Legendre spectral-Monte Carlo method and its error analysis for nonlinear stochastic Itô–Volterra integral equation

Ishtiaq Ali; Saeed Islam; Ishtiaq Ali; Saeed Islam

doi:10.3934/nhm.2025065

Networks and Heterogeneous Media

2025, Volume 20, Issue 5: 1524-1544. doi: 10.3934/nhm.2025065

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Legendre spectral-Monte Carlo method and its error analysis for nonlinear stochastic Itô–Volterra integral equation

Ishtiaq Ali ^{1
,
,},
Saeed Islam ^{2
,
,}

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa 31982, Saudi Arabia
2.
Department of Mechanical Engineering, Prince Mohammad Bin Fahd University, P. O. Box 1664, Al Khobar 31952, Saudi Arabia

Received: 15 October 2025 Revised: 16 December 2025 Accepted: 24 December 2025 Published: 05 January 2026

Nonlinear stochastic Itô–Volterra integral equations (NSIVIEs) represent systems whose current state is influenced by random fluctuations and is dependent on previous information. These equations appear in many real-world scenarios, including engineering systems, biological processes, financial markets, heterogeneous media, complex transport phenomena, and viscoelastic materials. Strong numerical frameworks are required because analytical solutions for these equations are rarely available, particularly when nonlinearities and random fluctuations are present. To effectively solve NSIVIEs, in this study we propose a new hybrid numerical framework that combines Monte Carlo simulation and Legendre spectral collocation. By using orthogonal polynomial basis functions to approximate the solution, this method provides spectral accuracy while handling the hereditary memory component of the Volterra equation through a high-order Legendre spectral collocation method. A precise statistical treatment of the random fluctuations is made possible by simultaneously addressing the stochastic Itô noise through Monte Carlo sampling across numerous independent realizations. We perform a thorough convergence analysis and obtain explicit error bounds that measure the decrease in approximation error with increasing spectral resolution and Monte Carlo sample count. Numerical experiments show that the method can accurately reproduce complex stochastic behaviors and validate theoretical predictions.
- stochastic Itô–Volterra integral equation,
- spectral methods,
- error estimates,
- Monte Carlo simulations,
- numerical examples
Citation: Ishtiaq Ali, Saeed Islam. Legendre spectral-Monte Carlo method and its error analysis for nonlinear stochastic Itô–Volterra integral equation[J]. Networks and Heterogeneous Media, 2025, 20(5): 1524-1544. doi: 10.3934/nhm.2025065

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Abstract

Nonlinear stochastic Itô–Volterra integral equations (NSIVIEs) represent systems whose current state is influenced by random fluctuations and is dependent on previous information. These equations appear in many real-world scenarios, including engineering systems, biological processes, financial markets, heterogeneous media, complex transport phenomena, and viscoelastic materials. Strong numerical frameworks are required because analytical solutions for these equations are rarely available, particularly when nonlinearities and random fluctuations are present. To effectively solve NSIVIEs, in this study we propose a new hybrid numerical framework that combines Monte Carlo simulation and Legendre spectral collocation. By using orthogonal polynomial basis functions to approximate the solution, this method provides spectral accuracy while handling the hereditary memory component of the Volterra equation through a high-order Legendre spectral collocation method. A precise statistical treatment of the random fluctuations is made possible by simultaneously addressing the stochastic Itô noise through Monte Carlo sampling across numerous independent realizations. We perform a thorough convergence analysis and obtain explicit error bounds that measure the decrease in approximation error with increasing spectral resolution and Monte Carlo sample count. Numerical experiments show that the method can accurately reproduce complex stochastic behaviors and validate theoretical predictions.

