马尔科夫切换型随机微分方程的数值稳定性
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摘要
马尔科夫切换型随机微分方程可用于解释环境突然发生变化的物理过程以及在不同市场条件下进行切换的金融模型。马尔科夫链的引入可以描述不同金融模型结构间的切换。例如,股票市场在熊市和牛市具有不同的特征,用一个常规的随机微分方程模型难以表达,这时我们就可以应用马尔科夫切换型随机微分方程将他们表达出来。因此带马尔科夫切换的随机微分方程方程在复杂的金融市场研究中发挥着重要的作用。
马尔科夫切换型随机微分方程往往不存在解析解,所以其主要解法是数值近似方法,如Euler-Maruyama方法,Milstein方法,强一阶显式方法等。数值稳定性是微分方程数值解法的一个重要性质。不稳定性的数值方法往往会造成舍入误差的恶性增长并导致数值解的失真,因此研究数值稳定性就显得非常重要。
本文先研究马尔科夫切换型随机微分方程的欧拉近似解的具体稳定区域。接着对马尔科夫切换型随机微分方程Milstein方法数值解的p阶矩指数稳定性问题做了研究。对于一维线性乘噪声试验方程,本文证明了存在一个步长范围使得Milstein方法的数值解是p阶矩指数稳定性的。特别的,对于均方稳定性而言,本文的存在性条件比现有文献要弱。同时,本文又进一步研究了一种高阶数值方法：强一阶显式方法。证明了存在一个步长范围使得强一阶显式方法的数值解是p阶矩指数稳定的。
本文还给出了一类时变的马尔科夫切换型随机微分方程的解的稳定性条件。
Stochastic differential equations (SDEs) with Markov switching can or Hybrid SDE be used to describe physical dynamics under environment change, and the finan-cial models fluctuating between various economic circumstances. The introduction of Markov chain into the SDEs illustrates the switching among different financial models. For instance, the stock market has different characteristics in bear market and bull market, making it difficult to describe the behavior using a simple model of stochastic differential equations. In this case, the SDEs with Markov switching is an excellent way to describe the market behaviors. Therefore, SDEs with Markov switching are taking an increasingly important part in modeling the complicated financial markets.
Generally, hybrid stochastic differential equations can not be solved analytically and hence numerical methods must be used. Such as Euler-Maruyama methods, Mil-stein methods, explicit order 1.0 strong scheme and so on. The numerical stability is very important characteristic of numerical methods, since an unstable numerical method may result in malignant growth of rounding error and distraction of the nu-merical solution, therefore studying on the numerical stability of stochastic differential equations is a very important issue.
In this paper, we first study the stability regions for Euler-Maruyama approxima-tions of hybrid stochastic differential equations. Then, the pth moment exponential stability of Milstein scheme is discussed. For scalar linear multiplied-noise equations, this paper proves that the Milstein scheme approximation is pth moment exponential stable if the stepsize is small enough. And for mean square exponential stability, in par-ticular, the existence conditions showed in this paper are weaker than the in literature. Meanwhile, we make a further study of higher order approximations:explicit order 1.0 strong scheme. The result is that there also exists an interval of stepsize where the pth moment exponential stability of explicit order 1.0 strong scheme approximations.
The sufficient conditions for stability of time-varying hybrid stochastic differential equations are given as well.
马尔科夫切换型随机微分方程往往不存在解析解,所以其主要解法是数值近似方法,如Euler-Maruyama方法,Milstein方法,强一阶显式方法等。数值稳定性是微分方程数值解法的一个重要性质。不稳定性的数值方法往往会造成舍入误差的恶性增长并导致数值解的失真,因此研究数值稳定性就显得非常重要。
本文先研究马尔科夫切换型随机微分方程的欧拉近似解的具体稳定区域。接着对马尔科夫切换型随机微分方程Milstein方法数值解的p阶矩指数稳定性问题做了研究。对于一维线性乘噪声试验方程,本文证明了存在一个步长范围使得Milstein方法的数值解是p阶矩指数稳定性的。特别的,对于均方稳定性而言,本文的存在性条件比现有文献要弱。同时,本文又进一步研究了一种高阶数值方法：强一阶显式方法。证明了存在一个步长范围使得强一阶显式方法的数值解是p阶矩指数稳定的。
本文还给出了一类时变的马尔科夫切换型随机微分方程的解的稳定性条件。
Stochastic differential equations (SDEs) with Markov switching can or Hybrid SDE be used to describe physical dynamics under environment change, and the finan-cial models fluctuating between various economic circumstances. The introduction of Markov chain into the SDEs illustrates the switching among different financial models. For instance, the stock market has different characteristics in bear market and bull market, making it difficult to describe the behavior using a simple model of stochastic differential equations. In this case, the SDEs with Markov switching is an excellent way to describe the market behaviors. Therefore, SDEs with Markov switching are taking an increasingly important part in modeling the complicated financial markets.
