Almost sure stability of stochastic theta methods with random variable stepsize for stochastic differential equations

References

• Blythe, S., Mao, X., & Liao, X. (2001). Stability of stochastic delay neural networks. Journal of the Franklin Institute, 338(4), 481–495. https://doi.org/10.1016/S0016-0032(01)00016-3
   Web of Science ®Google Scholar
• Burrage, P. M., & Burrage, K. (2003). A variable stepsize implementation for stochastic differential equations. SIAM Journal on Scientific Computing, 24(3), 848–864. https://doi.org/10.1137/S1064827500376922
   Google Scholar
• Chen, L., & Wu, F. (2012a). Almost sure exponential stability of the θ-method for stochastic differential equations. Statistics & Probability Letters, 82(9), 1669–1676. https://doi.org/10.1016/j.spl.2012.05.004
   Web of Science ®Google Scholar
• Chen, L., & Wu, F. (2012b). Choice of θ and its effects on stability in the stochastic θ-method of stochastic delay differential equations. International Journal of Computer Mathematics, 89(15), 2106–2122. https://doi.org/10.1080/00207160.2012.703320
   Web of Science ®Google Scholar
• Cox, D. R., & Miller, H. D. (2017). The theory of stochastic processes. Routledge.
   Google Scholar
• Gaines, J. G., & Lyons, T. J. (1997). Variable step size control in the numerical solution of stochastic differential equations. SIAM Journal on Applied Mathematics, 57(5), 1455–1484. https://doi.org/10.1137/S0036139995286515
   Google Scholar
• Higham, D. J. (2001). An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 43(3), 525–546. https://doi.org/10.1137/S0036144500378302
   Web of Science ®Google Scholar
• Higham, D. J., & Higham, N. J. (2016). MATLAB guide. SIAM.
   Google Scholar
• Hu, R. (2020). Almost sure exponential stability of the Milstein-type schemes for stochastic delay differential equations. Chaos Solitons & Fractals, 131(2), 109499. https://doi.org/10.1016/j.chaos.2019.109499
   Google Scholar
• Jansons, K. M., & Lythe, G. (2000). Efficient numerical solution of stochastic differential equations using exponential timestepping. Journal of Statistical Physics, 100(5/6), 1097–1109. https://doi.org/10.1023/A:1018711024740
   Google Scholar
• Kloeden, P. E., & Platen, E. (1992). Stochastic differential equations. In Numerical solution of stochastic differential equations (pp. 103–160). Springer.
   Google Scholar
• Li, M., & Deng, F. (2017). Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with Lévy noise. Nonlinear Analysis: Hybrid Systems, 24(2), 171–185. https://doi.org/10.1016/j.nahs.2017.01.001
   Google Scholar
• Liu, K., & Mao, X. (1998). Exponential stability of non-linear stochastic evolution equations. Stochastic Processes and Their Applications, 78(2), 173–193. https://doi.org/10.1016/S0304-4149(98)00048-9
   Web of Science ®Google Scholar
• Liu, L., Deng, F., & Hou, T. (2018). Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition. Applied Mathematics and Computation, 330, 201–212. https://doi.org/10.1016/j.amc.2018.02.031
   Web of Science ®Google Scholar
• Liu, W., & Mao, X. (2017). Almost sure stability of the Euler–Maruyama method with random variable stepsize for stochastic differential equations. Numerical Algorithms, 74(2), 573–592. https://doi.org/10.1007/s11075-016-0162-3
   Google Scholar
• Malham, S. J., & Wiese, A. (2010). An introduction to SDE simulation. Preprint arXiv:1004.0646.
   Google Scholar
• Mao, X. (1991). Almost sure exponential stability for delay stochastic differential equations with respect to semimartingales. Stochastic Analysis and Applications, 9(2), 177–194. https://doi.org/10.1080/07362999108809233
   Web of Science ®Google Scholar
• Mao, X. (2007). Stochastic differential equations and applications. Elsevier.
   Google Scholar
• Mao, X., & Szpruch, L. (2013). Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients. Journal of Computational and Applied Mathematics, 238(3), 14–28. https://doi.org/10.1016/j.cam.2012.08.015
   Google Scholar
• Römisch, W., & Winkler, R. (2006). Stepsize control for mean-square numerical methods for stochastic differential equations with small noise. SIAM Journal on Scientific Computing, 28(2), 604–625. https://doi.org/10.1137/030601429
   Google Scholar
• Rouz, O. F., Ahmadian, D., Akbarfam, A. J., & Milev, M. (2017). A note on the almost sure exponential stability of the Milstein method for stochastic delay differential equations with jumps. International Journal of Pure and Apllied Mathematics, 116(1), 201–216. https://doi.org/10.12732/ijpam.v116i1.22
   Google Scholar
• Tang, J., Liu, W., & Shen, Y. (2019). Stability of the drift-randomized Milstein method for a class of non-autonomous stochastic differential equations. In 2019 Chinese Control Conference (CCC) (pp. 1351–1355). IEEE.
   Google Scholar
• Taniguchi, T. (2007). The exponential stability for stochastic delay partial differential equations. Journal of Mathematical Analysis and Applications, 331(1), 191–205. https://doi.org/10.1016/j.jmaa.2006.08.055
   Google Scholar
• Wang, X., Gan, S., & Wang, D. (2012). A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise. BIT Numerical Mathematics, 52(3), 741–772. https://doi.org/10.1007/s10543-012-0370-8
   Google Scholar
• Wang, Z., & Zhang, C. (2006). An analysis of stability of Milstein method for stochastic differential equations with delay. Computers & Mathematics with Applications, 51(9–10), 1445–1452. https://doi.org/10.1016/j.camwa.2006.01.004
   Google Scholar
• Wu, F., Mao, X., & Kloeden, P. E. (2011). Almost sure exponential stability of the Euler–Maruyama approximations for stochastic functional differential equations. Random Operators and Stochastic Equations, 19, 165–186. https://doi.org/10.1515/ROSE.2011.010
   Google Scholar
• Wu, F., Mao, X., & Szpruch, L. (2010). Almost sure exponential stability of numerical solutions for stochastic delay differential equations. Numerische Mathematik, 115(4), 681–697. https://doi.org/10.1007/s00211-010-0294-7
   Web of Science ®Google Scholar
• Zhan, W., Jiang, Y., & Liu, W. (2018). A note on convergence and stability of the truncated Milstein method for stochastic differential equations. Preprint arXiv:1809.05993.
   Google Scholar
• Zhang, T., & Gao, C. (2021). Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise. Discrete & Continuous Dynamical Systems-B. https://doi.org/10.3934/dcdsb.2021204
   Google Scholar
• Zhang, Y., & Wang, P. (2011). Almost sure and moment exponential stability of Milstein methods for stochastic differential equations. Journal of Jilin University (Science Edition), 49(6), 1058–1060. https://doi.org/10.13413/j.cnki.jdxblxb.2011.06.026
   Google Scholar