References

[1]	K. Itô, On stochastic differential equations, Mem. Am. Math. Soc., 4 (1951), 1–51. http://dx.doi.org/10.1090/memo/0004 doi: 10.1090/memo/0004
[2]	P. Protter, Volterra equations driven by semimartingales, Ann. Probab., 13 (1985), 519–530. https://doi.org/10.1214/aop/1176993006 doi: 10.1214/aop/1176993006
[3]	E. Pardoux, P. Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab., 18 (1990), 1635–1655. https://doi.org/10.1214/aop/1176990638 doi: 10.1214/aop/1176990638
[4]	L. Decreusefond, Regularity properties of some stochastic Volterra integrals with singular kernel, Potential Anal., 16 (2002), 139–149. https://doi.org/10.1023/A:1012628013041 doi: 10.1023/A:1012628013041
[5]	F. Mirzaee, E. Hadadiyan, A collocation technique for solving nonlinear stochastic Itô–Volterra integral equations, Appl. Math. Comput., 247 (2014), 1011–1020. https://doi.org/10.1016/j.amc.2014.09.047 doi: 10.1016/j.amc.2014.09.047
[6]	J. Yong, Backward stochastic Volterra integral equations and some related problems, Stochastic Processes Appl., 116 (2006), 779–795. https://doi.org/10.1016/j.spa.2006.01.005 doi: 10.1016/j.spa.2006.01.005
[7]	X. Zhang, Euler schemes and large deviations for stochastic Volterra equations with singular kernels, J. Differ. Equ., 224 (2008), 2226–2250. https://doi.org/10.1016/j.jde.2008.02.019 doi: 10.1016/j.jde.2008.02.019
[8]	W. G. Cochran, J. S. Lee, J. Potthoff, Stochastic Volterra equations with singular kernels, Stochastic Processes Appl., 56 (1995), 337–349. https://doi.org/10.1016/0304-4149(94)00072-2 doi: 10.1016/0304-4149(94)00072-2
[9]	X. Zhang, Stochastic Volterra equations in Banach spaces and stochastic partial differential equations, J. Funct. Anal., 258 (2010), 1361–1425. https://doi.org/10.1016/j.jfa.2009.11.006 doi: 10.1016/j.jfa.2009.11.006
[10]	M. Khodabin, K. Maleknejad, M. Rostami, M. Nouri, Numerical solution of stochastic differential equations by second order Runge–Kutta methods, Math. Comput. Model., 53 (2011), 1910–1920. https://doi.org/10.1016/j.mcm.2011.01.018 doi: 10.1016/j.mcm.2011.01.018
[11]	J. C. Cortes, L. Jodar, L. Villafuerte, Numerical solution of random differential equations: A mean square approach, Math. Comput. Model., 45 (2007), 757–765. https://doi.org/10.1016/j.mcm.2006.07.017 doi: 10.1016/j.mcm.2006.07.017
[12]	S. Jankovic, D. Ilic, One linear analytic approximation for stochastic integro-differential equations, Acta Math. Sci., 30 (2010), 1073–1085. https://doi.org/10.1016/S0252-9602(10)60104-X doi: 10.1016/S0252-9602(10)60104-X
[13]	J. C. Cortés, L. Jódar, L. Villafuerte, Mean square numerical solution of random differential equations: Facts and possibilities, Comput. Math. Appl., 53 (2007), 1098–1106. https://doi.org/10.1016/j.camwa.2006.05.030 doi: 10.1016/j.camwa.2006.05.030
[14]	Y. Saito, T. Mitsui, Simulation of stochastic differential equations, Ann. Inst. Stat. Math., 45 (1993), 419–432. https://doi.org/10.1007/BF00773344 doi: 10.1007/BF00773344
[15]	A. Deb, A. Dasgupta, G. Sarkar, A new set of orthogonal functions and its applications to the analysis of dynamic systems, J. Franklin Inst., 343 (2006), 1–26. https://doi.org/10.1016/j.jfranklin.2005.06.005 doi: 10.1016/j.jfranklin.2005.06.005
[16]	E. Babolian, R. Mokhtari, M. Salmani, Using direct method for solving variational problems via triangular orthogonal functions, Appl. Math. Comput., 191 (2007), 206–217. https://doi.org/10.1016/j.amc.2007.02.080 doi: 10.1016/j.amc.2007.02.080
[17]	E. Babolian, Z. Masouri, S. Hatamzadeh-Varmazyar, Numerical solution of nonlinear Volterra–Fredholm integro-differential equations via direct method using triangular functions, Comput. Math. Appl., 58 (2009), 239–247. https://doi.org/10.1016/j.camwa.2009.03.087 doi: 10.1016/j.camwa.2009.03.087
[18]	K. Maleknejad, H. Almasieh, M. Roodaki, Triangular functions (TF) method for the solution of Volterra–Fredholm integral equations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3293–3298. https://doi.org/10.1016/j.cnsns.2009.12.015 doi: 10.1016/j.cnsns.2009.12.015
