# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022098
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## Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation

 1 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100049, China 2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Yulan Lu

Received  February 2020 Revised  November 2021 Early access June 2022

Fund Project: This work is funded by National Natural Science Foundation of China (No. 11971470, No. 11871068, No. 12031020, No. 12022118, No. 12026428), by Youth Innovation Promotion Association CAS, and by National Postdoctoral Program for Innovative Talents (No. BX20180347)

For the stochastic differential equation with piecewise continuous arguments, multiplicative noises and dissipative drift coefficients, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to numerically preserve the invariant measure, we apply the backward Euler method to the equation, and prove that the numerical solution at integer time is also a Markov chain and possesses a unique numerical invariant measure. By establishing several a priori estimations, we present the time-independent weak error analysis for the method via Malliavin calculus, which implies that the numerical invariant measure converges to the original one with weak order 1. Numerical experiments verify the theoretical analysis.

Citation: Chuchu Chen, Jialin Hong, Yulan Lu. Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022098
##### References:
 [1] A. Abdulle, G. Vilmart and K. C. Zygalakis, High order numerical approximation of the invariant measure of ergodic SDEs, SIAM J. Numer. Anal., 52 (2014), 1600-1622.  doi: 10.1137/130935616. [2] J. Bao, G. Yin and C. Yuan, Ergodicity for functional stochastic differential equations and applications, Nonlinear Anal., 98 (2014), 66-82.  doi: 10.1016/j.na.2013.12.001. [3] C.-E. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Anal., 40 (2014), 1-40.  doi: 10.1007/s11118-013-9338-9. [4] C. Chen, J. Hong and X. Wang, Approximation of invariant measure for damped stochastic nonlinear Schrödinger equation via an ergodic numerical scheme, Potential Anal., 46 (2017), 323-367.  doi: 10.1007/s11118-016-9583-9. [5] J. Cui, J. Hong and L. Sun, Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE, Stochastic Process. Appl., 134 (2021), 55-93.  doi: 10.1016/j.spa.2020.12.003. [6] G. Da Prato, An Introduction to Infinite-Dimensional Analysis, Universitext, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-29021-4. [7] L. Dai, Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812818515. [8] D. Gusak, A. Kukush, A. Kulik, Y. Mishura and A. Pilipenko, Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory, Springer, New York, 2010. doi: 10.1007/978-0-387-87862-1. [9] J. Hong, L. Sun and X. Wang, High order conformal symplectic and ergodic schemes for the stochastic Langevin equation via generating functions, SIAM J. Numer. Anal., 55 (2017), 3006-3029.  doi: 10.1137/17M111691X. [10] J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations: Numerical Approximations and Symplectic Structures, Lecture Notes in Mathematics, 2251, Springer, Singapore, 2019. doi: 10.1007/978-981-32-9069-3. [11] K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705. [12] L. Liu, A. Wu, Z. Zeng and T. Huang, Global mean square exponential stability of stochastic neural networks with retarded and advanced argument, Neurocomputing, 247 (2017), 156-164.  doi: 10.1016/j.neucom.2017.03.057. [13] Y. L. Lu, M. H. Song and M. Z. Liu, Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, J. Comput. Appl. Math., 317 (2017), 55-71.  doi: 10.1016/j.cam.2016.11.033. [14] Y. Lu, M. Song and M. Liu, Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 695-717.  doi: 10.3934/dcdsb.2018203. [15] X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Trans. Automat. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696. [16] J. C. Mattingly, A. M. Stuart and M. V. Tretyakov, Convergence of numerical time-averaging and stationary measures via Poisson equations, SIAM J. Numer. Anal., 48 (2010), 552-577.  doi: 10.1137/090770527. [17] M. Milošević, The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments, J. Comput. Appl. Math., 298 (2016), 1-12.  doi: 10.1016/j.cam.2015.11.019. [18] D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin, 2006. [19] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York-London 1970. [20] I. Ozturk and F. Bozkurt, Stability analysis of a population model with piecewise constant arguments, Nonlinear Anal. Real World Appl., 12 (2011), 1532-1545.  doi: 10.1016/j.nonrwa.2010.10.011. [21] J. Ren, J. Wu and X. Zhang, Exponential ergodicity of non-Lipschitz multivalued stochastic differential equations, Bull. Sci. Math., 134 (2010), 391-404.  doi: 10.1016/j.bulsci.2009.01.003. [22] D. Talay, Second-order discretization schemes of stochastic differential systems for the computation of the invariant law, Rapports de Recherche, Institut National de Recherche en Informatique et en Automatique, (1987), 13-26.  doi: 10.1080/17442509008833606. [23] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. doi: 10.1142/1860. [24] F. Wu, G. Yin and H. Mei, Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity, J. Differential Equations, 262 (2017), 1226-1252.  doi: 10.1016/j.jde.2016.10.006. [25] Y. Xie and C. Zhang, A class of stochastic one-parameter methods for nonlinear SFDEs with piecewise continuous arguments, Appl. Numer. Math., 135 (2019), 1-14.  doi: 10.1016/j.apnum.2018.08.007.

