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September  2016, 21(7): 2293-2319. doi: 10.3934/dcdsb.2016048

Kolmogorov-type systems with regime-switching jump diffusion perturbations

Fuke Wu 1, , George Yin 2, and  Zhuo Jin 3,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
2. 

Department of Mathematics, Wayne State University, Detroit, Michigan 48202
3. 

Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia

Received  August 2015 Revised  November 2015 Published  August 2016

Population systems are often subject to various different types of environmental noises. This paper considers a class of Kolmogorov-type systems perturbed by three different types of noise including Brownian motions, Markovian switching processes, and Poisson jumps, which is described by a regime-switching jump diffusion process. This paper examines these three different types of noises and determines their effects on the properties of the systems. The properties to be studied include existence and uniqueness of global positive solutions, boundedness of this positive solution, and asymptotic growth property, and extinction in the senses of the almost sure and the $p$th moment. Finally, this paper also considers a stochastic Lotka-Volterra system with regime-switching jump diffusion processes as a special case.
Keywords: extinction., jump process, regime switching, global solution, Kolmogorov-type system, bounedeness, Brownian motion.
Mathematics Subject Classification: Primary: 60H10, 60J70, 60J28; Secondary: 92D25, 93E0.
Citation: Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048
References:
[1]

W. J. Anderson, Continuous-Time Markov Chains,, Springer-Verlag, (1991).  doi: 10.1007/978-1-4612-3038-0.  Google Scholar

[2]

D. Applebaum, Levy Processes and Stochastic Calculus,, $2^{nd}$ Edition, (2009).  doi: 10.1017/CBO9780511809781.  Google Scholar

[3]

D. Applebaum and M. Siakalli, Asymptotic stability properties of stochastic differential equations driven by Lévy noise,, Journal of Applied Probability, 46 (2009), 1116.   Google Scholar

[4]

A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364.   Google Scholar

[5]

A. Bahar and X. Mao, Stochastic delay population dynamcis,, International Journal of pure and applied mathematics, 11 (2004), 377.   Google Scholar

[6]

J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps,, Nonlinear Analysis: Theory, 74 (2011), 6601.  doi: 10.1016/j.na.2011.06.043.  Google Scholar

[7]

J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise,, Journal of Mathematical Analysis and Applications, 391 (2012), 363.  doi: 10.1016/j.jmaa.2012.02.043.  Google Scholar

[8]

N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise,, Journal of Differential Equations, 257 (2014), 2078.  doi: 10.1016/j.jde.2014.05.029.  Google Scholar

[9]

A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications,, $2^{nd}$, (1998).  doi: 10.1007/978-1-4612-5320-4.  Google Scholar

[10]

J. D. Deuschel and D. W. Stroock, Large Deviations,, Academic Press, (1989).   Google Scholar

[11]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, I,, Communications on Pure and Applied Mathematics, 28 (1975), 1.   Google Scholar

[12]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, II,, Communications on Pure and Applied Mathematics, 28 (1975), 279.   Google Scholar

[13]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, III,, Communications on Pure and Applied Mathematics, 29 (1976), 389.  doi: 10.1002/cpa.3160290405.  Google Scholar

[14]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, IV,, Communications on Pure and Applied Mathematics, 36 (1983), 183.  doi: 10.1002/cpa.3160360204.  Google Scholar

[15]

N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise,, Journal of Differential Equations, 250 (2011), 386.  doi: 10.1016/j.jde.2010.08.023.  Google Scholar

[16]

N. H. Du, D. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models,, to appear in Journal of Applied Probability., ().  doi: 10.1017/jpr.2015.18.  Google Scholar

[17]

T. Faria and J. J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks,, Journal of Differential Equations, 244 (2008), 1049.  doi: 10.1016/j.jde.2007.12.005.  Google Scholar

[18]

B. M. Gary, A functional equation characterizing monomial functions used in permanence theory for ecological differential equation,, Universitatis Iagellonicae acta mathematica, 42 (2004), 69.   Google Scholar

[19]

