November  2020, 25(11): 4493-4513. doi: 10.3934/dcdsb.2020109

Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps

1. 

School of Mathematics, Southeast University, Nanjing 211189, China

2. 

Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 211189, China

* Corresponding author: Jinde Cao

Received  January 2019 Revised  November 2019 Published  March 2020

Fund Project: This work was jointly supported by the Key Project of National Science Foundation of China under Grant No. 61833005

The main aim of this article is to examine almost sure exponential stabilization and suppression of nonlinear systems by periodically intermittent stochastic perturbation with jumps. On the one hand, some sufficient criteria ensure almost sure stabilization of the unstable deterministic system by applying exponential martingale inequality with jumps. On the other hand, sufficient conditions of destabilization are provided under which the system is stable by the well-known strong law of large numbers of local martingale and Poisson process. Both the sample Lyapunov exponents are closely related to the control period $ T $ and noise width $ \theta $. As for applications, the well-known Lorenz chaotic systems and nonlinear Liénard equation with jumps are discussed. Finally, two simulation examples demonstrating the effectiveness of the results are provided.

Citation: Wensheng Yin, Jinde Cao. Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4493-4513. doi: 10.3934/dcdsb.2020109
References:
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2009.  doi: 10.1017/CBO9780511755323.  Google Scholar
[2]

D. Applebaum, Extending stochastic resonance for neuron models to general Lévy noise, IEEE Trans. Neural Netw., 20 (2009), 1993-1995.  doi: 10.1109/TNN.2009.2033183.  Google Scholar

[3]

D. Applebaum and M. Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Probab., 46 (2009), 1116-1129.  doi: 10.1239/jap/1261670692.  Google Scholar

[4]

D. Applebaum and M. Siakalli, Stochastic stabilization of dynamical systems using Lévy noise, Stoch. Dyn., 10 (2010), 509-527.  doi: 10.1142/S0219493710003066.  Google Scholar

[5]

A. D. ApplebyX. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Tran. Automat. Control, 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[6]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203485217.  Google Scholar

[7]

F. DengQ. Luo and X. Mao, Stochastic stabilization of hybrid differential equations, Automatica J. IFAC, 48 (2012), 2321-2328.  doi: 10.1016/j.automatica.2012.06.044.  Google Scholar

[8]

Y. FengX. YangQ. Song and J. Cao, Synchronization of memristive neural networks with mixed delays via quantized intermittent control, Appl. Math. Comput., 339 (2018), 874-887.  doi: 10.1016/j.amc.2018.08.009.  Google Scholar

[9]

R. Fernholz and I. Karatzas, Relative arbitrage in volatility-stabilized markets, Ann. Financ., 1 (2005), 149-177.  doi: 10.1007/s10436-004-0011-6.  Google Scholar

[10]

H. Gao and Y. Wang, Stochastic mutualism model under regime switching with Lévy jumps, Phys. A, 515 (2019), 355-375.  doi: 10.1016/j.physa.2018.09.189.  Google Scholar

[11]

B. GuoY. WuY. Xiao and C. Zhang, Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control, Appl. Math. Comput., 331 (2018), 341-357.  doi: 10.1016/j.amc.2018.03.020.  Google Scholar

[12]

R. Hasminskii, Stochastic Stability of Differential Equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[13]

N. Li and J. Cao, Intermittent control on switched networks via $\omega$-matrix measure method, Nonlinear Dynam., 77 (2014), 1363-1375.  doi: 10.1007/s11071-014-1385-2.  Google Scholar

[14]

S. Li, J. Cao and Y. He, Pinning controllability scheme of directed complex delayed dynamical networks via periodically intermittent control, Discrete Dyn. Nat. Soc., (2016), 10 pp. doi: 10.1155/2016/1585928.  Google Scholar

[15]

C. LiZ. Dong and R. Situ, Almost sure stability of linear stochastic differential equations with jumps, Probab. Theory Related Fields, 123 (2002), 121-155.  doi: 10.1007/s004400200198.  Google Scholar

[16]

L. LiuY. Shen and F. Jiang, The almost sure asymptotic stability and $p$th moment asymptotic stability of nonlinear stochastic differential systems with polynomial growth, IEEE Trans. Automat. Control, 56 (2011), 1985-1990.  doi: 10.1109/TAC.2011.2146970.  Google Scholar

