随机微分方程数值解的收敛性及泰勒展式方法的应用
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摘要
对于随机微分方程而言,一般很难求出其具体解过程,有时,即使解能求出,其形式也是隐式或者太复杂,而难于估计。因此,许多学者提出不同的数值策略,如Euler-Maruyama,向前、向后Euler-Maruyama,θ方法,Taylor展式等以讨论数值解的稳定性与精确解的稳定性间的关系及精确解与数值解间的各种收敛。
而大多数文献是在全局或局部Lipschitz条件下讨论随机微分方程的精确解与数值解间的收敛,有时全局或局部Lipschitz条件也显得较为苛刻,许多随机微分方程的漂移系数及扩散系数并不满足,故对其精确解与数值解间的收敛带来一系列困难,就作者而言,对这方面的讨论较少。
本文用Euler-Maruyama方法研究马尔可夫调制及带Poisson跳随机时滞微分方程在Non-Lipschitz条件下的精确解与数值解间的L~1及L~p收敛。再者,由于不同的数值策略,精确解与数值解间的收敛速度不同,本文考虑利用Taylor展式研究马尔可夫调制随机时滞微分方程精确解与数值解间的L~p收敛。由证明过程可知,此种方法较Euler-Maruyama方法优,其精确解与数值解间的收敛速度较快。
本文共三章。
第一章绪论。
第二章马尔可夫调制及带跳随机时滞微分方程精确解与数值解间的收敛。
第三章泰勒展式方法在马尔可夫调制随机时滞微分方程数值解与精确解间收敛中的应用。
Generally, explicit solutions can hardly be obtained for stochastic differential equations.Even when such a solution can be found,it may be only in implicit form or too complicated to visualize and evaluate numerically.So, several numerical schemes such as Euler- Maruyama method ,backward Euler-Maruyama,θmethod, stochastic Taylor expansion e.t. have been developed to produce approximate solutions to study relations between the stability of numerical solutions and true solutions and to investigate all kinds of convergences between numerical solutions and true solutions.
However,most literatures have studied the convergence between numerical solutions and true solutions of stochastic differential equations under the global or local Lipschitz condition. Sometimes,the global or local Lipschitz condition is so strong that the drift coefficient and diffusion coefficient of many stochastic differential equations can not satisfy.This brings about difficulties to investigate the convergence of between numerical solutions and true solutions. As far as I am concerned, there is few papers concerning with non-Lipschitz condition.
In this paper, utilizing the Euler-Maruyama method, we investigate the convergences between numerical solution and true solution of stochastic differential delay equations with Markovian switching and Poisson jump in the L~l and L~p senses. Furthermore, by means of different numerical schemes, the rate of convergence between numerical solutions and true solutions is different. In this article, we also employ Taylor expansion method to consider the convergence between numerical solutions and exact solutions of stochastic differential delay equations with Markovian switching in If sense. Through the process of proof, we can see the rate of convergence between numerical solutions and true solutions is faster than the Euler-Maruyama method. Therefore, the Taylor expansion method is superior to the Euler-Maruyama method.
The paper contains four chapters.
Chapter 1 is the preface.
In chapter 2, we investigate the convergence between numerical solutions and exact solutions of stochastic differential delay equation with Markovian switching and Poisson jump under non-Lipschitz condition in the L~l and L~p senses.
In chapter 3, the convergence between numerical solutions and exact solutions of stochastic differential delay equation with Markovian switching is discussed under local Lipschitz condition by Taylor expansion method.
而大多数文献是在全局或局部Lipschitz条件下讨论随机微分方程的精确解与数值解间的收敛,有时全局或局部Lipschitz条件也显得较为苛刻,许多随机微分方程的漂移系数及扩散系数并不满足,故对其精确解与数值解间的收敛带来一系列困难,就作者而言,对这方面的讨论较少。
本文用Euler-Maruyama方法研究马尔可夫调制及带Poisson跳随机时滞微分方程在Non-Lipschitz条件下的精确解与数值解间的L~1及L~p收敛。再者,由于不同的数值策略,精确解与数值解间的收敛速度不同,本文考虑利用Taylor展式研究马尔可夫调制随机时滞微分方程精确解与数值解间的L~p收敛。由证明过程可知,此种方法较Euler-Maruyama方法优,其精确解与数值解间的收敛速度较快。
本文共三章。
第一章绪论。
第二章马尔可夫调制及带跳随机时滞微分方程精确解与数值解间的收敛。
第三章泰勒展式方法在马尔可夫调制随机时滞微分方程数值解与精确解间收敛中的应用。
Generally, explicit solutions can hardly be obtained for stochastic differential equations.Even when such a solution can be found,it may be only in implicit form or too complicated to visualize and evaluate numerically.So, several numerical schemes such as Euler- Maruyama method ,backward Euler-Maruyama,θmethod, stochastic Taylor expansion e.t. have been developed to produce approximate solutions to study relations between the stability of numerical solutions and true solutions and to investigate all kinds of convergences between numerical solutions and true solutions.