T. C. Gard, Persistence in stochastic food web models,, Bulletin of Mathematical Biology, 46 (1984), 357.  doi: 10.1007/BF02462011.  Google Scholar

[20]

T. C. Gard, Stability for multispecies population models in random environments,, Nonlinear Analysis, 10 (1986), 1411.  doi: 10.1016/0362-546X(86)90111-2.  Google Scholar

[21]

T. C. Gard, Introduction to Stochastic Differential Equations,, Dekker, (1988).   Google Scholar

[22]

K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system,, The Journal of the Australian Mathematical Society. Series B, 27 (1985), 66.  doi: 10.1017/S0334270000004768.  Google Scholar

[23]

X. Han, Z. Teng and D. Xiao, Persistence and average persistence of a nonautonomous Kolmogorov system,, Chaos Solitons Fractals, 30 (2006), 748.  doi: 10.1016/j.chaos.2006.04.026.  Google Scholar

[24]

O. Kallenberg, Foundations of Modern Probability,, $2^{nd}$ Edition, (2002).   Google Scholar

[25]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic press, (1993).   Google Scholar

[26]

Y. Li and Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems,, Journal of Mathematical Analysis and Applications, 255 (2001), 260.  doi: 10.1006/jmaa.2000.7248.  Google Scholar

[27]

M. Liu and K. Wang, Stochastic Lotka-Volterra systems with Lévy noise,, Journal of Mathematical Analysis and Applications, 410 (2014), 750.  doi: 10.1016/j.jmaa.2013.07.078.  Google Scholar

[28]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching,, Journal of Mathematical Analysis and applications, 334 (2007), 69.  doi: 10.1016/j.jmaa.2006.12.032.  Google Scholar

[29]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching II,, Journal of Mathematical Analysis and Applications, 355 (2009), 577.  doi: 10.1016/j.jmaa.2009.02.010.  Google Scholar

[30]

E. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment,, Mathematical Biosciences, 145 (1997), 47.  doi: 10.1016/S0025-5564(97)00029-1.  Google Scholar

[31]

E. Lungu and B. Øksendal, Optimal Harvesting from Interacting Populations in a Stochastic Environment,, Bernoulli, 7 (2001), 527.  doi: 10.2307/3318500.  Google Scholar

[32]

X. Mao, G. Marion and E. Renshaw, Environmental noise supresses explosion in population dynamics,, Stochastic Processes and their Applications, 97 (2002), 95.  doi: 10.1016/S0304-4149(01)00126-0.  Google Scholar

[33]

X. Mao, Delay population dynamics and environmental noise,, Stochastics and Dynamics, 5 (2005), 149.  doi: 10.1142/S021949370500133X.  Google Scholar

[34]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).  doi: 10.1142/p473.  Google Scholar

[35]

X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ Edition, (2008).  doi: 10.1533/9780857099402.  Google Scholar

[36]

J. D. Murray, Mathematical Biology, I. an Introduction,, $3^{rd}$ Edition, (2002).   Google Scholar

[37]

H. D. Nguyen, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise,, Journal of Differential Equations, 257 (2014), 2078.  doi: 10.1016/j.jde.2014.05.029.  Google Scholar

[38]

S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics,, Dynamics of Continuous Discrete and Impulsive Systems Series A, 15 (2008), 603.   Google Scholar

[39]

P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ Edition, (2004).   Google Scholar

[40]

M. Slatkin, The dynamics of a population in a Markovian environment,, Ecology, 59 (1978), 249.  doi: 10.2307/1936370.  Google Scholar

[41]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific, (1996).  doi: 10.1142/9789812830548.  Google Scholar

[42]

Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment,, Journal of Mathematical Analysis and applications, 323 (2006), 938.  doi: 10.1016/j.jmaa.2005.11.009.  Google Scholar

[43]

B. Tang and Y. Kuang, Permanence in Kolmogorov-type systems of nonautonomous functional differential equations,, Journal of Mathematical Analysis and Applications, 197 (1996), 427.  doi: 10.1006/jmaa.1996.0030.  Google Scholar

