August  2022, 27(8): 4231-4253. doi: 10.3934/dcdsb.2021225

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, China

* Corresponding author: Aiguo Xiao

Received  May 2020 Revised  April 2021 Published  August 2022 Early access  September 2021

Fund Project: This research is supported by the National Natural Science Foundation of China (Nos. 12071403, 11671343), the Research Foundation of Education Commission of Hunan Province of China (No. 19B565) and the Postgraduate Innovation Fund of Hunan Province in China (No. CX20190420)

This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when $ \alpha = 1 $. Finally, several numerical tests are reported for verification of the theoretical findings.

Citation: Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4231-4253. doi: 10.3934/dcdsb.2021225
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P. T. AnhT. S. Doan and P. T. Huong, A variation of constant formula for Caputo fractional stochastic differential equations, Statist. Probab. Lett., 145 (2019), 351-358.  doi: 10.1016/j.spl.2018.10.010.

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P. Balasubramaniam and P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators, J. Optim. Theory Appl., 174 (2017), 139-155.  doi: 10.1007/s10957-016-0865-6.

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Z.-Q. ChenK.-H. Kim and P. Kim, Fractional time stochastic partial differential equations, Stochastic Process. Appl., 125 (2015), 1470-1499.  doi: 10.1016/j.spa.2014.11.005.

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R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, 2004. doi: 10.1201/9780203485217.

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[24]

L. LiJ.-G. Liu and J. Lu, Fractional stochastic differential equations satisfying fluctuation-dissipation theorem, J. Stat. Phys., 169 (2017), 316-339.  doi: 10.1007/s10955-017-1866-z.

[25]

Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.

[26]

H. LiangZ. Yang and J. Gao, Strong superconvergence of the Euler–Maruyama method for linear stochastic Volterra integral equations, J. Comput. Appl. Math., 317 (2017), 447-457.  doi: 10.1016/j.cam.2016.11.005.

[27]

M. Maleki and M. Tavassoli Kajani, Numerical approximations for Volterra's population growth model with fractional order via a multi-domain pseudospectral method, Appl. Math. Model., 39 (2015), 4300-4308.  doi: 10.1016/j.apm.2014.12.045.

[28]

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2008. doi: 10.1533/9780857099402.

[29]

S. A. McKinley and H. D. Nguyen, Anomalous diffusion and the generalized Langevin equation, SIAM J. Math. Anal., 50 (2018), 5119-5160.  doi: 10.1137/17M115517X.

[30]

F. Mirzaee and N. Samadyar, Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation, Optik Int. J. Light Electron Opt., 132 (2017), 262-273.  doi: 10.1016/j.ijleo.2016.12.029.

[31]

F. Mirzaee and N. Samadyar, On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions, Eng. Anal. Bound. Elem., 100 (2019), 246-255.  doi: 10.1016/j.enganabound.2018.05.006.

[32]

F. Mohammadi, Efficient Galerkin solution of stochastic fractional differential equations using second kind Chebyshev wavelets, Bol. Soc. Parana. Mat., 35 (2017), 195-215.  doi: 10.5269/bspm.v35i1.28262.

[33]

S. M. Momani, Local and global existence theorems on fractional integro-differential equations, J. Fract. Calc., 18 (2000), 81-86. 

[34]

J.-C. Pedjeu and G. S. Ladde, Stochastic fractional differential equations: Modeling, method and analysis, Chaos Solitons Fractals, 45 (2012), 279-293.  doi: 10.1016/j.chaos.2011.12.009.

[35]

A. N. V. Rao and C. P. Tsokos, On the existence, uniqueness, and stability behavior of a random solution to a nonlinear perturbed stochastic integro-differential equation, Information and Control, 27 (1975), 61-74.  doi: 10.1016/S0019-9958(75)90074-1.

[36]

F. M. Scudo, Vito Volterra and theoretical ecology, Theoret. Population Biol., 2 (1971), 1-23.  doi: 10.1016/0040-5809(71)90002-5.

[37]

D. T. SonP. T. HuongP. E. Kloeden and H. T. Tuan, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stoch. Anal. Appl., 36 (2018), 654-664.  doi: 10.1080/07362994.2018.1440243.

[38]

Z. TaheriS. Javadi and E. Babolian, Numerical solution of stochastic fractional integro-differential equation by the spectral collocation method, J. Comput. Appl. Math., 321 (2017), 336-347.  doi: 10.1016/j.cam.2017.02.027.

[39]

V. E. Tarasov, Fractional integro-differential equations for electromagnetic waves in dielectric media, Theoret. and Math. Phys., 158 (2009), 355-359.  doi: 10.1007/s11232-009-0029-z.

[40]

K. G. TeBeest, Classroom Note: Numerical and analytical solutions of Volterra's population model, SIAM Rev., 39 (1997), 484-493.  doi: 10.1137/S0036144595294850.

[41]

H. T. Tuan, On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1749-1762.  doi: 10.3934/dcdsb.2020318.

[42]

Y. WangJ. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.

[43]

Z. Wang, Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients, Statist. Probab. Lett., 78 (2008), 1062-1071.  doi: 10.1016/j.spl.2007.10.007.

