Convergence Rate for a CIR Model with Fixed Delay Driven by Poisson Jumps

Abstract

In this paper, we consider a fixed delay CIR process with Poisson jumps, which serves as an extended model proposed by [Stoch. Anal. Appl., 2019, 37(4): 550–573]. We rigorously present the existence, uniqueness and nonnegativeness of the exact solution. Furthermore, we develop a backward Euler–Maruyama (EM) method for the fixed delay model and show that the numerical solution converges strongly to the exact solution with rate

\frac{1}{2}

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.

Keywords

CIR model / Poisson jump / delay / strong convergence rate / backward EM scheme / 65C30 / 65L20

References

[1] Akgiray V. Conditional heteroscedasticity in time series of stock returns: evidence and forcast. The Journal of Business, 1989, 62(1): 55-80

[2] Applebaum D. Lévy Processes and Stochastic Calculus, 2009Second EditionCambridge, Cambridge University Press 116

[3] Bao J, Böttcher B, Mao X, Yuan C. Convergence rate of numerical solutions to SFDEs with jumps. J. Comput. Appl. Math., 2011, 236(2): 119-131

[4] Cont R, Tankov P. Financial Modelling with Jump Processes, 2004, Boca Raton, FL, Chapman & Hall/CRC

[5] Cox JC, Ingersoll JEJr., Ross SA. A theory of the term structure of interest rates. Econometrica, 1985, 53(2): 385-407

[6] Dereich S, Neuenkirch A, Szpruch L. An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2012, 468(2140): 1105-1115

[7] Fatemion Aghda AS, Hosseini SM, Tahmasebi M. Analysis of non-negativity and convergence of solution of the balanced implicit method for the delay Cox–Ingersoll–Ross model. Appl. Numer. Math., 2017, 118: 249-265

[8] Flore F., Nappo G., Strong convergence of a positive preserving drift-implicit Euler scheme for the fixed delay CIR process. 2018, arXiv:1807.06474

[9] Flore F, Nappo G. A Feynman–Kac type formula for a fixed delay CIR model. Stoch. Anal. Appl., 2019, 37(4): 550-573

[10] Higham DJ, Kloeden PE. Numerical methods for nonlinear stochastic differential equations with jumps. Numer. Math., 2005, 101(1): 101-119

[11] Higham DJ, Kloeden PE. Convergence and stability of implicit methods for jump-diffusion systems. Int. J. Numer. Anal. Model., 2006, 3(2): 125-140

[12] Johannes M. The statistical and economic role of jumps in continuous-time interest rate models. The Journal of Finance, 2004, 59(1): 227-260

[13] Maghsoodi Y. Exact solutions and doubly efficient approximations of jump-diffusion Itô equations. Stochastic Anal. Appl., 1998, 16(6): 1049-1072

[14] Mao X. Exponential Stability of Stochastic Differential Equations, 1994, New York, Marcel Dekker, Inc.182

[15] Tan J, Chen Y, Men W, Guo Y. Positivity and convergence of the balanced implicit method for the nonlinear jump-extended CIR model. Math. Comput. Simulation, 2021, 182: 195-210

[16] Tan L, Wang S, Luo L. Strong convergence rate of implicit Euler scheme to a CIR model with delay. Appl. Numer. Math., 2023, 190: 15-26

[17] Wu F, Mao X, Chen K. The Cox–Ingersoll–Ross model with delay and strong convergence of its Euler–Maruyama approximate solutions. Appl. Numer. Math., 2009, 59(10): 2641-2658

[18] Yamada T, Watanabe S. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ., 1971, 11: 155-167

[19] Yang X, Wang X. A transformed jump-adapted backward Euler method for jump-extended CIR and CEV models. Numer. Algorithms, 2017, 74(1): 39-57

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