[17]

L. Liu and Y. Shen, Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition, Automatica J. IFAC, 48 (2012), 619-624.  doi: 10.1016/j.automatica.2012.01.022.  Google Scholar

[18]

X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[19]

X. MaoG. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica J. IFAC, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar

[20]

X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Tran. Automat. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.  Google Scholar

[21]

L. Pan and J. Cao, Stochastic quasi-synchronization for delayed dynamical networks via intermittent control, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1332-1343.  doi: 10.1016/j.cnsns.2011.07.010.  Google Scholar

[22]

A. Patel and B. Kosko, Stochastic resonance in continuous and spiking neuron models with Lévy noise, IEEE Trans. Neural Netw., 19 (2008), 1993-2008.  doi: 10.1109/TNN.2008.2005610.  Google Scholar

[23]

M. Scheutzow, Stabilisation and destabilisation by noise in the plane, Stochastic Anal. Appl., 11 (1993), 97-113.  doi: 10.1080/07362999308809304.  Google Scholar

[24]

M. Siakalli, Stability Properties of Stochastic Differential Equations Driven by Lévy Noise, Ph.D thesis, University of Sheffield, 2009. Google Scholar

[25]

Y. Wan and J. Cao, Distributed robust stabilization of linear multi-agent systems with intermittent control, J. Franklin Inst., 352 (2015), 4515-4527.  doi: 10.1016/j.jfranklin.2015.06.024.  Google Scholar

[26]

P. WangY. Hong and H. Su, Stabilization of stochastic complex-valued coupled delayed systems with Markovian switching via periodically intermittent control, Nonlinear Anal. Hybrid Syst., 29 (2018), 395-413.  doi: 10.1016/j.nahs.2018.03.006.  Google Scholar

[27]

F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control, 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108.  Google Scholar

[28]

F. WuG. Yin and Z. Jin, Kolmogorov-type systems with regime-switching jump diffusion perturbations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2293-2319.  doi: 10.3934/dcdsb.2016048.  Google Scholar

[29]

Y. XuX. WangH. Zhang and W. Xu, Stochastic stability for nonlinear systems driven by Lévy noise, Nonlinear Dynam., 68 (2012), 7-15.  doi: 10.1007/s11071-011-0199-8.  Google Scholar

[30]

Y. Xu, H. Zhou and W. Li, Stabilization of stochastic delayed systems with Lévy noise on networks via periodically intermittent control, Internat. J. Control, (2018). doi: 10.1080/00207179.2018.1479538.  Google Scholar

[31]

G. YinY. Talafha and F. Xi, Stochastic Liénard equations with random switching and two-time scales, Comm. Statist. Theory Methods, 43 (2014), 1533-1547.  doi: 10.1080/03610926.2012.741741.  Google Scholar

[32]

Q. Zhu, Asymptotic stability in the $p$th moment for stochastic differential equations with Lévy noise, J. Math. Anal. Appl., 416 (2014), 126-142.  doi: 10.1016/j.jmaa.2014.02.016.  Google Scholar

[33]

Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching, Internat. J. Control, 90 (2017), 1703-1712.  doi: 10.1080/00207179.2016.1219069.  Google Scholar

[34]

Q. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Systems Control Lett., 118 (2018), 62-68.  doi: 10.1016/j.sysconle.2018.05.015.  Google Scholar

[35]

B. ZhangF. DengS. Peng and S. Xie, Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system, J. Franklin Inst., 355 (2018), 3829-3852.  doi: 10.1016/j.jfranklin.2017.12.033.  Google Scholar

[36]

S. ZhuK. SunS. Zhou and Y. Shi, Stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial, J. Franklin Inst., 354 (2017), 6550-6566.  doi: 10.1016/j.jfranklin.2017.08.007.  Google Scholar

[37]

X. ZongT. Li and J. Zhang, Consensus conditions for continuous-time multi-agent systems with additive and multiplicative measurement noises, SIAM J. Control Optim., 56 (2018), 19-52.  doi: 10.1137/15M1019775.  Google Scholar

[38]