[44]

Z. Teng, The almost periodic Kolmogorov competitive sysems,, Nonlinear Analysis, 42 (2000), 1221.  doi: 10.1016/S0362-546X(99)00149-2.  Google Scholar

[45]

F. Wu and S. Hu, Stochastic functional Kolmogorov-type population dynamics,, Journal of Mathematical analysis and applications, 347 (2008), 534.  doi: 10.1016/j.jmaa.2008.06.038.  Google Scholar

[46]

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

[47]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay,, SIAM Journal on Applied Mathematics, 70 (2009), 641.  doi: 10.1137/080719194.  Google Scholar

[48]

F. Wu, S. Hu and Y. Liu, Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system,, Journal of Mathematical Analysis and Applications, 364 (2010), 104.  doi: 10.1016/j.jmaa.2009.10.072.  Google Scholar

[49]

F. Wu and G. Yin, Environmental noise impact on regularity and extinction of population systems with infinite delay,, Journal of Mathematical Analysis and Applications, 396 (2012), 772.  doi: 10.1016/j.jmaa.2012.07.017.  Google Scholar

[50]

G. Yin and F. Xi, Stablity of regime-switching jump diffusions,, SIAM Journal on Control and Optimization, 48 (2010), 4525.  doi: 10.1137/080738301.  Google Scholar

[51]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010).  doi: 10.1007/978-1-4419-1105-6.  Google Scholar

[52]

G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems,, SIAM Journal on Applied Mathematics, 72 (2012), 1361.  doi: 10.1137/110851171.  Google Scholar

[53]

C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems,, Nonlinear Analysis: Theory, 71 (2009).  doi: 10.1016/j.na.2009.01.166.  Google Scholar

[54]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, Journal of Mathematical Analysis and Applications, 357 (2009), 154.  doi: 10.1016/j.jmaa.2009.03.066.  Google Scholar

[55]

X. Zong, F. Wu, G. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems,, SIAM Journal on Control and Optimization, 52 (2014), 2595.  doi: 10.1137/14095251X.  Google Scholar

show all references

References:
[1]

W. J. Anderson, Continuous-Time Markov Chains,, Springer-Verlag, (1991).  doi: 10.1007/978-1-4612-3038-0.  Google Scholar

[2]

D. Applebaum, Levy Processes and Stochastic Calculus,, $2^{nd}$ Edition, (2009).  doi: 10.1017/CBO9780511809781.  Google Scholar

[3]

D. Applebaum and M. Siakalli, Asymptotic stability properties of stochastic differential equations driven by Lévy noise,, Journal of Applied Probability, 46 (2009), 1116.   Google Scholar

[4]

A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model,, Journal of Mathematical Analysis and Applications, 292 (2004), 364.   Google Scholar

[5]

A. Bahar and X. Mao, Stochastic delay population dynamcis,, International Journal of pure and applied mathematics, 11 (2004), 377.   Google Scholar

[6]

J. Bao, X. Mao, G. Yin and C. Yuan, Competitive Lotka-Volterra population dynamics with jumps,, Nonlinear Analysis: Theory, 74 (2011), 6601.  doi: 10.1016/j.na.2011.06.043.  Google Scholar

[7]

J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise,, Journal of Mathematical Analysis and Applications, 391 (2012), 363.  doi: 10.1016/j.jmaa.2012.02.043.  Google Scholar

[8]

N. H. Dang, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise,, Journal of Differential Equations, 257 (2014), 2078.  doi: 10.1016/j.jde.2014.05.029.  Google Scholar

[9]

A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications,, $2^{nd}$, (1998).  doi: 10.1007/978-1-4612-5320-4.  Google Scholar

[10]

J. D. Deuschel and D. W. Stroock, Large Deviations,, Academic Press, (1989).   Google Scholar

[11]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, I,, Communications on Pure and Applied Mathematics, 28 (1975), 1.   Google Scholar

[12]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, II,, Communications on Pure and Applied Mathematics, 28 (1975), 279.   Google Scholar

