# American Institute of Mathematical Sciences

## The averaging method for multivalued SDEs with jumps and non-Lipschitz coefficients

 1 College of Information Sciences and Technology, Donghua University, Shanghai, 201620, China 2 School of mathematics and information technology, Jiangsu Second Normal University, Nanjing, 210013, China 3 Department of Applied Mathematics, Donghua University, Shanghai 201620, China 4 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, U.K

* Corresponding author: Liangjian Hu

Received  May 2018 Revised  July 2018 Published  February 2019

Fund Project: The research of W.Mao was supported by the National Natural Science Foundation of China (11401261) and "333 High-level Project" of Jiangsu Province. The research of L.Hu was supported by the National Natural Science Foundation of China (11471071). The research of S.You was supported by the Natural Science Foundation of Shanghai (17ZR1401300). The research of X.Mao was supported by the Leverhulme Trust (RF-2015-385), the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship), the EPSRC (EP/K503174/1)

In this paper, we study the averaging principle for multivalued SDEs with jumps and non-Lipschitz coefficients. By the Bihari's inequality and the properties of the concave function, we prove that the solution of averaged multivalued SDE with jumps converges to that of the standard one in the sense of mean square and also in probability. Finally, two examples are presented to illustrate our theory.

Citation: Wei Mao, Liangjian Hu, Surong You, Xuerong Mao. The averaging method for multivalued SDEs with jumps and non-Lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019039
##### References:
 [1] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. [2] D. D. Bainov and S. D. Milusheva, Justification of the averaging method for a system of functional differential equations with variable structure and impulses, Appl. Math. and Optimization, 16 (1987), 19-36. doi: 10.1007/BF01442183. [3] F. Bernardin, Multivalued stochastic differential equations: Convergence of a numerical scheme, Set-Valued Analysis, 11 (2003), 393-415. doi: 10.1023/A:1025656814701. [4] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problem of differential equations, Acta Math. Acad. Sci. Hungar., 7 (1956), 71-94. doi: 10.1007/BF02022967. [5] E. Cépa, Equations differentielles stochastiques multivoques, in: Seminaire de Probabilites, Lecture Notes in Mathematics, Springer, Berlin, 46 (1995), 86-107. doi: 10.1007/BFb0094202. [6] E. Cépa, Probleme de Skorohod multivoque, Ann. Probab., 26 (1998), 500-532. doi: 10.1214/aop/1022855642. [7] M. Federson and J. G. Mesquita, Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses, Journal of Differential Equations, 255 (2013), 3098-3126. doi: 10.1016/j.jde.2013.07.026. [8] D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, SIAM J. Multiscale Model, Simul., 6 (2007), 577-594. doi: 10.1137/060673345. [9] J. Golec and G. Ladde, Averaging principle and systems of singularly perturbed stochastic differential equations, J. Math. Phys., 31 (1990), 1116-1123. doi: 10.1063/1.528792. [10] R. Guo and B. Pei, Stochastic averaging principles for multi-valued stochastic differential equations driven by Poisson Point Processes, Stochastic Analysis and Applications, 36 (2018), 751-766. doi: 10.1080/07362994.2018.1461567. [11] J. K. Hale, Averaging methods for differential equations with retarded arguments with a small parameter, J. Differential Equations, 2 (1996), 57-73. doi: 10.1016/0022-0396(66)90063-5. [12] R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations, Kibernet, 4 (1968), 260-279. [13] R. Z. Khasminskii and G. Yin, On averaging principles: An asymptotic expansion approach, SIAM J. Math. Anal., 35 (2004), 1534-1560. doi: 10.1137/S0036141002403973. [14] V. G. Kolomiets and A. I. Melnikov, Averaging of stochastic systems of integral-differential equations with Poisson noise, Ukr. Math. J., 43 (1991), 242-246. doi: 10.1007/BF01060515. [15] P. Krée, Diffusion for multivalued stochastic differential equations, J. Funct. Anal., 49 (1982), 73-90. doi: 10.1016/0022-1236(82)90086-6. [16] N. M. Krylov and N. N. Bogolyubov, Les proprietes ergodiques des suites des probabilites en chaine, C. R. Math. Acad. Sci., 204 (1937), 1454-1546. [17] D. Lépingle