X. ZongF. WuG. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.  Google Scholar

show all references

References:
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2009.  doi: 10.1017/CBO9780511755323.  Google Scholar
[2]

D. Applebaum, Extending stochastic resonance for neuron models to general Lévy noise, IEEE Trans. Neural Netw., 20 (2009), 1993-1995.  doi: 10.1109/TNN.2009.2033183.  Google Scholar

[3]

D. Applebaum and M. Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Probab., 46 (2009), 1116-1129.  doi: 10.1239/jap/1261670692.  Google Scholar

[4]

D. Applebaum and M. Siakalli, Stochastic stabilization of dynamical systems using Lévy noise, Stoch. Dyn., 10 (2010), 509-527.  doi: 10.1142/S0219493710003066.  Google Scholar

[5]

A. D. ApplebyX. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Tran. Automat. Control, 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[6]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203485217.  Google Scholar

[7]

F. DengQ. Luo and X. Mao, Stochastic stabilization of hybrid differential equations, Automatica J. IFAC, 48 (2012), 2321-2328.  doi: 10.1016/j.automatica.2012.06.044.  Google Scholar

[8]

Y. FengX. YangQ. Song and J. Cao, Synchronization of memristive neural networks with mixed delays via quantized intermittent control, Appl. Math. Comput., 339 (2018), 874-887.  doi: 10.1016/j.amc.2018.08.009.  Google Scholar

[9]

R. Fernholz and I. Karatzas, Relative arbitrage in volatility-stabilized markets, Ann. Financ., 1 (2005), 149-177.  doi: 10.1007/s10436-004-0011-6.  Google Scholar

[10]

H. Gao and Y. Wang, Stochastic mutualism model under regime switching with Lévy jumps, Phys. A, 515 (2019), 355-375.  doi: 10.1016/j.physa.2018.09.189.  Google Scholar

[11]

B. GuoY. WuY. Xiao and C. Zhang, Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control, Appl. Math. Comput., 331 (2018), 341-357.  doi: 10.1016/j.amc.2018.03.020.  Google Scholar

[12]

R. Hasminskii, Stochastic Stability of Differential Equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[13]

N. Li and J. Cao, Intermittent control on switched networks via $\omega$-matrix measure method, Nonlinear Dynam., 77 (2014), 1363-1375.  doi: 10.1007/s11071-014-1385-2.  Google Scholar

[14]

S. Li, J. Cao and Y. He, Pinning controllability scheme of directed complex delayed dynamical networks via periodically intermittent control, Discrete Dyn. Nat. Soc., (2016), 10 pp. doi: 10.1155/2016/1585928.  Google Scholar

[15]

C. LiZ. Dong and R. Situ, Almost sure stability of linear stochastic differential equations with jumps, Probab. Theory Related Fields, 123 (2002), 121-155.  doi: 10.1007/s004400200198.  Google Scholar

[16]

L. LiuY. Shen and F. Jiang, The almost sure asymptotic stability and $p$th moment asymptotic stability of nonlinear stochastic differential systems with polynomial growth, IEEE Trans. Automat. Control, 56 (2011), 1985-1990.  doi: 10.1109/TAC.2011.2146970.  Google Scholar

[17]

L. Liu and Y. Shen, Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition, Automatica J. IFAC, 48 (2012), 619-624.  doi: 10.1016/j.automatica.2012.01.022.  Google Scholar

[18]

X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[19]

X. MaoG. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica J. IFAC, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.  Google Scholar

[20]

X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Tran. Automat. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.  Google Scholar

[21]

L. Pan and J. Cao, Stochastic quasi-synchronization for delayed dynamical networks via intermittent control, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1332-1343.  doi: 10.1016/j.cnsns.2011.07.010.  Google Scholar

[22]

A. Patel and B. Kosko, Stochastic resonance in continuous and spiking neuron models with Lévy noise, IEEE Trans. Neural Netw., 19 (2008), 1993-2008.  doi: 10.1109/TNN.2008.2005610.  Google Scholar

[23]

M. Scheutzow, Stabilisation and destabilisation by noise in the plane, Stochastic Anal. Appl., 11 (1993), 97-113.  doi: 10.1080/07362999308809304.  Google Scholar

[24]