However,most literatures have studied the convergence between numerical solutions and true solutions of stochastic differential equations under the global or local Lipschitz condition. Sometimes,the global or local Lipschitz condition is so strong that the drift coefficient and diffusion coefficient of many stochastic differential equations can not satisfy.This brings about difficulties to investigate the convergence of between numerical solutions and true solutions. As far as I am concerned, there is few papers concerning with non-Lipschitz condition.
In this paper, utilizing the Euler-Maruyama method, we investigate the convergences between numerical solution and true solution of stochastic differential delay equations with Markovian switching and Poisson jump in the L~l and L~p senses. Furthermore, by means of different numerical schemes, the rate of convergence between numerical solutions and true solutions is different. In this article, we also employ Taylor expansion method to consider the convergence between numerical solutions and exact solutions of stochastic differential delay equations with Markovian switching in If sense. Through the process of proof, we can see the rate of convergence between numerical solutions and true solutions is faster than the Euler-Maruyama method. Therefore, the Taylor expansion method is superior to the Euler-Maruyama method.
The paper contains four chapters.
Chapter 1 is the preface.
In chapter 2, we investigate the convergence between numerical solutions and exact solutions of stochastic differential delay equation with Markovian switching and Poisson jump under non-Lipschitz condition in the L~l and L~p senses.
In chapter 3, the convergence between numerical solutions and exact solutions of stochastic differential delay equation with Markovian switching is discussed under local Lipschitz condition by Taylor expansion method.
引文
[1] X. Mao, C. Yuan, G.Yin, Approximations of Euler-Maruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions , Journal of Computational and Applied Mathematics, 205 (2007) 936-948.
[2] W. J. Anderson, Continuous-Time Markov Chains, Springer, Berlin, 1991.
[3] C. T. H. Baker, E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, Technical Report, University of Manchester, Department of Mathematics, Numerical Analysis Report 345, 1999.
[4]P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.
[5] U. KMuchler, E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput.Simulation 54 (2000) 189-205.
[6] P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, 3rd Printing, Springer, New York, 1998.
[7] Oksendal,-Bernt, Applied Stochastic Control of Jump Diffusions,Springer, 2004. 17,
[8] U. Kucher, E. Platen, Weak discrete time approximation of stochastic differential differential equations with time delay, Math. Comput. Simulat. 59 (2002) 497-507.
[9]Xuerong Mao, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, Journal of Computational and Applied Mathematics151 (2003) 215-227.
[10] D. J. Higham, X. Mao, A.M. Stuart, Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J. Num. Anal., 2002, to appear.
[11] X. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays, Numer. Funct. Anal. Optim. 15 (1 & 2 )(1994) 65-76.
[12] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.
[13] X. Mao, Stochastic Differential Equations and Applications, Horwood, England, 1997.
[14] D. J. Higham, X. Mao, A.M. Stuart, Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J. Num.Anal. 40 (2002) 1041-1063.
[15] X. Mao, Stability of stochastic differential equations with Markovian switching, Stoch. Proc. Their Appl. 79 (1999) 45 - 67.
[16] X. Mao, A. Matasov, A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli 6(1) (1) (2000) 73-90.
[17] C. Yuan, X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process.Appl. 103 (2003) 277-291.
[18] Chenggui Yuan, Jiezhong Zou, Xuerong Mao, Stability in distribution of stochastic differential delay equations with Markovian switching, System and Control Letters 50 (2003) 195-207.
[19] G. Marion, X. Mao, E. Renshaw, Convergence of the Euler scheme for a class of stochastic differential equations, Int. Math. J. 1 (1)(2002) 9-22.
[20] A. Bahar, X. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal.Appl.292 (2004) 364-380
[21] Xuerong Mao, Exponential stability in p-th mean of solutions, an of convergent Euler-type Solutions, of stochastic delay differential equations. Journal of Computational and Applied Mathematics 184 (2005) 404-427.