and C. Marois, Equations différentielles stochastiques multivoques unidimensionnelles, Séminaire de Probabilités XXI, Springer, Berlin, Heidelberg, 1247 (1987), 520-533. doi: 10.1007/BFb0077653. [18] Z. Liang, Existence and pathwise uniqueness of solutions for stochastic differential equations with respect to martingales in the plane, Stochastic Processes and their Applications, 83 (1999), 303-317. doi: 10.1016/S0304-4149(99)00040-X. [19] W. Liu and M. Stephan, Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple, Journal of Mathematical Analysis and Applications, 410 (2014), 158-178. doi: 10.1016/j.jmaa.2013.08.016. [20] C. Marois, Equations differentielles stochastiques multivoques discontinues avec frontiere mobile, Stochastics., 30 (1990), 105-121. doi: 10.1080/17442509008833636. [21] L. Maticiuc, A. Rascanu and L. Slominski, Multivalued monotone stochastic differential equations with jumps, Stochastics and Dynamics, 17 (2017), 1750018, 25 pp. doi: 10.1142/S0219493717500186. [22] K. Matthies, Time-averaging under fast periodic forcing of parabolic partial differential equations: Exponential estimates, J. Differ. Equ., 174 (2011), 133-180. doi: 10.1006/jdeq.2000.3934. [23] L. Ngoran and N. Z. Modeste, Averaging principle for multivalued stochastic differential equations, Random Operators and Stochastic Equations, 9 (2001), 399-407. doi: 10.1515/rose.2001.9.4.399. [24] P. H. Protter, Stochastic Integration and Differential Equations, second ed., Applications of Mathematics, Springer-Verlag, Berlin., 2004. [25] J. Ren and S. Xu, A transfer principle for multivalued stochastic differential equations, Journal of Functional Analysis, 256 (2009), 2780-2814. doi: 10.1016/j.jfa.2008.09.016. [26] Y. Ren, J. Wang and L. Hu, Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems, International Journal of Control., 90 (2017), 1132-1154. doi: 10.1080/00207179.2016.1204560. [27] J. Ren and J. Wu, Multi-valued Stochastic Differential Equations Driven by Poisson Point Processes, Stochastic Analysis with Financial Applications, Springer, 65 (2011), 191-205. doi: 10.1007/978-3-0348-0097-6_13. [28] A. Y. Veretennikov, On the averaging principle for systems of stochastic differential equations, Math. USSR-Sb., 69 (1991), 271-284. doi: 10.1070/SM1991v069n01ABEH001237. [29] V. M. Volosov, Averaging in systems of ordinary differential equations, Russian. Math. Surveys, 17 (1962), 1-126. [30] J. Wu, Uniform large deviations for multivalued stochastic differential equations with Poisson jumps, Kyoto Journal of Mathematics, 51 (2011), 535-559. doi: 10.1215/21562261-1299891. [31] Y. Xu, J. Q. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D., 240 (2011), 1395-1401. doi: 10.1016/j.physd.2011.06.001. [32] Y. Xu, B. Pei and J. L. Wu, Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stochastics and Dynamics, 17 (2017), 1750013, 16 pp. doi: 10.1142/S0219493717500137. [33] Y. Xu, B. Pei and R. Guo, Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst., Ser. B., 20 (2015), 2257-2267. doi: 10.3934/dcdsb.2015.20.2257. [34] J. Xu and J. Liu, An averaging principle for multivalued stochastic differential equations, Stoch. Anal. Appl., 32 (2014), 962-974. doi: 10.1080/07362994.2014.959594. [35] Y. Xu, B. Pei and Y. Li, Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Mathematical Methods in the Applied Sciences, 38 (2015), 2120-2131. doi: 10.1002/mma.3208. [36] Y. Xu, B. Pei and G. Guo, Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise, Applied Mathematics and Computation, 263 (2015), 398-409. doi: 10.1016/j.amc.2015.04.070. [37] G. Yin and K. M. Ramachandran, A differential delay equation with wideband noise perturbation, Stochastic Processes and Their Applications, 35 (1990), 231-249. doi: 10.1016/0304-4149(90)90004-C. [38] A. Zalinescu, Stochastic variational inequalities with jumps, Stochastic Processes and their Applications, 124 (2014), 785-811. doi: 10.1016/j.spa.2013.09.005. [39] H. Zhang, Strong convergence rate for multivalued stochastic differential equations via stochastic theta method, Stochastics, 90 (2018), 762-781. doi: 10.1080/17442508.2017.1416117. [40] X. Zhang, Skorohod problem and multivalued stochastic evolution equations in Banach spaces, Bulletin des sciences mathematiques, 131 (2007), 175-217. doi: 10.1016/j.bulsci.2006.05.009.