[13]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, III,, Communications on Pure and Applied Mathematics, 29 (1976), 389.  doi: 10.1002/cpa.3160290405.  Google Scholar

[14]

M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time, IV,, Communications on Pure and Applied Mathematics, 36 (1983), 183.  doi: 10.1002/cpa.3160360204.  Google Scholar

[15]

N. H. Du and N. H. Dang, Dynamics of Kolmogorov systems of competitive type under the telegraph noise,, Journal of Differential Equations, 250 (2011), 386.  doi: 10.1016/j.jde.2010.08.023.  Google Scholar

[16]

N. H. Du, D. H. Nguyen and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models,, to appear in Journal of Applied Probability., ().  doi: 10.1017/jpr.2015.18.  Google Scholar

[17]

T. Faria and J. J. Oliveira, Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks,, Journal of Differential Equations, 244 (2008), 1049.  doi: 10.1016/j.jde.2007.12.005.  Google Scholar

[18]

B. M. Gary, A functional equation characterizing monomial functions used in permanence theory for ecological differential equation,, Universitatis Iagellonicae acta mathematica, 42 (2004), 69.   Google Scholar

[19]

T. C. Gard, Persistence in stochastic food web models,, Bulletin of Mathematical Biology, 46 (1984), 357.  doi: 10.1007/BF02462011.  Google Scholar

[20]

T. C. Gard, Stability for multispecies population models in random environments,, Nonlinear Analysis, 10 (1986), 1411.  doi: 10.1016/0362-546X(86)90111-2.  Google Scholar

[21]

T. C. Gard, Introduction to Stochastic Differential Equations,, Dekker, (1988).   Google Scholar

[22]

K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system,, The Journal of the Australian Mathematical Society. Series B, 27 (1985), 66.  doi: 10.1017/S0334270000004768.  Google Scholar

[23]

X. Han, Z. Teng and D. Xiao, Persistence and average persistence of a nonautonomous Kolmogorov system,, Chaos Solitons Fractals, 30 (2006), 748.  doi: 10.1016/j.chaos.2006.04.026.  Google Scholar

[24]

O. Kallenberg, Foundations of Modern Probability,, $2^{nd}$ Edition, (2002).   Google Scholar

[25]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic press, (1993).   Google Scholar

[26]

Y. Li and Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems,, Journal of Mathematical Analysis and Applications, 255 (2001), 260.  doi: 10.1006/jmaa.2000.7248.  Google Scholar

[27]

M. Liu and K. Wang, Stochastic Lotka-Volterra systems with Lévy noise,, Journal of Mathematical Analysis and Applications, 410 (2014), 750.  doi: 10.1016/j.jmaa.2013.07.078.  Google Scholar

[28]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching,, Journal of Mathematical Analysis and applications, 334 (2007), 69.  doi: 10.1016/j.jmaa.2006.12.032.  Google Scholar

[29]

Q. Luo and X. Mao, Stochastic population dynamics under regime switching II,, Journal of Mathematical Analysis and Applications, 355 (2009), 577.  doi: 10.1016/j.jmaa.2009.02.010.  Google Scholar

[30]

E. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment,, Mathematical Biosciences, 145 (1997), 47.  doi: 10.1016/S0025-5564(97)00029-1.  Google Scholar

[31]

E. Lungu and B. Øksendal, Optimal Harvesting from Interacting Populations in a Stochastic Environment,, Bernoulli, 7 (2001), 527.  doi: 10.2307/3318500.  Google Scholar

[32]

X. Mao, G. Marion and E. Renshaw, Environmental noise supresses explosion in population dynamics,, Stochastic Processes and their Applications, 97 (2002), 95.  doi: 10.1016/S0304-4149(01)00126-0.  Google Scholar

[33]

X. Mao, Delay population dynamics and environmental noise,, Stochastics and Dynamics, 5 (2005), 149.  doi: 10.1142/S021949370500133X.  Google Scholar

[34]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).  doi: 10.1142/p473.  Google Scholar