[22] C. Yuan, William Glover, Approximate solutions of stochastic differential delay equations with Markovian switching, Journal of Computational and Applied Mathematics 194 (2006) 207-226.
[23] S. Jankovic and D. Ilic, An Analytic Approximation of Solutions of Stochastic Differential Equations, computers and Mathematics with Application 47(2004), 903-912.
[24] Li Ronghua, Convergence of numerical solutions to stochastic delay differential equations with jumps, Journal of Computational and Applied Mathematics 172 (2006) 584-602.
[25] G. K. Basak, A. Bisi, M. K. Ghosh, Stability of a random diffusion with linear drift, J. Math. Anal. Appl. 202 (1996) 604-622.
[26] A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society 1989.
[27] K. Ito, Stochastic integral, Proc. Imp. Acad. Tokyo 20 (1944) 519-524.
[28] K. Ito, On a formula concerning stochastic differential equations, Nagoya Math. J. 3 (1951) 55-65.
[29] J. K. Hale, V. S.M. Lunel, Introduction to Functional Differential Equations, Springer, Berlin, 1993.
[30] Y. Hu, S. E.A. Mohammed, F. Yan, Numerical solution of stochastic differential systems with memory, Preprint, Southern Illinois University, 2001.
[31] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992.
[32] Y. Komori, Y. Saito, T. Mitsui, Some issues in discrete approximate solution for stochastic differential equations, Comput. Math. Appl. 28 (1994) 269-278.
[33] U. KMuchler, E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput.Simulation 54 (2000) 189-205.
[34] G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers Group, Dordrecht, 1995.
[35] E. Platen, An introduction to numerical methods for stochastic differential equations, Acta Numer. 8 (1999) 195-244.
[36] Y. Saito, T. Mitsui, T-stability of numerical schemes for stochastic differential equations, World Sci. Ser. Appl. Anal. 2 (1993) 333-344.
[37] Y. Saito, T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal. 33 (1996)2254-2267.
[38] Jiaowan Luo,Comparison principle and stability of stochastic differential delay equations with Poisson jump and Markovian switching, Nonlinear Analysis 64 (2006).
[39] M. Athans, Command and control (C2) theory: a challenge to control science, IEEE Trans. Automat. Control 32 (1987) 286 - 293.
[40] E.K. Bouks, Control of system with controlled jump Markov disturbances, Control Theory Adv. Technol. 9 (1993) 577 - 597.
[41] V. Dragan, T. Morozan, Stability and robust stabilization to linear stochastic systems described by differential equationswith Markovian jumping and multiplicative white noise, Stoch. Anal. Appl. 20 (2002)33 - 92.
[42] X. Feng, K. A. Loparo, Y. Ji, H. J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control 37 (1992) 38-53.
[43] A. Friedman, Stochastic Differential Equations and Their Applications, Academic Press, New York, 1976.
[44] R. Z. Has' minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1981.
[45] Y. Hu, Semi-implicit Euler -Maruyama scheme for stiff stochastic equations, in: H. Koerezlioglu (Ed.), Proceedings of the Silvri Workshop on Progress and Probability, Birkhauser, Boston, Stochastic Analysis and Related Topics V, vol. 38, 1996, pp. 183-302.
[46] Y. Ji, H. J. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automat. Control 35 (1990) 777 - 788.
[47] T. Kazangey, D. D. Sworder, Effective federal policies for regulating residential housing, in: Proceedings of the Summer Computer Simulation Conference, 1971, pp. 1120-1128.
[48] M. Lewin, On the boundedness recurrence and stability of solutions of an Ito equation perturbed by a Markov chain, Stoch. Anal. Appl. 4 (1986) 431-487.
[50] M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker,New York, 1990.
[51] T. Morozan, Parametrized Riccati equations for controlled linear differential systems with jump Markov perturbations, Stoch. Anal.Appl. 16 (1998) 661 - 682.
[52] G. Pan, Y. Bar-Shalom, Stabilization of jump linear Gaussian systems without mode observations, Int. J. Control 64 (1996) 631-661.
[53] W. P. Petersen, Numerical simulation of It δ stochastic differential equations on supercomputers, Random Media Springer IMA Ser. 7 (1987) 215-228.
[54] C. E. Souza de, M. D. Fragoso, H~∞ control for linear systems with Markovian juming parameters, Control Theory Adv. Technol. 9 (1993) 457- 466.