Generally, hybrid stochastic differential equations can not be solved analytically and hence numerical methods must be used. Such as Euler-Maruyama methods, Mil-stein methods, explicit order 1.0 strong scheme and so on. The numerical stability is very important characteristic of numerical methods, since an unstable numerical method may result in malignant growth of rounding error and distraction of the nu-merical solution, therefore studying on the numerical stability of stochastic differential equations is a very important issue.
In this paper, we first study the stability regions for Euler-Maruyama approxima-tions of hybrid stochastic differential equations. Then, the pth moment exponential stability of Milstein scheme is discussed. For scalar linear multiplied-noise equations, this paper proves that the Milstein scheme approximation is pth moment exponential stable if the stepsize is small enough. And for mean square exponential stability, in par-ticular, the existence conditions showed in this paper are weaker than the in literature. Meanwhile, we make a further study of higher order approximations:explicit order 1.0 strong scheme. The result is that there also exists an interval of stepsize where the pth moment exponential stability of explicit order 1.0 strong scheme approximations.
The sufficient conditions for stability of time-varying hybrid stochastic differential equations are given as well.
引文
[1]J.A.D.Appleby.G.Berkolaiko and A.Rodkina,On the Asymptotic Behavior of the Moments of Stochastic Difference Equations,Technical report MS-05-21,School of Mathematical Sciences,Dublin City University,Dublin,Ireland,(2005).
[2]L.Arnold,Stochastic Differential Equations, Theory and Applications, Wiley, Chichester, UK, (1972).
[3]A.Bryden and D.J.Higham,On the boundedness of asymptotic stability regions for the stochastic theta method,BIT,43(2003),PP.1-6.
[4]E.Buckwar,R.Horvath-Bokor and R.Winkler,Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations,BIT,46(2006),PP.261-282.
[5]K.Burrage and T.Tian,A note on the stability properties of the Euler methods for solving stochastic differential equations,New Zealand J.Math.29(2000),pp.115-127.
[6]D.J.Higham,Mean-square and asymptotic stability of the stochastic theta method,SIAM J.Numer.Anal.,38(2000),pp.753-769.
[7]X.Mao, Exponential Stability of Stochastic Differential Equations[M]. Marcel Dekker, (1994).
[8]X.Mao,Stability of Stochastic Differential Equations with Respect to Semi-martingales,Pitman Res.Notes Math.Ser.251,Longman Scientific and Techni-cal,Harlow,UK,(1991).
[9]D.J.Higham.X.Mao and C.Yuan,Almost Sure and Moment Exponential Sta-bility in the Numercal Simulation of Stochastic Differential Equations,SIAM J.Numer.Anal.,45(2007),pp.592-609.
[10]P.E.Kloeden and E.Platen,Numerical Solutions of Stochastic Differential Equations,Springer-Verlag, Berlin (1992),298-310.
[11]田增峰,魏跃春,胡良剑.随机微分方程Euler法的均方稳定性和指数稳定性[J].自然杂志,(2002),24：369-370.
[12]田增锋,胡良剑.随机微分方程数值求解的两种插值法的比较[J].纺织高校基础科学学报,(2003).
[13]L. ARNOLD, Stochastic Differential Equations, Wiley, New York, (1974).
[14]Y.Komori,T.Mitsui, Stable ROW type weak scheme for stochastic differential equations,Monte Carlo Methods Appl.1(1995)279-300
[15]Y.Saito,T.Mitsui, T-stability of numerical scheme for stochastic differential equa-tions,World Sci.Ser.Appl.Anal.2(1993)333-334.
