM Methods for Neutral Stochastic Differential Delay Equations[J].Journal of Computational and Applied Mathematics,2019,362:83-98.