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##### References:
 [1] A. Abdulle, G. Vilmart and K. C. Zygalakis, High order numerical approximation of the invariant measure of ergodic SDEs, SIAM J. Numer. Anal., 52 (2014), 1600-1622.  doi: 10.1137/130935616. [2] J. Bao, G. Yin and C. Yuan, Ergodicity for functional stochastic differential equations and applications, Nonlinear Anal., 98 (2014), 66-82.  doi: 10.1016/j.na.2013.12.001. [3] C.-E. Bréhier, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Anal., 40 (2014), 1-40.  doi: 10.1007/s11118-013-9338-9. [4] C. Chen, J. Hong and X. Wang, Approximation of invariant measure for damped stochastic nonlinear Schrödinger equation via an ergodic numerical scheme, Potential Anal., 46 (2017), 323-367.  doi: 10.1007/s11118-016-9583-9. [5] J. Cui, J. Hong and L. Sun, Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE, Stochastic Process. Appl., 134 (2021), 55-93.  doi: 10.1016/j.spa.2020.12.003. [6] G. Da Prato, An Introduction to Infinite-Dimensional Analysis, Universitext, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-29021-4. [7] L. Dai, Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. doi: 10.1142/9789812818515. [8] D. Gusak, A. Kukush, A. Kulik, Y. Mishura and A. Pilipenko, Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory, Springer, New York, 2010. doi: 10.1007/978-0-387-87862-1. [9] J. Hong, L. Sun and X. Wang, High order conformal symplectic and ergodic schemes for the stochastic Langevin equation via generating functions, SIAM J. Numer. Anal., 55 (2017), 3006-3029.  doi: 10.1137/17M111691X. [10] J. Hong and X. Wang, Invariant Measures for Stochastic Nonlinear Schrödinger Equations: Numerical Approximations and Symplectic Structures, Lecture Notes in Mathematics, 2251, Springer, Singapore, 2019. doi: 10.1007/978-981-32-9069-3. [11] K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705. [12] L. Liu, A. Wu, Z. Zeng and T. Huang, Global mean square exponential stability of stochastic neural networks with retarded and advanced argument, Neurocomputing, 247 (2017), 156-164.  doi: 10.1016/j.neucom.2017.03.057. [13] Y. L. Lu, M. H. Song and M. Z. Liu, Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, J. Comput. Appl. Math., 317 (2017), 55-71.  doi: 10.1016/j.cam.2016.11.033. [14] Y. Lu, M. Song and M. Liu, Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 695-717.  doi: 10.3934/dcdsb.2018203. [15] X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Trans. Automat. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696. [16] J. C. Mattingly, A. M. Stuart and M. V. Tretyakov, Convergence of numerical time-averaging and stationary measures via Poisson equations, SIAM J. Numer. Anal., 48 (2010), 552-577.  doi: 10.1137/090770527. [17] M. Milošević, The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments, J. Comput. Appl. Math., 298 (2016), 1-12.  doi: 10.1016/j.cam.2015.11.019. [18] D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin, 2006. [19] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York-London 1970. [20] I. Ozturk and F. Bozkurt, Stability analysis of a population model with piecewise constant arguments, Nonlinear Anal. Real World Appl., 12 (2011), 1532-1545.  doi: 10.1016/j.nonrwa.2010.10.011. [21] J. Ren, J. Wu and X. Zhang, Exponential ergodicity of non-Lipschitz multivalued stochastic differential equations, Bull. Sci. Math., 134 (2010), 391-404.  doi: 10.1016/j.bulsci.2009.01.003. [22] D. Talay, Second-order discretization schemes of stochastic differential systems for the computation of the invariant law, Rapports de Recherche, Institut National de Recherche en Informatique et en Automatique, (1987), 13-26.  doi: 10.1080/17442509008833606. [23] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. doi: 10.1142/1860. [24] F. Wu, G. Yin and H. Mei, Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity, J. Differential Equations, 262 (2017), 1226-1252.  doi: 10.1016/j.jde.2016.10.006. [25] Y. Xie and C. Zhang, A class of stochastic one-parameter methods for nonlinear SFDEs with piecewise continuous arguments, Appl. Numer. Math., 135 (2019), 1-14.  doi: 10.1016/j.apnum.2018.08.007.
The expectations and variances of $X(t)$ and $X(k)$
The expectations (left column) and variances (right column) of $X(t)$ with different $\theta_2$
Order of weak convergence of BE method
The evolution of $\mathbb{E}\phi(Y_k)$ started from different initial data
Order of weak convergence of the BE method
The evolution of $\mathbb{E}\phi(Y_k)$ started from different initial data
Order of weak convergence of the BE method
The evolution of $\mathbb{E}\phi(Y_k)$ started from different initial data
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