[19]	S. Nemati, P. M. Lima, Y. Ordokhani, Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J. Comput. Appl. Math., 242 (2013), 53–69. https://doi.org/10.1016/j.cam.2012.10.021 doi: 10.1016/j.cam.2012.10.021
[20]	M. Samar, K. E. Yao, X. Zhu, Numerical solution of nonlinear backward stochastic Volterra integral equations, Axioms, 12 (2023), 888. https://doi.org/10.3390/axioms12090888 doi: 10.3390/axioms12090888
[21]	Y. Karaca, D. Baleanu, Evolutionary mathematical science, fractional modeling and artificial intelligence of nonlinear dynamics in complex systems, Chaos Theory Appl., 4 (2022), 111–118. https://dergipark.org.tr/en/pub/chaos/article/1188154
[22]	K. Rybakov, Spectral representations of iterated stochastic integrals and their application for modeling nonlinear stochastic dynamics, Mathematics, 11 (2023), 4047. https://doi.org/10.3390/math11194047 doi: 10.3390/math11194047
[23]	A. M. S. Mahdy, A. S. Nagdy, K. M. Hashem, D. S. Mohamed, A computational technique for solving three-dimensional mixed Volterra–Fredholm integral equations, Fractal Fract., 7 (2023), 196. https://doi.org/10.3390/fractalfract7020196 doi: 10.3390/fractalfract7020196
[24]	S. Saha Ray, P. Singh, Numerical solution of stochastic Itô–Volterra integral equation by using shifted Jacobi operational matrix method, Appl. Math. Comput., 410 (2021), 126440. https://doi.org/10.1016/j.amc.2021.126440 doi: 10.1016/j.amc.2021.126440
[25]	P. K. Singh, S. Saha Ray, An efficient numerical method based on Lucas polynomials to solve multi-dimensional stochastic Itô–Volterra integral equations, Math. Comput. Simul., 203 (2023), 826–845. https://doi.org/10.1016/j.matcom.2022.06.029 doi: 10.1016/j.matcom.2022.06.029
[26]	P. K. Singh, S. Saha Ray, A collocation method for nonlinear stochastic differential equations driven by fractional Brownian motion and its application to mathematical finance, Methodol. Comput. Appl. Probab., 26 (2024), 1–23. https://doi.org/10.1007/s11009-024-10087-w doi: 10.1007/s11009-024-10087-w
[27]	M. A. Abdelkawy, Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations, Int. J. Nonlinear Sci. Numer. Simul., 24 (2023), 123–136. https://doi.org/10.1515/ijnsns-2020-0144 doi: 10.1515/ijnsns-2020-0144
[28]	M. Ahmadinia, H. Afshariarjmand, M. Salehi, Numerical solution of multi-dimensional Itô–Volterra integral equations by the second kind Chebyshev wavelets and parallel computing process, Appl. Math. Comput., 450 (2023), 127988. https://doi.org/10.1016/j.amc.2023.127988 doi: 10.1016/j.amc.2023.127988
[29]	S. Alipour, F. Mirzaee, An iterative algorithm for solving two-dimensional nonlinear stochastic integral equations: A combined successive approximations method with bilinear spline interpolation, Appl. Math. Comput., 371 (2020), 124947. https://doi.org/10.1016/j.amc.2019.124947 doi: 10.1016/j.amc.2019.124947
[30]	F. Mirzaee, E. Solhi, S. Naserifar, Approximate solution of stochastic Volterra integro-differential equations by using moving least squares scheme and spectral collocation method, Appl. Math. Comput., 410 (2021), 126447. https://doi.org/10.1016/j.amc.2021.126447 doi: 10.1016/j.amc.2021.126447
[31]	E. Solhi, F. Mirzaee, S. Naserifar, Approximate solution of two-dimensional linear and nonlinear stochastic Itô–Volterra integral equations via meshless scheme, Math. Comput. Simul., 207 (2023), 369–387. https://doi.org/10.1016/j.matcom.2023.01.009 doi: 10.1016/j.matcom.2023.01.009
[32]	M. Kamiński, M. Kleiber, Numerical homogenization of N-component composites including stochastic interface defects, Int. J. Numer. Methods Eng., 47 (2000), 1001–1027. https://doi.org/10.1002/(SICI)1097-0207(20000220)47:5<1001::AID-NME814>3.0.CO;2-V doi: 10.1002/(SICI)1097-0207(20000220)47:5<1001::AID-NME814>3.0.CO;2-V
