September  2019, 24(9): 4937-4954. doi: 10.3934/dcdsb.2019039

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, 2019, 24 (9) : 4937-4954. 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.  Google Scholar
[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.  Google Scholar

[3]

F. Bernardin, Multivalued stochastic differential equations: Convergence of a numerical scheme, Set-Valued Analysis, 11 (2003), 393-415.  doi: 10.1023/A:1025656814701.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. Cépa, Probleme de Skorohod multivoque, Ann. Probab., 26 (1998), 500-532.  doi: 10.1214/aop/1022855642.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations, Kibernet, 4 (1968), 260-279.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Krée, Diffusion for multivalued stochastic differential equations, J. Funct. Anal., 49 (1982), 73-90.  doi: 10.1016/0022-1236(82)90086-6.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

C. Marois, Equations differentielles stochastiques multivoques discontinues avec frontiere mobile, Stochastics., 30 (1990), 105-121.  doi: 10.1080/17442509008833636.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

P. H. Protter, Stochastic Integration and Differential Equations, second ed., Applications of Mathematics, Springer-Verlag, Berlin., 2004.  Google Scholar

[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.  Google Scholar

[26]

Y. RenJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

V. M. Volosov, Averaging in systems of ordinary differential equations, Russian. Math. Surveys, 17 (1962), 1-126.   Google Scholar

[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.  Google Scholar

[31]

Y. XuJ. 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.  Google Scholar

[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.  Google Scholar

[33]

Y. XuB. 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.  Google Scholar

[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.  Google Scholar

[35]

Y. XuB. 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.  Google Scholar

[36]

Y. XuB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984.  doi: 10.1007/978-3-642-69512-4.  Google Scholar
[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.  Google Scholar

[3]

F. Bernardin, Multivalued stochastic differential equations: Convergence of a numerical scheme, Set-Valued Analysis, 11 (2003), 393-415.  doi: 10.1023/A:1025656814701.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. Cépa, Probleme de Skorohod multivoque, Ann. Probab., 26 (1998), 500-532.  doi: 10.1214/aop/1022855642.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations, Kibernet, 4 (1968), 260-279.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Krée, Diffusion for multivalued stochastic differential equations, J. Funct. Anal., 49 (1982), 73-90.  doi: 10.1016/0022-1236(82)90086-6.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

C. Marois, Equations differentielles stochastiques multivoques discontinues avec frontiere mobile, Stochastics., 30 (1990), 105-121.  doi: 10.1080/17442509008833636.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

P. H. Protter, Stochastic Integration and Differential Equations, second ed., Applications of Mathematics, Springer-Verlag, Berlin., 2004.  Google Scholar

[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.  Google Scholar

[26]

Y. RenJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

V. M. Volosov, Averaging in systems of ordinary differential equations, Russian. Math. Surveys, 17 (1962), 1-126.   Google Scholar

[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.  Google Scholar

[31]

Y. XuJ. 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.  Google Scholar

[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.  Google Scholar

[33]

Y. XuB. 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.  Google Scholar

[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.  Google Scholar

[35]

Y. XuB. 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.  Google Scholar

[36]

Y. XuB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3299-3318. doi: 10.3934/dcdsb.2018321

[2]

Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233

[3]

Mikhail Krastanov, Michael Malisoff, Peter Wolenski. On the strong invariance property for non-Lipschitz dynamics. Communications on Pure & Applied Analysis, 2006, 5 (1) : 107-124. doi: 10.3934/cpaa.2006.5.107

[4]

Boris Hasselblatt and Amie Wilkinson. Prevalence of non-Lipschitz Anosov foliations. Electronic Research Announcements, 1997, 3: 93-98.

[5]

Alexander Veretennikov. On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 523-549. doi: 10.3934/dcdsb.2013.18.523

[6]

Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698

[7]

Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119

[8]

Peng Gao, Yong Li. Averaging principle for the Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089

[9]

Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245

[10]

Janusz Mierczyński. Averaging in random systems of nonnegative matrices. Conference Publications, 2015, 2015 (special) : 835-840. doi: 10.3934/proc.2015.0835

[11]

Yaofeng Su. Almost surely invariance principle for non-stationary and random intermittent dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6585-6597. doi: 10.3934/dcds.2019286

[12]

Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic & Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669

[13]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035

[14]

Jérôme Coville, Nicolas Dirr, Stephan Luckhaus. Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients. Networks & Heterogeneous Media, 2010, 5 (4) : 745-763. doi: 10.3934/nhm.2010.5.745

[15]

Janusz Mierczyński, Wenxian Shen. Time averaging for nonautonomous/random linear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 661-699. doi: 10.3934/dcdsb.2008.9.661

[16]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216

[17]

Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197

[18]

Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247

[19]

Guillaume Bal, Olivier Pinaud. Self-averaging of kinetic models for waves in random media. Kinetic & Related Models, 2008, 1 (1) : 85-100. doi: 10.3934/krm.2008.1.85

[20]

Gechun Liang, Wei Wei. Optimal switching at Poisson random intervention times. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1483-1505. doi: 10.3934/dcdsb.2016008

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (320)
  • HTML views (447)
  • Cited by (0)

Other articles
by authors

[Back to Top]