[55] L. Shaikhet, Stability of stochastic hereditary systems with Markov switching, Theory Stoc. Process. 2(18) (18) (1996) 180-184.
[56] D. D. Sworder, V. G. Robinson, Feedback regulators for jump parameter systems with state and control depend transition rates, IEEE Trans.Automat. Control 18 (1973) 355-360.
[57] D. Talay, Approximation of upper Lyapunov exponents of bilinear stochastic differential systems, SIMA J. Num. Anal. 28 (1991) 1141 - 1164.
[58] A. S. Willsky, B.C. Rogers, Stochastic stability research for complex power systems, DOE Contract, LIDS, MIT, Rep. ET-76-C-01-2295, 1979.
[59] W. M. Wonham, Random differential equations in control theory, Probab. Methods Appl. Math. 2 (1970) 131-212.
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[68]H.Schurz,Numerical analysis of SDEs without teras,in:V.Lakshmikantham,D.Kannan(Eds.),Handbook of Stochastic Analysis and Applications,Marcel Dekker,Basel,2002,pp.237-359.
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[75]H.Gilsing,On the stability of the Euler scheme for an affine stochastic delay differential equation with time delay.Technical Report,Humboldt University,Berlin(in German)Diskussion Paper No.20,SFB 373,2001.
[76]A.V.Svishchuk,Yu.I.Kazmerchuk,Stability of stochastic delay equations of Ito form with jumps and Markovian switchings,and their applications in finance,Theor.Probab.Math.Stat.64(2002)167-178.
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[2] W. J. Anderson, Continuous-Time Markov Chains, Springer, Berlin, 1991.
[3] C. T. H. Baker, E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, Technical Report, University of Manchester, Department of Mathematics, Numerical Analysis Report 345, 1999.
[4]P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.
[5] U. KMuchler, E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput.Simulation 54 (2000) 189-205.
[6] P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, 3rd Printing, Springer, New York, 1998.
[7] Oksendal,-Bernt, Applied Stochastic Control of Jump Diffusions,Springer, 2004. 17,
[8] U. Kucher, E. Platen, Weak discrete time approximation of stochastic differential differential equations with time delay, Math. Comput. Simulat. 59 (2002) 497-507.
[9]Xuerong Mao, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, Journal of Computational and Applied Mathematics151 (2003) 215-227.
[10] D. J. Higham, X. Mao, A.M. Stuart, Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J. Num. Anal., 2002, to appear.
[11] X. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays, Numer. Funct. Anal. Optim. 15 (1 & 2 )(1994) 65-76.
[12] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.
[13] X. Mao, Stochastic Differential Equations and Applications, Horwood, England, 1997.
[14] D. J. Higham, X. Mao, A.M. Stuart, Strong convergence of numerical methods for nonlinear stochastic differential equations, SIAM J. Num.Anal. 40 (2002) 1041-1063.
[15] X. Mao, Stability of stochastic differential equations with Markovian switching, Stoch. Proc. Their Appl. 79 (1999) 45 - 67.
[16] X. Mao, A. Matasov, A. B. Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli 6(1) (1) (2000) 73-90.
[17] C. Yuan, X. Mao, Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process.Appl. 103 (2003) 277-291.
[18] Chenggui Yuan, Jiezhong Zou, Xuerong Mao, Stability in distribution of stochastic differential delay equations with Markovian switching, System and Control Letters 50 (2003) 195-207.
[19] G. Marion, X. Mao, E. Renshaw, Convergence of the Euler scheme for a class of stochastic differential equations, Int. Math. J. 1 (1)(2002) 9-22.
[20] A. Bahar, X. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal.Appl.292 (2004) 364-380
[21] Xuerong Mao, Exponential stability in p-th mean of solutions, an of convergent Euler-type Solutions, of stochastic delay differential equations. Journal of Computational and Applied Mathematics 184 (2005) 404-427.
[22] C. Yuan, William Glover, Approximate solutions of stochastic differential delay equations with Markovian switching, Journal of Computational and Applied Mathematics 194 (2006) 207-226.
[23] S. Jankovic and D. Ilic, An Analytic Approximation of Solutions of Stochastic Differential Equations, computers and Mathematics with Application 47(2004), 903-912.
[24] Li Ronghua, Convergence of numerical solutions to stochastic delay differential equations with jumps, Journal of Computational and Applied Mathematics 172 (2006) 584-602.
[25] G. K. Basak, A. Bisi, M. K. Ghosh, Stability of a random diffusion with linear drift, J. Math. Anal. Appl. 202 (1996) 604-622.
