Modified nadaraya-watson estimation for uncertain differential equations with financial market applications

Lihong Shao; Zilong Wang; Anshui Li; Lihong Shao; Zilong Wang; Anshui Li

doi:10.3934/jimo.2026167

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 10: 4825-4852. doi: 10.3934/jimo.2026167

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Research article

Modified nadaraya-watson estimation for uncertain differential equations with financial market applications

Department of Statistics, Shaoxing University, Shaoxing, Zhejiang, China

Received: 10 July 2026 Revised: 21 August 2026 Accepted: 28 August 2026 Published: 09 September 2026
62P25, 65D10

Uncertain differential equations act as powerful modeling instruments across numerous research sectors, and they are particularly prevalent in modern financial research. Parameter estimation poses a notably challenging issue throughout studies on uncertain differential equations. This work established an estimation framework dedicated to homogeneous uncertain differential equations where neither drift nor diffusion function had an explicit analytical form. To tackle this obstacle, we designed a nonparametric refined Nadaraya-Watson estimation approach to fit the two unknown functional mappings. Extensive simulation experiments and an empirical study based on the Shanghai Interbank Offered Rate (SHIBOR) time series are conducted to evaluate the stability and practical value of our estimator. The experimental outcomes demonstrated that the proposed method possessed strong applicability for real financial modeling scenarios. The paper concluded with a summary of the main contributions and possible future directions.
- uncertainty theory,
- uncertain differential equations,
- nonparametric estimation,
- modified nadaraya-watson estimation,
- residuals analysis
Citation: Lihong Shao, Zilong Wang, Anshui Li. Modified nadaraya-watson estimation for uncertain differential equations with financial market applications[J]. Journal of Industrial and Management Optimization, 2026, 22(10): 4825-4852. doi: 10.3934/jimo.2026167

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Abstract

Uncertain differential equations act as powerful modeling instruments across numerous research sectors, and they are particularly prevalent in modern financial research. Parameter estimation poses a notably challenging issue throughout studies on uncertain differential equations. This work established an estimation framework dedicated to homogeneous uncertain differential equations where neither drift nor diffusion function had an explicit analytical form. To tackle this obstacle, we designed a nonparametric refined Nadaraya-Watson estimation approach to fit the two unknown functional mappings. Extensive simulation experiments and an empirical study based on the Shanghai Interbank Offered Rate (SHIBOR) time series are conducted to evaluate the stability and practical value of our estimator. The experimental outcomes demonstrated that the proposed method possessed strong applicability for real financial modeling scenarios. The paper concluded with a summary of the main contributions and possible future directions.

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