[35]

X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ Edition, (2008).  doi: 10.1533/9780857099402.  Google Scholar

[36]

J. D. Murray, Mathematical Biology, I. an Introduction,, $3^{rd}$ Edition, (2002).   Google Scholar

[37]

H. D. Nguyen, N. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise,, Journal of Differential Equations, 257 (2014), 2078.  doi: 10.1016/j.jde.2014.05.029.  Google Scholar

[38]

S. Pang, F. Deng and X. Mao, Asymptotic properties of stochastic population dynamics,, Dynamics of Continuous Discrete and Impulsive Systems Series A, 15 (2008), 603.   Google Scholar

[39]

P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ Edition, (2004).   Google Scholar

[40]

M. Slatkin, The dynamics of a population in a Markovian environment,, Ecology, 59 (1978), 249.  doi: 10.2307/1936370.  Google Scholar

[41]

Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems,, World Scientific, (1996).  doi: 10.1142/9789812830548.  Google Scholar

[42]

Y. Takeuchi, N. H. Du, N. T. Hieu and K. Sato, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment,, Journal of Mathematical Analysis and applications, 323 (2006), 938.  doi: 10.1016/j.jmaa.2005.11.009.  Google Scholar

[43]

B. Tang and Y. Kuang, Permanence in Kolmogorov-type systems of nonautonomous functional differential equations,, Journal of Mathematical Analysis and Applications, 197 (1996), 427.  doi: 10.1006/jmaa.1996.0030.  Google Scholar

[44]

Z. Teng, The almost periodic Kolmogorov competitive sysems,, Nonlinear Analysis, 42 (2000), 1221.  doi: 10.1016/S0362-546X(99)00149-2.  Google Scholar

[45]

F. Wu and S. Hu, Stochastic functional Kolmogorov-type population dynamics,, Journal of Mathematical analysis and applications, 347 (2008), 534.  doi: 10.1016/j.jmaa.2008.06.038.  Google Scholar

[46]

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

[47]

F. Wu and Y. Xu, Stochastic Lotka-Volterra population dynamics with infinite delay,, SIAM Journal on Applied Mathematics, 70 (2009), 641.  doi: 10.1137/080719194.  Google Scholar

[48]

F. Wu, S. Hu and Y. Liu, Positive solution and its asymptotic behaviour of stochastic functional Kolmogorov-type system,, Journal of Mathematical Analysis and Applications, 364 (2010), 104.  doi: 10.1016/j.jmaa.2009.10.072.  Google Scholar

[49]

F. Wu and G. Yin, Environmental noise impact on regularity and extinction of population systems with infinite delay,, Journal of Mathematical Analysis and Applications, 396 (2012), 772.  doi: 10.1016/j.jmaa.2012.07.017.  Google Scholar

[50]

G. Yin and F. Xi, Stablity of regime-switching jump diffusions,, SIAM Journal on Control and Optimization, 48 (2010), 4525.  doi: 10.1137/080738301.  Google Scholar

[51]

G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications,, Springer, (2010).  doi: 10.1007/978-1-4419-1105-6.  Google Scholar

[52]

G. Yin, G. Zhao and F. Wu, Regularization and stabilization of randomly switching dynamic systems,, SIAM Journal on Applied Mathematics, 72 (2012), 1361.  doi: 10.1137/110851171.  Google Scholar

[53]

C. Zhu and G. Yin, On hybrid competitive Lotka-Volterra ecosystems,, Nonlinear Analysis: Theory, 71 (2009).  doi: 10.1016/j.na.2009.01.166.  Google Scholar

[54]

C. Zhu and G. Yin, On competitive Lotka-Volterra model in random environments,, Journal of Mathematical Analysis and Applications, 357 (2009), 154.  doi: 10.1016/j.jmaa.2009.03.066.  Google Scholar

[55]

X. Zong, F. Wu, G. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems,, SIAM Journal on Control and Optimization, 52 (2014), 2595.  doi: 10.1137/14095251X.  Google Scholar

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