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##### References:
 [1] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. [2] D. D. Bainov and S. D. Milusheva, Justification of the averaging method for a system of functional differential equations with variable structure and impulses, Appl. Math. and Optimization, 16 (1987), 19-36. doi: 10.1007/BF01442183. [3] F. Bernardin, Multivalued stochastic differential equations: Convergence of a numerical scheme, Set-Valued Analysis, 11 (2003), 393-415. doi: 10.1023/A:1025656814701. [4] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problem of differential equations, Acta Math. Acad. Sci. Hungar., 7 (1956), 71-94. doi: 10.1007/BF02022967. [5] E. Cépa, Equations differentielles stochastiques multivoques, in: Seminaire de Probabilites, Lecture Notes in Mathematics, Springer, Berlin, 46 (1995), 86-107. doi: 10.1007/BFb0094202. [6] E. Cépa, Probleme de Skorohod multivoque, Ann. Probab., 26 (1998), 500-532. doi: 10.1214/aop/1022855642. [7] M. Federson and J. G. Mesquita, Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses, Journal of Differential Equations, 255 (2013), 3098-3126. doi: 10.1016/j.jde.2013.07.026. [8] D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, SIAM J. Multiscale Model, Simul., 6 (2007), 577-594. doi: 10.1137/060673345. [9] J. Golec and G. Ladde, Averaging principle and systems of singularly perturbed stochastic differential equations, J. Math. Phys., 31 (1990), 1116-1123. doi: 10.1063/1.528792. [10] R. Guo and B. Pei, Stochastic averaging principles for multi-valued stochastic differential equations driven by Poisson Point Processes, Stochastic Analysis and Applications, 36 (2018), 751-766. doi: 10.1080/07362994.2018.1461567. [11] J. K. Hale, Averaging methods for differential equations with retarded arguments with a small parameter, J. Differential Equations, 2 (1996), 57-73. doi: 10.1016/0022-0396(66)90063-5. [12] R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations, Kibernet, 4 (1968), 260-279. [13] R. Z. Khasminskii and G. Yin, On averaging principles: An asymptotic expansion approach, SIAM J. Math. Anal., 35 (2004), 1534-1560. doi: 10.1137/S0036141002403973. [14] V. G. Kolomiets and A. I. Melnikov, Averaging of stochastic systems of integral-differential equations with Poisson noise, Ukr. Math. J., 43 (1991), 242-246. doi: 10.1007/BF01060515. [15] P. Krée, Diffusion for multivalued stochastic differential equations, J. Funct. Anal., 49 (1982), 73-90. doi: 10.1016/0022-1236(82)90086-6. [16] N. M. Krylov and N. N. Bogolyubov, Les proprietes ergodiques des suites des probabilites en chaine, C. R. Math. Acad. Sci., 204 (1937), 1454-1546. [17] D. Lépingle and C. Marois, Equations différentielles stochastiques multivoques unidimensionnelles, Séminaire de Probabilités XXI, Springer, Berlin, Heidelberg, 1247 (1987), 520-533. doi: 10.1007/BFb0077653. [18] Z. Liang, Existence and pathwise uniqueness of solutions for stochastic differential equations with respect to martingales in the plane, Stochastic Processes and their Applications, 83 (1999), 303-317. doi: 10.1016/S0304-4149(99)00040-X. [19] W. Liu and M. Stephan, Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple, Journal of Mathematical Analysis and Applications, 410 (2014), 158-178. doi: 10.1016/j.jmaa.2013.08.016. [20] C. Marois, Equations differentielles stochastiques multivoques discontinues avec frontiere mobile, Stochastics., 30 (1990), 105-121. doi: 10.1080/17442509008833636. [21] L. Maticiuc, A. Rascanu and L. Slominski, Multivalued