[16]Y.Saito,T.Mitsui, Stability analysis of numerical schemes for stochastic differential equations,SIAM J.Numer.Anal.33(1996)2254-2267.
[17]X.Mao, Stochastic Differential Equation and Applications, Horwood, Chichester ,UK,(1997)
[18]Chenggui Yuan,Xurong Convergence of the Euler-Manuyama method for stochas-tic differential equation with Markovian switching. Mathematics and computers in Simulation64(2004)223-235
[19]C. Yuan, X. Mao, Convergence of the Euler-Maruyama method for stochas-tic differential equations with Markovian switching, Math. Comput.Simulation 64(2004)223-235,
[20]Sulin Pang,Feiqi Deng, Xuerong Mao, Almost sure and moment exponential stabil-ity of Euler-Maruyama discietizations for hybrid stochastic differential equations, Journal of Computational and Applied Mathematics213(2008)127-141
[21]Milstein,G.N., Numerical Integration of Stochastic Differential Equations, Math-ematics and Its Applications.Kluwer (1995).
[22]朱霞,等,随机微分方程Milstein方法的稳定性,华中科技大学学报(自然科学版),(2003),31(3)：111-113.
[23]Z.Wang, C.Zhang, An analysis of stability of Milsiein method for stochas-tic differential equations with delay, Computers and Mathematics with Applications(2006),51:1445-1452.
[24]A.Rathinasamy,K.Balachandran, Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations, Nonlinear Analy-sis:Hybrid System(2008),2:1256-1263.
[25]王梓坤,随机过程通论(上卷)(1996),39-63.
[26]W.J.Anderson, Continuous-Time Markov Chains, Springer, Berlin, (1991),345-351.
[27]D,J,Higham,X.Mao,A.M. Stuart.Exponential mean-square stability of numerical solutions to stochastic differential equations,LMS J.Comput.Math.6(2003) 297-313.
[28]W.J.Anderson,Continuous-Time Markov Chains,Springer,Berlin,(1991)
[29]D.R.Smith.Markov-switching and stochastic volatility diffusion models of shorterm interest rates,J.Bussiness Econom.Statist,20(2002)183-197.
[30]C.T.H.B. Baker,E.Buckwar. Exponential stability in pin mean of solutions ,and of convergent Euler -type solutions,of stochastic delay differential equa-tions, J.Comput.Appl.Math.184(2005)404-427.
[31]X.Mao.A.Matasov.A.B.Piunovkiv. Stochastic differential delay equations with Markovian switching.Bernoulli 6(2000)73-90
[32]X. Mao, Robustness of stability of stochastic differential delay equations with Markovian switching, SACTA 3 (2000) 48-61.
[33]X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, (2006).
[34]A. Rathinasamy, K. Balachandran, Mean-square stability of second-order Runge-Kutta methods for multi-dimensional linear stochastic differential systems, Journal of Computational and Applied Mathematics 219 (2008) 170-197.
[35]郭海山,胡良剑,随机微分方程数值解的几乎必然稳定区域,纺织高校基础科学学报(2010),23(1)：54-58
[36]G.N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer, Dordrecht, (1995).
[2]L.Arnold,Stochastic Differential Equations, Theory and Applications, Wiley, Chichester, UK, (1972).
[3]A.Bryden and D.J.Higham,On the boundedness of asymptotic stability regions for the stochastic theta method,BIT,43(2003),PP.1-6.
[4]E.Buckwar,R.Horvath-Bokor and R.Winkler,Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations,BIT,46(2006),PP.261-282.
[5]K.Burrage and T.Tian,A note on the stability properties of the Euler methods for solving stochastic differential equations,New Zealand J.Math.29(2000),pp.115-127.
[6]D.J.Higham,Mean-square and asymptotic stability of the stochastic theta method,SIAM J.Numer.Anal.,38(2000),pp.753-769.
[7]X.Mao, Exponential Stability of Stochastic Differential Equations[M]. Marcel Dekker, (1994).
[8]X.Mao,Stability of Stochastic Differential Equations with Respect to Semi-martingales,Pitman Res.Notes Math.Ser.251,Longman Scientific and Techni-cal,Harlow,UK,(1991).
