Existence results for the Cox–Ingersoll–Ross model with variable exponent diffusion

Mustafa Avci; Mustafa Avci

doi:10.3934/math.2025984

AIMS Mathematics

2025, Volume 10, Issue 9: 22106-22126. doi: 10.3934/math.2025984

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Research article

Existence results for the Cox–Ingersoll–Ross model with variable exponent diffusion

Mustafa Avci ^,

Faculty of Science and Technology, Applied Mathematics, Athabasca University, Athabasca, AB, T9S3A3, Canada

Received: 05 July 2025 Revised: 08 September 2025 Accepted: 18 September 2025 Published: 23 September 2025
MSC : 60G07, 60H15, 60H20, 60H30

This study proposes a new stochastic model where the diffusion coefficient involves a state-dependent variable exponent function $ p(\cdot) $. This new theoretically flexible framework generalizes the classical Cox–Ingersoll–Ross model. The existence, uniqueness, and higher moment properties of solutions are analyzed. The validity and efficiency of the model is illustrated with numerical experiments. Finally, a detailed analysis for Itô vs. Stratonovich interpretations of the proposed model is provided.
- stochastic process,
- state-dependent variable exponent,
- Cox-Ingersoll-Ross (CIR) model,
- truncation procedure,
- Picard iterations,
- non-Lipschitz diffusion coefficient,
- moment estimates
Citation: Mustafa Avci. Existence results for the Cox–Ingersoll–Ross model with variable exponent diffusion[J]. AIMS Mathematics, 2025, 10(9): 22106-22126. doi: 10.3934/math.2025984

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Abstract

This study proposes a new stochastic model where the diffusion coefficient involves a state-dependent variable exponent function $ p(\cdot) $. This new theoretically flexible framework generalizes the classical Cox–Ingersoll–Ross model. The existence, uniqueness, and higher moment properties of solutions are analyzed. The validity and efficiency of the model is illustrated with numerical experiments. Finally, a detailed analysis for Itô vs. Stratonovich interpretations of the proposed model is provided.

References

[1]	A. Alfonsi, High-order discretization schemes for the CIR process: Application to affine processes and Heston models, Math. Comput., 79 (2010), 209–237.
[2]	Z. Brzeźniak, T. Zastawniak, Basic Stochastic Processes: A Course Through Exercises, Springer: London, 1998. https://doi.org/10.1007/978-1-4471-0533-6
[3]	O. Calin, An informal introduction to stochastic calculus with applications, 2Ed., World Scientific Publishing Co.: Singapore, 2022. https://doi.org/10.1142/12545
[4]	P. Chrisoffersen, K. Jacobs, K. Mimouni, Volatility dynamics for the SandP500: evidence from realized volatility, daily returns and option prices, Rev. Financ. Stud., 23 (2010), 3141–3189. https://doi.org/10.1093/rfs/hhq032 doi: 10.1093/rfs/hhq032
[5]	J. C. Cox, J. E. Ingersoll Jr., S. A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica, 53 (1985), 385–407. https://doi.org/10.2307/1911242 doi: 10.2307/1911242
[6]	A. Cozma, C. Reisinger, Strong order 1/2 convergence of full truncation Euler approximations to the Cox–Ingersoll–Ross process, IMA J. Numer. Anal., 40 (2020), 358–376. https://doi.org/10.2307/1911242 doi: 10.2307/1911242
[7]	W. Feller, The parabolic differential equations and the associated Semi-Groups of transformations, Ann. Math., 55 (1952), 468–519. https://doi.org/10.2307/1969644 doi: 10.2307/1969644
[8]	W. Feller, Two Singular Diffusion Problems, Ann. Math., 54 (1951), 173–182. https://doi.org/10.2307/1969318 doi: 10.2307/1969318
[9]	A. Friedman, Stochastic Differential Equations and Applications, Vol. 1, Academic Press Inc.: New York, 1975. https://doi.org/10.1016/C2013-0-07332-X
[10]	S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327–343. https://doi.org/10.1093/rfs/6.2.327 doi: 10.1093/rfs/6.2.327
[11]	D. J. Higham, X. Mao, A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041–1063. https://doi.org/10.1137/S0036142901389530 doi: 10.1137/S0036142901389530
[12]	M. Hutzenthaler, A. Jentzen, P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, P. Roy. Soc. A: Math. Phy., 467 (2011), 1563–1576. https://doi.org/10.1098/rspa.2010.0348 doi: 10.1098/rspa.2010.0348
[13]	I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Springer: New York, 1991. https://doi.org/10.1007/978-1-4612-0949-2
[14]	F. C. Klebaner, Introduction to stochastic calculus with applications, 3rd ed., World Scientific Publishing Co., London, 2012. https://doi.org/10.1142/p821
[15]	X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Woodhead Publishing Ltd: Cambridge, 2007.
[16]	R. Mendoza-Arriaga, V. Linetsky, Time-changed CIR default intensities with two-sided mean-reverting jumps, Ann. Appl. Probab., 24 (2014), 811–56. https://doi.org/10.1214/13-AAP936 doi: 10.1214/13-AAP936
[17]	B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 6th ed., Springer: New York, 2003. https://doi.org/10.1007/978-3-642-14394-6
[18]	S. Sabanis, A note on tamed Euler approximations, Electron. Commun. Probab., 18 (2013), 1–10. https://doi.org/10.1214/ECP.v18-2824 doi: 10.1214/ECP.v18-2824
[19]	E. Salavati, An extension of the Yamada-Watanabe theorem, Math. Meth. Appl. Sci., 40 (2017), 7022–7025. https://doi.org/10.1002/mma.4509 doi: 10.1002/mma.4509

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