Analytical formula for fractional-order conditional moments of a nonlinear drift CEV process with regime switching: Hybrid approach with applications to finance

Kittisak Chumpong; Khamron Mekchay; Fukiat Nualsri; Phiraphat Sutthimat; Kittisak Chumpong; Khamron Mekchay; Fukiat Nualsri; Phiraphat Sutthimat

doi:10.3934/math.2026113

AIMS Mathematics

2026, Volume 11, Issue 1: 2816-2851. doi: 10.3934/math.2026113

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Analytical formula for fractional-order conditional moments of a nonlinear drift CEV process with regime switching: Hybrid approach with applications to finance

1.
Division of Computational Science, Faculty of Science, Prince of Songkla University, Songkhla 90110, Thailand
2.
Research Center in Mathematics and Statistics with Applications, Prince of Songkla University, Songkhla 90110, Thailand
3.
Financial Mathematics, Data Science and Computational Innovations Research Unit (FDC), Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand
4.
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
5.
Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand

Received: 06 November 2025 Revised: 10 January 2026 Accepted: 21 January 2026 Published: 28 January 2026
MSC : 91G20, 91G80

This paper presents the development of an analytical methodology for computing fractional-order conditional moments in nonlinear drift constant elasticity of variance (NLD-CEV) processes, where regime transitions are governed by continuous-time finite-state irreducible Markov chains. Through the implementation of a hybrid systems framework, we establish closed-form solutions for conditional moments spanning arbitrary fractional orders across multiple regime states, significantly advancing the analytical tractability of NLD-CEV processes under stochastic regime conditions. The theoretical foundation of our approach relies on the construction and resolution of an intricate system of coupled partial differential equations, derived through the application of the Feynman–Kac formula in the context of switching diffusions. Our analysis extends to a detailed examination of asymptotic behaviors exhibited by fractional-order conditional moments within a two-state regime-switching framework, particularly emphasizing the interplay between the Markov chain intensity matrix's symmetry properties and various parametric configurations in determining the process' evolution. To illustrate the practical relevance of our approach, Monte Carlo simulations for the process with regime-switching are applied to validate the accuracy and computational efficiency of the analytical formulas. Furthermore, to demonstrate significant improvements over traditional methods, we apply our findings for the valuation of financial derivatives within a dynamic nonlinear mean-reverting regime-switching process. This work offers substantial contributions to financial modeling and derivative pricing by providing a robust tool for practitioners and researchers who are dealing with complex stochastic environments.
- nonlinear drift CEV process,
- regime switching,
- Markov chain,
- Markovian switching,
- hybrid system,
- VIX futures
Citation: Kittisak Chumpong, Khamron Mekchay, Fukiat Nualsri, Phiraphat Sutthimat. Analytical formula for fractional-order conditional moments of a nonlinear drift CEV process with regime switching: Hybrid approach with applications to finance[J]. AIMS Mathematics, 2026, 11(1): 2816-2851. doi: 10.3934/math.2026113

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Abstract

This paper presents the development of an analytical methodology for computing fractional-order conditional moments in nonlinear drift constant elasticity of variance (NLD-CEV) processes, where regime transitions are governed by continuous-time finite-state irreducible Markov chains. Through the implementation of a hybrid systems framework, we establish closed-form solutions for conditional moments spanning arbitrary fractional orders across multiple regime states, significantly advancing the analytical tractability of NLD-CEV processes under stochastic regime conditions. The theoretical foundation of our approach relies on the construction and resolution of an intricate system of coupled partial differential equations, derived through the application of the Feynman–Kac formula in the context of switching diffusions. Our analysis extends to a detailed examination of asymptotic behaviors exhibited by fractional-order conditional moments within a two-state regime-switching framework, particularly emphasizing the interplay between the Markov chain intensity matrix's symmetry properties and various parametric configurations in determining the process' evolution. To illustrate the practical relevance of our approach, Monte Carlo simulations for the process with regime-switching are applied to validate the accuracy and computational efficiency of the analytical formulas. Furthermore, to demonstrate significant improvements over traditional methods, we apply our findings for the valuation of financial derivatives within a dynamic nonlinear mean-reverting regime-switching process. This work offers substantial contributions to financial modeling and derivative pricing by providing a robust tool for practitioners and researchers who are dealing with complex stochastic environments.

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