[9]D.J.Higham.X.Mao and C.Yuan,Almost Sure and Moment Exponential Sta-bility in the Numercal Simulation of Stochastic Differential Equations,SIAM J.Numer.Anal.,45(2007),pp.592-609.
[10]P.E.Kloeden and E.Platen,Numerical Solutions of Stochastic Differential Equations,Springer-Verlag, Berlin (1992),298-310.
[11]田增峰,魏跃春,胡良剑.随机微分方程Euler法的均方稳定性和指数稳定性[J].自然杂志,(2002),24：369-370.
[12]田增锋,胡良剑.随机微分方程数值求解的两种插值法的比较[J].纺织高校基础科学学报,(2003).
[13]L. ARNOLD, Stochastic Differential Equations, Wiley, New York, (1974).
[14]Y.Komori,T.Mitsui, Stable ROW type weak scheme for stochastic differential equations,Monte Carlo Methods Appl.1(1995)279-300
[15]Y.Saito,T.Mitsui, T-stability of numerical scheme for stochastic differential equa-tions,World Sci.Ser.Appl.Anal.2(1993)333-334.
[16]Y.Saito,T.Mitsui, Stability analysis of numerical schemes for stochastic differential equations,SIAM J.Numer.Anal.33(1996)2254-2267.
[17]X.Mao, Stochastic Differential Equation and Applications, Horwood, Chichester ,UK,(1997)
[18]Chenggui Yuan,Xurong Convergence of the Euler-Manuyama method for stochas-tic differential equation with Markovian switching. Mathematics and computers in Simulation64(2004)223-235
[19]C. Yuan, X. Mao, Convergence of the Euler-Maruyama method for stochas-tic differential equations with Markovian switching, Math. Comput.Simulation 64(2004)223-235,
[20]Sulin Pang,Feiqi Deng, Xuerong Mao, Almost sure and moment exponential stabil-ity of Euler-Maruyama discietizations for hybrid stochastic differential equations, Journal of Computational and Applied Mathematics213(2008)127-141
[21]Milstein,G.N., Numerical Integration of Stochastic Differential Equations, Math-ematics and Its Applications.Kluwer (1995).
[22]朱霞,等,随机微分方程Milstein方法的稳定性,华中科技大学学报(自然科学版),(2003),31(3)：111-113.
[23]Z.Wang, C.Zhang, An analysis of stability of Milsiein method for stochas-tic differential equations with delay, Computers and Mathematics with Applications(2006),51:1445-1452.
[24]A.Rathinasamy,K.Balachandran, Mean-square stability of Milstein method for linear hybrid stochastic delay integro-differential equations, Nonlinear Analy-sis:Hybrid System(2008),2:1256-1263.
[25]王梓坤,随机过程通论(上卷)(1996),39-63.
[26]W.J.Anderson, Continuous-Time Markov Chains, Springer, Berlin, (1991),345-351.
[27]D,J,Higham,X.Mao,A.M. Stuart.Exponential mean-square stability of numerical solutions to stochastic differential equations,LMS J.Comput.Math.6(2003) 297-313.
[28]W.J.Anderson,Continuous-Time Markov Chains,Springer,Berlin,(1991)
[29]D.R.Smith.Markov-switching and stochastic volatility diffusion models of shorterm interest rates,J.Bussiness Econom.Statist,20(2002)183-197.
[30]C.T.H.B. Baker,E.Buckwar. Exponential stability in pin mean of solutions ,and of convergent Euler -type solutions,of stochastic delay differential equa-tions, J.Comput.Appl.Math.184(2005)404-427.
[31]X.Mao.A.Matasov.A.B.Piunovkiv. Stochastic differential delay equations with Markovian switching.Bernoulli 6(2000)73-90
[32]X. Mao, Robustness of stability of stochastic differential delay equations with Markovian switching, SACTA 3 (2000) 48-61.
[33]X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, (2006).
[34]A. Rathinasamy, K. Balachandran, Mean-square stability of second-order Runge-Kutta methods for multi-dimensional linear stochastic differential systems, Journal of Computational and Applied Mathematics 219 (2008) 170-197.
[35]郭海山,胡良剑,随机微分方程数值解的几乎必然稳定区域,纺织高校基础科学学报(2010),23(1)：54-58
[36]G.N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer, Dordrecht, (1995).