[44]

Z. Yang, H. Yang and Z. Yao, Strong convergence analysis for Volterra integro-differential equations with fractional Brownian motions, J. Comput. Appl. Math., 383 (2021), 113156. doi: 10.1016/j.cam.2020.113156.

[45]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

[46]

Ş. Yüzbaşı, A numerical approximation for Volterra's population growth model with fractional order, Appl. Math. Model., 37 (2013), 3216-3227.  doi: 10.1016/j.apm.2012.07.041.

[47]

G. Zhang and R. Zhu, Runge–Kutta convolution quadrature methods with convergence and stability analysis for nonlinear singular fractional integro-differential equations, Commun. Nonlinear Sci. Numer. Simul., 84 (2020), 105132. doi: 10.1016/j.cnsns.2019.105132.

[48]

G. Zou, Numerical solutions to time-fractional stochastic partial differential equations, Numer. Algorithms, 82 (2019), 553-571.  doi: 10.1007/s11075-018-0613-0.

show all references

References:
[1]

A. AghajaniY. Jalilian and J. J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fract. Calc. Appl. Anal., 15 (2012), 44-69.  doi: 10.2478/s13540-012-0005-4.

[2]

P. T. AnhT. S. Doan and P. T. Huong, A variation of constant formula for Caputo fractional stochastic differential equations, Statist. Probab. Lett., 145 (2019), 351-358.  doi: 10.1016/j.spl.2018.10.010.

[3]

M. Asgari, Block pulse approximation of fractional stochastic integro-differential equation, Commun. Numer. Anal., 2014 (2014), 1-7. 

[4]

A. A. Badr and H. S. El-Hoety, Monte–Carlo Galerkin approximation of fractional stochastic integro-differential equation, Math. Probl. Eng., 2012 (2012), 709106. doi: 10.1155/2012/709106.

[5]

P. Balasubramaniam and P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators, J. Optim. Theory Appl., 174 (2017), 139-155.  doi: 10.1007/s10957-016-0865-6.

[6]

W. Cao, Z. Zhang and G. E. Karniadakis, Numerical methods for stochastic delay differential equations via the Wong–Zakai approximation, SIAM J. Sci. Comput., 37 (2015), A295–A318. doi: 10.1137/130942024.

[7]

Z.-Q. ChenK.-H. Kim and P. Kim, Fractional time stochastic partial differential equations, Stochastic Process. Appl., 125 (2015), 1470-1499.  doi: 10.1016/j.spa.2014.11.005.

[8]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC, 2004. doi: 10.1201/9780203485217.

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.
[10]

X. DaiW. Bu and A. Xiao, Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations, J. Comput. Appl. Math., 356 (2019), 377-390.  doi: 10.1016/j.cam.2019.02.002.

[11]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. doi: 10.1007/978-3-642-14574-2.

[12]

T. S. Doan, P. T. Huong, P. E. Kloeden and A. M. Vu, Euler–Maruyama scheme for Caputo stochastic fractional differential equations, J. Comput. Appl. Math., 380 (2020), 112989. doi: 10.1016/j.cam.2020.112989.

[13]

N. T. Dung, Fractional stochastic differential equations with applications to finance, J. Math. Anal. Appl., 397 (2013), 334-348.  doi: 10.1016/j.jmaa.2012.07.062.

[14]

D. J. HighamX. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.  doi: 10.1137/S0036142901389530.

[15]

M. HutzenthalerA. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 1563-1576.  doi: 10.1098/rspa.2010.0348.

[16]

M. HutzenthalerA. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.  doi: 10.1214/11-AAP803.

[17]

G. Izzo, E. Messina and A. Vecchio, Stability of numerical solutions for Abel–Volterra integral equations of the second kind, Mediterr. J. Math., 15 (2018), Paper No. 113. doi: 10.1007/s00009-018-1149-1.

[18]

B. JinY. Yan and Z. Zhou, Numerical approximation of stochastic time-fractional diffusion, ESAIM Math. Model. Numer. Anal., 53 (2019), 1245-1268.  doi: 10.1051/m2an/2019025.

[19]

M. Kamrani, Numerical solution of stochastic fractional differential equations, Numer. Algorithms, 68 (2015), 81-93.  doi: 10.1007/s11075-014-9839-7.

[20]

M. Kamrani, Convergence of Galerkin method for the solution of stochastic fractional integro differential equations, Optik Int. J. Light Electron Opt., 127 (2016), 10049-10057. 

[21]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. doi: 10.1007/978-3-662-12616-5.

[22] V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations, CRC Press, 1995. 
[23]

J. J. Levin and J. A. Nohel, On a system of integrodifferential equations occuring in reactor dynamics, J. Math. Mech., 9 (1960), 347-368.  doi: 10.1512/iumj.1960.9.59020.

[24]

L. LiJ.-G. Liu and J. Lu, Fractional stochastic differential equations satisfying fluctuation-dissipation theorem, J. Stat. Phys., 169 (2017), 316-339.  doi: 10.1007/s10955-017-1866-z.

