Stability of the analytic solution and the partially truncated Euler-Maruyama method for a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. (English) Zbl 1524.65048
Summary: In this paper, we develop the partially truncated Euler-Maruyama method of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The mean-square exponential stability and the mean-square stability of the analytic solution and the partially truncated Euler-Maruyama method are investigated. Moreover, the theoretical results are illustrated by some numerical examples.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H20 | Stochastic integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 45D05 | Volterra integral equations |
| 45J05 | Integro-ordinary differential equations |
Keywords:
stochastic Volterra integro-differential equations; non-globally Lipschitz condition; partially truncated Euler-Maruyama method; mean-square exponential stability; mean-square stabilityReferences:
| [1] | Burton, T., Volterra Integral and Differential Equations (1983), Academic Press: Academic Press, London · Zbl 0515.45001 |
| [2] | Guo, Q.; Liu, W.; Mao, X.; Yue, R., The partially truncated Euler-Maruyama method and its stability and boundedness, Appl. Numer. Math., 115, 235-251 (2003) · Zbl 1358.65008 |
| [3] | Hara, T.; Yoneyama, T.; Miyazaki, R., Some refinements of Razumikhin’s method and their applications, Funkcial. Ekvac. Ser. Int., 35, 2, 279-305 (1992) · Zbl 0770.34051 |
| [4] | Higham, D. J.; Mao, X.; Stuart, A. M., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40, 3, 1041-1063 (2002) · Zbl 1026.65003 |
| [5] | Hu, P.; Huang, C. M., Stability of Euler-Maruyama method for linear stochastic delay integro-differential equations, Math. Numer. Sin., 32, 1, 105-112 (2010) · Zbl 1224.65011 |
| [6] | Hu, P.; Huang, C. M., Stability of stochastic θ-methods for stochastic delay integro-differential equations, Int. J. Comput. Math., 88, 7, 1417-1429 (2011) · Zbl 1222.65010 |
| [7] | Hu, P.; Huang, C. M., The stochastic Θ-method for nonlinear stochastic Volterra integro-differential equations, Abstr. Appl. Anal., 2014, 1, 137-151 (2014) |
| [8] | Hutzenthaler, M.; Jentzen, A.; Kloeden, P. E., Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. Roy. Soc. A-Math. Phys., 467, 2130, 1563-1576 (2011) · Zbl 1228.65014 |
| [9] | Hutzenthaler, M.; Jentzen, A.; Kloeden, P. E., Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22, 4, 1611-1641 (2012) · Zbl 1256.65003 |
| [10] | Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations (1992), Springer-Verlag: Springer-Verlag, New York · Zbl 0925.65261 |
| [11] | Liang, H.; Yang, Z. W.; Gao, J. F., Strong superconvergence of Euler-Maruyama method for linear stochastic Volterral integral equations, J. Comput. Appl. Math., 317, 447-457 (2017) · Zbl 1357.65011 |
| [12] | Liao, J.; Liu, W.; Wang, X., Truncated Milstein method for non-autonomous stochastic differential equations and its modification, J. Comput. Appl. Math., 402 (2022) · Zbl 1482.65009 |
| [13] | Mao, X., Stability of stochastic integro-differential equations, Stoch. Anal. Appl., 18, 6, 1005-1017 (2000) · Zbl 0969.60068 |
| [14] | Mao, X., Stochastic Differential Equations and Applications, 2nd ed. (2007), Horwood: Horwood, Chichester (UK) · Zbl 1138.60005 |
| [15] | Mao, X., The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290, 370-384 (2015) · Zbl 1330.65016 |
| [16] | Mao, X., Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 296, 362-375 (2016) · Zbl 1378.65036 |
| [17] | Mao, X.; Riedle, M., Mean square stability of stochastic Volterra integro-differential equations, Syst. Control Lett., 55, 6, 459-465 (2006) · Zbl 1129.34332 |
| [18] | Mao, X.; Wei, F.; Wiriyakraikul, T., Positivity preserving truncated Euler-Maruyama method for stochastic Lotka-Volterra competition model, J. Comput. Appl. Math., 394, 3 (2021) · Zbl 1465.65008 |
| [19] | Song, M.; Hu, L.; Mao, X.; Zhang, L., Khasminskii-type theorems for stochastic functional differential equations, Discrete Contin. Dyn. Syst. Ser. B., 18, 6, 1697-1714 (2013) · Zbl 1281.34128 |
| [20] | Subramaniam, R.; Balachandran, K., Existence of solutions of a class of stochastic Volterra integral equations with applications to chemotherapy, J. Aust. Math. Soc. Series B. Appl. Math, 41, 1, 93-104 (1999) · Zbl 0945.60055 |
| [21] | Szynal, D.; Wedrychowicz, S., On solutions of a stochastic integral equation of the Volterra type with applications for chemotherapy, J. Appl. Probab., 25, 257-267 (1988) · Zbl 0643.60048 |
| [22] | Tretyakov, M. V.; Zhang, Z., A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal., 51, 6, 3135-3162 (2013) · Zbl 1293.60069 |
| [23] | Tsokos, C. P.; Padgett, W. J., Random Integral Equations with Applications to Life Sciences and Engineering (1974), Academic Press: Academic Press, New York · Zbl 0287.60065 |
| [24] | Wen, C. H.; Zhang, T. S., Rectangular method on stochastic Volterra equations, Int. J. Appl. Math. Stat., 14, J09, 12-26 (2009) |
| [25] | Wen, C. H.; Zhang, T. S., Improved rectangular method on stochastic Volterra equations, J. Comput. Appl. Math., 235, 8, 2492-2501 (2011) · Zbl 1221.65023 |
| [26] | Wu, Q.; Hu, L.; Zhang, Z., Convergence and stability of balanced methods for stochastic delay integro-differential equations, Appl. Math. Comput., 237, 11, 446-460 (2014) · Zbl 1334.65017 |
| [27] | Zhang, W., Convergence of the balanced Euler method for a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients, Appl. Numer. Math., 154, 17-35 (2020) · Zbl 1498.65026 |
| [28] | Zhang, W., Theoretical and numerical analysis of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients, Appl. Numer. Math., 147, 254-276 (2020) · Zbl 1448.65012 |
| [29] | Zhang, W.; Liang, H.; Gao, Jianfang, Theoretical and numerical analysis of the Euler-Maruyama method for generalized stochastic Volterra integro-differential equations, J. Comput. Appl. Math., 365 (2020) · Zbl 1524.65047 |
| [30] | Zhang, W.; Song, M. H.; Liu, M. Z., Strong convergence of the partially truncated Euler-Maruyama method for a class of stochastic differential delay equations, J. Comput. Appl. Math., 335, 114-128 (2018) · Zbl 1444.34099 |
| [31] | Zhang, Z. Q.; Ma, H. P., Order-preserving strong scheme for SDEs with locally Lipschitz coefficients, Appl. Numer. Math., 112, 1-16 (2017) · Zbl 1354.65017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