monotone stochastic differential equations with jumps, Stochastics and Dynamics, 17 (2017), 1750018, 25 pp. doi: 10.1142/S0219493717500186. [22] K. Matthies, Time-averaging under fast periodic forcing of parabolic partial differential equations: Exponential estimates, J. Differ. Equ., 174 (2011), 133-180. doi: 10.1006/jdeq.2000.3934. [23] L. Ngoran and N. Z. Modeste, Averaging principle for multivalued stochastic differential equations, Random Operators and Stochastic Equations, 9 (2001), 399-407. doi: 10.1515/rose.2001.9.4.399. [24] P. H. Protter, Stochastic Integration and Differential Equations, second ed., Applications of Mathematics, Springer-Verlag, Berlin., 2004. [25] J. Ren and S. Xu, A transfer principle for multivalued stochastic differential equations, Journal of Functional Analysis, 256 (2009), 2780-2814. doi: 10.1016/j.jfa.2008.09.016. [26] Y. Ren, J. Wang and L. Hu, Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems, International Journal of Control., 90 (2017), 1132-1154. doi: 10.1080/00207179.2016.1204560. [27] J. Ren and J. Wu, Multi-valued Stochastic Differential Equations Driven by Poisson Point Processes, Stochastic Analysis with Financial Applications, Springer, 65 (2011), 191-205. doi: 10.1007/978-3-0348-0097-6_13. [28] A. Y. Veretennikov, On the averaging principle for systems of stochastic differential equations, Math. USSR-Sb., 69 (1991), 271-284. doi: 10.1070/SM1991v069n01ABEH001237. [29] V. M. Volosov, Averaging in systems of ordinary differential equations, Russian. Math. Surveys, 17 (1962), 1-126. [30] J. Wu, Uniform large deviations for multivalued stochastic differential equations with Poisson jumps, Kyoto Journal of Mathematics, 51 (2011), 535-559. doi: 10.1215/21562261-1299891. [31] Y. Xu, J. Q. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D., 240 (2011), 1395-1401. doi: 10.1016/j.physd.2011.06.001. [32] Y. Xu, B. Pei and J. L. Wu, Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion, Stochastics and Dynamics, 17 (2017), 1750013, 16 pp. doi: 10.1142/S0219493717500137. [33] Y. Xu, B. Pei and R. Guo, Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion, Discrete Contin. Dyn. Syst., Ser. B., 20 (2015), 2257-2267. doi: 10.3934/dcdsb.2015.20.2257. [34] J. Xu and J. Liu, An averaging principle for multivalued stochastic differential equations, Stoch. Anal. Appl., 32 (2014), 962-974. doi: 10.1080/07362994.2014.959594. [35] Y. Xu, B. Pei and Y. Li, Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Mathematical Methods in the Applied Sciences, 38 (2015), 2120-2131. doi: 10.1002/mma.3208. [36] Y. Xu, B. Pei and G. Guo, Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise, Applied Mathematics and Computation, 263 (2015), 398-409. doi: 10.1016/j.amc.2015.04.070. [37] G. Yin and K. M. Ramachandran, A differential delay equation with wideband noise perturbation, Stochastic Processes and Their Applications, 35 (1990), 231-249. doi: 10.1016/0304-4149(90)90004-C. [38] A. Zalinescu, Stochastic variational inequalities with jumps, Stochastic Processes and their Applications, 124 (2014), 785-811. doi: 10.1016/j.spa.2013.09.005. [39] H. Zhang, Strong convergence rate for multivalued stochastic differential equations via stochastic theta method, Stochastics, 90 (2018), 762-781. doi: 10.1080/17442508.2017.1416117. [40] X. Zhang, Skorohod problem and multivalued stochastic evolution equations in Banach spaces, Bulletin des sciences mathematiques, 131 (2007), 175-217. doi: 10.1016/j.bulsci.2006.05.009.
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