M. Siakalli, Stability Properties of Stochastic Differential Equations Driven by Lévy Noise, Ph.D thesis, University of Sheffield, 2009. Google Scholar

[25]

Y. Wan and J. Cao, Distributed robust stabilization of linear multi-agent systems with intermittent control, J. Franklin Inst., 352 (2015), 4515-4527.  doi: 10.1016/j.jfranklin.2015.06.024.  Google Scholar

[26]

P. WangY. Hong and H. Su, Stabilization of stochastic complex-valued coupled delayed systems with Markovian switching via periodically intermittent control, Nonlinear Anal. Hybrid Syst., 29 (2018), 395-413.  doi: 10.1016/j.nahs.2018.03.006.  Google Scholar

[27]

F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control, 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108.  Google Scholar

[28]

F. WuG. Yin and Z. Jin, Kolmogorov-type systems with regime-switching jump diffusion perturbations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2293-2319.  doi: 10.3934/dcdsb.2016048.  Google Scholar

[29]

Y. XuX. WangH. Zhang and W. Xu, Stochastic stability for nonlinear systems driven by Lévy noise, Nonlinear Dynam., 68 (2012), 7-15.  doi: 10.1007/s11071-011-0199-8.  Google Scholar

[30]

Y. Xu, H. Zhou and W. Li, Stabilization of stochastic delayed systems with Lévy noise on networks via periodically intermittent control, Internat. J. Control, (2018). doi: 10.1080/00207179.2018.1479538.  Google Scholar

[31]

G. YinY. Talafha and F. Xi, Stochastic Liénard equations with random switching and two-time scales, Comm. Statist. Theory Methods, 43 (2014), 1533-1547.  doi: 10.1080/03610926.2012.741741.  Google Scholar

[32]

Q. Zhu, Asymptotic stability in the $p$th moment for stochastic differential equations with Lévy noise, J. Math. Anal. Appl., 416 (2014), 126-142.  doi: 10.1016/j.jmaa.2014.02.016.  Google Scholar

[33]

Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching, Internat. J. Control, 90 (2017), 1703-1712.  doi: 10.1080/00207179.2016.1219069.  Google Scholar

[34]

Q. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Systems Control Lett., 118 (2018), 62-68.  doi: 10.1016/j.sysconle.2018.05.015.  Google Scholar

[35]

B. ZhangF. DengS. Peng and S. Xie, Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system, J. Franklin Inst., 355 (2018), 3829-3852.  doi: 10.1016/j.jfranklin.2017.12.033.  Google Scholar

[36]

S. ZhuK. SunS. Zhou and Y. Shi, Stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial, J. Franklin Inst., 354 (2017), 6550-6566.  doi: 10.1016/j.jfranklin.2017.08.007.  Google Scholar

[37]

X. ZongT. Li and J. Zhang, Consensus conditions for continuous-time multi-agent systems with additive and multiplicative measurement noises, SIAM J. Control Optim., 56 (2018), 19-52.  doi: 10.1137/15M1019775.  Google Scholar

[38]

X. ZongF. WuG. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.  Google Scholar

Figure 1.  The state $ x(t) $ of the system (29)
Figure 2.  The state $ x(t) $ of system (30)
Figure 5.  The exponent dynamical behaviors of state $ x(t) $ of the system (34)
Figure 3.  The state $ x(t) $ of the system (32)
Figure 4.  The state $ x(t) $ of system (33)
Figure 6.  The state $ x(t) $ of system (34)
[1]

Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021009

[2]

Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226

[3]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079

[4]

Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021072

[5]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[6]

Imene Aicha Djebour, Takéo Takahashi, Julie Valein. Feedback stabilization of parabolic systems with input delay. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021027

[7]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004

[8]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026

[9]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[10]

Sel Ly, Nicolas Privault. Stochastic ordering by g-expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 61-98. doi: 10.3934/puqr.2021004

[11]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008

[12]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[13]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208

[14]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[15]

Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039

[16]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[17]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201

[18]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[19]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[20]

Yingxu Tian, Junyi Guo, Zhongyang Sun. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1887-1912. doi: 10.3934/jimo.2020051

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (192)
  • HTML views (278)
  • Cited by (0)

Other articles
by authors

[Back to Top]