[33]	J. Zhu, F. Khan, S. U. Khan, W. Sumelka, F. U. Khan, S. A. AlQahtani, Computational investigation of stochastic Zika virus optimal control model using Legendre spectral method, Sci. Rep., 14 (2024), 18112. https://doi.org/10.1038/s41598-024-69096-x doi: 10.1038/s41598-024-69096-x
[34]	S. Lin, J. Zhang, C. Qiu, Asymptotic analysis for one-stage stochastic linear complementarity problems and applications, Mathematics, 11 (2023), 482. https://doi.org/10.3390/math11020482 doi: 10.3390/math11020482
[35]	M. Xie, S. U. Khan, W. Sumelka, A. M. Alamri, S. A. AlQahtani, Advanced stability analysis of a fractional delay differential system with stochastic phenomena using spectral collocation method, Sci. Rep., 14 (2024), 12047. https://doi.org/10.1038/s41598-024-62851-0 doi: 10.1038/s41598-024-62851-0
[36]	S. U. Khan, I. Ali, Numerical analysis of stochastic SIR model by Legendre spectral collocation method, Adv. Mech. Eng., 11 (2019), 7. https://doi.org/10.1177/1687814019862918 doi: 10.1177/1687814019862918
[37]	P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, NY, USA, 2004. https://doi.org/10.1007/978-0-387-21617-1
[38]	R. E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numer., 7 (1998), 1–49. http://dx.doi.org/10.1017/S0962492900002804 doi: 10.1017/S0962492900002804
[39]	I. Ali, S. U. Khan, A dynamic competition analysis of stochastic fractional differential equation arising in finance via pseudospectral method, Mathematics, 11 (2023), 1328. https://doi.org/10.3390/math11061328 doi: 10.3390/math11061328
[40]	S. U. Khan, M. Ali, I. Ali, A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis, Adv. Differ. Equ., 2019 (2019), 161. https://doi.org/10.1186/s13662-019-2096-2 doi: 10.1186/s13662-019-2096-2
[41]	I. Ali, S. U. Khan, Dynamics and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method, AIMS Math., 8 (2023), 4220–4236. https://doi.org/10.3934/math.2023210 doi: 10.3934/math.2023210
[42]	S. U. Khan, I. Ali, Application of Legendre spectral-collocation method to delay differential and stochastic delay differential equation, AIP Adv., 8 (2018), 035301. https://doi.org/10.1063/1.5016680 doi: 10.1063/1.5016680
[43]	I. Ali, H. Brunner, T. Tang, A spectral method for pantograph-type delay differential equations and its convergence analysis, J. Comput. Math., 27 (2009), 254–265. Available from: https://global-sci.com/index.php/JCM/article/view/11934.
[44]	I. Ali, Convergence analysis of spectral methods for integro–differential equations with vanishing proportional delays, J. Comput. Math., 28 (2011), 962–973. https://doi.org/10.4208/jcm.1006-m3150 doi: 10.4208/jcm.1006-m3150
[45]	X. Wen, J. Huang, A combination method for numerical solution of the nonlinear stochastic Itô–Volterra integral equation, Appl. Math. Comput., 407 (2021), 126302. https://doi.org/10.1016/j.amc.2021.126302 doi: 10.1016/j.amc.2021.126302
[46]	F. Mirzaee, E. Hadadiyan, A collocation technique for solving nonlinear stochastic Itô–Volterra integral equations, Appl. Math. Comput., 247 (2014), 1011–1020. https://doi.org/10.1016/j.amc.2014.09.047 doi: 10.1016/j.amc.2014.09.047
[47]	Z. Taheri, S. Javadi, E. Babolian, Numerical solution of stochastic fractional integro–differential equation by the spectral collocation method, J. Comput. Appl. Math., 321 (2017), 336–347. https://doi.org/10.1016/j.cam.2017.02.027 doi: 10.1016/j.cam.2017.02.027
[48]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods, Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006. https://doi.org/10.1007/978-3-540-30726-6
[49]	G. Mastroianni, D. Occorsio, Optimal systems of nodes for Lagrange interpolation on bounded intervals: A survey, J. Comput. Appl. Math., 134 (2001), 325–341. https://doi.org/10.1016/S0377-0427(00)00557-4 doi: 10.1016/S0377-0427(00)00557-4
[50]	P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, Germany, 1999. https://doi.org/10.1007/978-3-662-12616-5

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