[25]

Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differential Equations, 266 (2019), 3514-3558.  doi: 10.1016/j.jde.2018.09.009.

[26]

H. LiangZ. Yang and J. Gao, Strong superconvergence of the Euler–Maruyama method for linear stochastic Volterra integral equations, J. Comput. Appl. Math., 317 (2017), 447-457.  doi: 10.1016/j.cam.2016.11.005.

[27]

M. Maleki and M. Tavassoli Kajani, Numerical approximations for Volterra's population growth model with fractional order via a multi-domain pseudospectral method, Appl. Math. Model., 39 (2015), 4300-4308.  doi: 10.1016/j.apm.2014.12.045.

[28]

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2008. doi: 10.1533/9780857099402.

[29]

S. A. McKinley and H. D. Nguyen, Anomalous diffusion and the generalized Langevin equation, SIAM J. Math. Anal., 50 (2018), 5119-5160.  doi: 10.1137/17M115517X.

[30]

F. Mirzaee and N. Samadyar, Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation, Optik Int. J. Light Electron Opt., 132 (2017), 262-273.  doi: 10.1016/j.ijleo.2016.12.029.

[31]

F. Mirzaee and N. Samadyar, On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions, Eng. Anal. Bound. Elem., 100 (2019), 246-255.  doi: 10.1016/j.enganabound.2018.05.006.

[32]

F. Mohammadi, Efficient Galerkin solution of stochastic fractional differential equations using second kind Chebyshev wavelets, Bol. Soc. Parana. Mat., 35 (2017), 195-215.  doi: 10.5269/bspm.v35i1.28262.

[33]

S. M. Momani, Local and global existence theorems on fractional integro-differential equations, J. Fract. Calc., 18 (2000), 81-86. 

[34]

J.-C. Pedjeu and G. S. Ladde, Stochastic fractional differential equations: Modeling, method and analysis, Chaos Solitons Fractals, 45 (2012), 279-293.  doi: 10.1016/j.chaos.2011.12.009.

[35]

A. N. V. Rao and C. P. Tsokos, On the existence, uniqueness, and stability behavior of a random solution to a nonlinear perturbed stochastic integro-differential equation, Information and Control, 27 (1975), 61-74.  doi: 10.1016/S0019-9958(75)90074-1.

[36]

F. M. Scudo, Vito Volterra and theoretical ecology, Theoret. Population Biol., 2 (1971), 1-23.  doi: 10.1016/0040-5809(71)90002-5.

[37]

D. T. SonP. T. HuongP. E. Kloeden and H. T. Tuan, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stoch. Anal. Appl., 36 (2018), 654-664.  doi: 10.1080/07362994.2018.1440243.

[38]

Z. TaheriS. Javadi and E. Babolian, Numerical solution of stochastic fractional integro-differential equation by the spectral collocation method, J. Comput. Appl. Math., 321 (2017), 336-347.  doi: 10.1016/j.cam.2017.02.027.

[39]

V. E. Tarasov, Fractional integro-differential equations for electromagnetic waves in dielectric media, Theoret. and Math. Phys., 158 (2009), 355-359.  doi: 10.1007/s11232-009-0029-z.

[40]

K. G. TeBeest, Classroom Note: Numerical and analytical solutions of Volterra's population model, SIAM Rev., 39 (1997), 484-493.  doi: 10.1137/S0036144595294850.

[41]

H. T. Tuan, On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1749-1762.  doi: 10.3934/dcdsb.2020318.

[42]

Y. WangJ. Xu and P. E. Kloeden, Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative, Nonlinear Anal., 135 (2016), 205-222.  doi: 10.1016/j.na.2016.01.020.

[43]

Z. Wang, Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients, Statist. Probab. Lett., 78 (2008), 1062-1071.  doi: 10.1016/j.spl.2007.10.007.

[44]

Z. Yang, H. Yang and Z. Yao, Strong convergence analysis for Volterra integro-differential equations with fractional Brownian motions, J. Comput. Appl. Math., 383 (2021), 113156. doi: 10.1016/j.cam.2020.113156.

[45]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

[46]

Ş. Yüzbaşı, A numerical approximation for Volterra's population growth model with fractional order, Appl. Math. Model., 37 (2013), 3216-3227.  doi: 10.1016/j.apm.2012.07.041.

[47]

G. Zhang and R. Zhu, Runge–Kutta convolution quadrature methods with convergence and stability analysis for nonlinear singular fractional integro-differential equations, Commun. Nonlinear Sci. Numer. Simul., 84 (2020), 105132. doi: 10.1016/j.cnsns.2019.105132.

[48]

G. Zou, Numerical solutions to time-fractional stochastic partial differential equations, Numer. Algorithms, 82 (2019), 553-571.  doi: 10.1007/s11075-018-0613-0.

Figure 1.  The mean square errors of the EM scheme (11) for Example 5.1
Figure 2.  The mean square errors of the EM scheme (11) for Example 5.2
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