[26] A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society 1989.
[27] K. Ito, Stochastic integral, Proc. Imp. Acad. Tokyo 20 (1944) 519-524.
[28] K. Ito, On a formula concerning stochastic differential equations, Nagoya Math. J. 3 (1951) 55-65.
[29] J. K. Hale, V. S.M. Lunel, Introduction to Functional Differential Equations, Springer, Berlin, 1993.
[30] Y. Hu, S. E.A. Mohammed, F. Yan, Numerical solution of stochastic differential systems with memory, Preprint, Southern Illinois University, 2001.
[31] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992.
[32] Y. Komori, Y. Saito, T. Mitsui, Some issues in discrete approximate solution for stochastic differential equations, Comput. Math. Appl. 28 (1994) 269-278.
[33] U. KMuchler, E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput.Simulation 54 (2000) 189-205.
[34] G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers Group, Dordrecht, 1995.
[35] E. Platen, An introduction to numerical methods for stochastic differential equations, Acta Numer. 8 (1999) 195-244.
[36] Y. Saito, T. Mitsui, T-stability of numerical schemes for stochastic differential equations, World Sci. Ser. Appl. Anal. 2 (1993) 333-344.
[37] Y. Saito, T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal. 33 (1996)2254-2267.
[38] Jiaowan Luo,Comparison principle and stability of stochastic differential delay equations with Poisson jump and Markovian switching, Nonlinear Analysis 64 (2006).
[39] M. Athans, Command and control (C2) theory: a challenge to control science, IEEE Trans. Automat. Control 32 (1987) 286 - 293.
[40] E.K. Bouks, Control of system with controlled jump Markov disturbances, Control Theory Adv. Technol. 9 (1993) 577 - 597.
[41] V. Dragan, T. Morozan, Stability and robust stabilization to linear stochastic systems described by differential equationswith Markovian jumping and multiplicative white noise, Stoch. Anal. Appl. 20 (2002)33 - 92.
[42] X. Feng, K. A. Loparo, Y. Ji, H. J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control 37 (1992) 38-53.
[43] A. Friedman, Stochastic Differential Equations and Their Applications, Academic Press, New York, 1976.
[44] R. Z. Has' minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1981.
[45] Y. Hu, Semi-implicit Euler -Maruyama scheme for stiff stochastic equations, in: H. Koerezlioglu (Ed.), Proceedings of the Silvri Workshop on Progress and Probability, Birkhauser, Boston, Stochastic Analysis and Related Topics V, vol. 38, 1996, pp. 183-302.
[46] Y. Ji, H. J. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automat. Control 35 (1990) 777 - 788.
[47] T. Kazangey, D. D. Sworder, Effective federal policies for regulating residential housing, in: Proceedings of the Summer Computer Simulation Conference, 1971, pp. 1120-1128.
[48] M. Lewin, On the boundedness recurrence and stability of solutions of an Ito equation perturbed by a Markov chain, Stoch. Anal. Appl. 4 (1986) 431-487.
[50] M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker,New York, 1990.
[51] T. Morozan, Parametrized Riccati equations for controlled linear differential systems with jump Markov perturbations, Stoch. Anal.Appl. 16 (1998) 661 - 682.
[52] G. Pan, Y. Bar-Shalom, Stabilization of jump linear Gaussian systems without mode observations, Int. J. Control 64 (1996) 631-661.
[53] W. P. Petersen, Numerical simulation of It δ stochastic differential equations on supercomputers, Random Media Springer IMA Ser. 7 (1987) 215-228.
[54] C. E. Souza de, M. D. Fragoso, H~∞ control for linear systems with Markovian juming parameters, Control Theory Adv. Technol. 9 (1993) 457- 466.
[55] L. Shaikhet, Stability of stochastic hereditary systems with Markov switching, Theory Stoc. Process. 2(18) (18) (1996) 180-184.
[56] D. D. Sworder, V. G. Robinson, Feedback regulators for jump parameter systems with state and control depend transition rates, IEEE Trans.Automat. Control 18 (1973) 355-360.
[57] D. Talay, Approximation of upper Lyapunov exponents of bilinear stochastic differential systems, SIMA J. Num. Anal. 28 (1991) 1141 - 1164.
[58] A. S. Willsky, B.C. Rogers, Stochastic stability research for complex power systems, DOE Contract, LIDS, MIT, Rep. ET-76-C-01-2295, 1979.
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