几种随机微分方程数值方法与数值模拟
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摘要
随机微分方程的理论广泛应用于经济、生物、物理、自动化等领域，然而在很长一段时间里，由于缺乏有效的求解随机系统的数值方法以及足够强大的计算机计算能力，在实际问题中，以随机微分方程(组)为代表的描述物理现象的许多复杂的数学模型或者被束之高阁，或者被迫通过忽略随机因素而简化，均不能得到很好的应用。可喜的是近十年来，在随机微分方程数值解方面已取得了一些成就，这意味着由某些随机微分方程描述的数学模型可以借助于计算机进行研究。
本文首先介绍了随机微分方程的背景知识及其理论解的重要性质。其中通过随机积分导出了Ito型和Stratonovich型两种重要形式的随机微分方程，并给出了计算随机积分期望的相关引理；介绍了随机微分方程强解的存在唯一性定理，对于线性随机微分方程，给出了解的解析表达式；推导了解的随机Taylor展开式。
由于随机系统的复杂性，一般情况很难得到方程理论解的解析表达式。这样一来，数值方法的构造显得尤为重要。现在对随机微分方程数值解的研究还处在初级阶段。为了构造有效的数值方法，首先要考虑到数值方法的收敛性和稳定性。本文介绍了随机微分方程理论解的随机渐进稳定性和均方(MS)稳定性，同时介绍了数值解的MS-稳定性和T-稳定性。
在主体部分，本文分别通过直接截断随机Taylor展开式和比较理论解与随机Runge-Kutta格式的Taylor展开式的方法分别得到了数值求解随机微分方程的Taylor方法和Runge-Kutta方法，并对具体方法进行了MS-稳定性分析，对实际算例进行了数值模拟。
其中显式Euler-Mayaruma方法和Milstein方法是求解Ito型随机微分方程的基本方法。本文在此基础上介绍了相应的半隐式Euler-Mayaruma方法、Milstein方法和隐式Euler-Taylor方法、Milstein方法，并通过截断随机Taylor展开式的方式推导了1.5阶Taylor方法。
在推导具体的Runge-Kutta方法时，本文首先介绍了Runge-Kutta方法在常微分方程中的应用，形式上类比得到了随机Runge-Kutta方法。通过应用有根树理论简化了Runge-Kutta格式的Taylor展开式，应用阶条件构造了3级显式(M2)和3级半隐式(SIM1)两个具体的Runge-Kutta格式。
稳定性分析表明各种数值方法的隐式格式稳定性优于相应的显式格式和半隐式格式。数值模拟表明新格式M2和SIM1与经典的Runge-Kutta格式(如4级显式(M3)和2级对角隐式(DIM1))一样具有较高的数值精度。
The theory of stochastic differential equation (SDE) was widely applied in the fields of economy, biology, physics and automatization. However, during quite a long period of time, due to the lack of efficient numerical methods for solving stochastic systems and computers with sufficient power, many complicated mathematical models that attempt to represent physical phenomena, such as SDE(s), had been put aside or simplified when applied in practical problems by omitting stochastic factors. Thus these models were just beautiful in form and never fully utilized. Fortunately, in the past decade or so numerical methods for SDE(s) have made some cheering achievements, which predicate some mathematical models represented by SDE(s) are being researched with computers.
First, the background of SDE and the importance of its theoretical solution are introduced. Two of the very important forms of SDE, Ito SDE and Stratonovich SDE, are deduced by stochastic integrals and several lemmas about the moments of stochastic integrals are also given in the paper. In addition, I mention the theorem giving necessary and sufficient conditions for the existence and uniqueness of a solution to SDE and I give representation formulae of solutions of linear SDEs. And the stochastic Taylor series of solution are deduced.
For the complexity of stochastic systems, it's very difficult to calculate the representation formulae of solutions of generic SDE. Thus constructing numeric methods is paramount. Nowadays, the research of numerical solution of SDE is still in its nascent state. Convergence and stability need to be considered before developing efficient numerical methods. Stochastic asymptotical stability and that in mean-square sense (MS-stability) of the theoretical solution is introduced in the paper, as well as MS-stability and T-stability.
In the body of the paper, both direct truncation of stochastic Taylor series and a comparison of the Taylor series of the theoretical solution and its corresponding Runge-Kutta form are considered, which lead to Taylor methods and Runge-Kutta methods. For Taylor methods, explicit Euler-Mayaruma method and Milstein method are basic for solving Ito SDE(s), on which basis Semi-implicit Euler-Mayaruma method, Semi-implicit Milstein method, implicit Euler-Taylor method and implicit Milstein method are introduced and order 1.5 Taylor method are obtain in the similar way. For Runge-Kutta methods, their application to ordinary differential equation are mentioned at first and the stochastic settings are constructed by comparison. Rooted tree theory simplifies the form of Runge-Kutta methods and two new Runge-Kutta methods of 3 stage explicit (M2) and 3 stage semi-implicit (SIM1) are designed.
In the end, stability analyses under mean-square sense are performed on concrete methods and numerical simulations are implemented, which illustrate implicit form out-performs semi-implicit, and semi-implicit is better than explicit in stability for every method, and new methods M2, SIM1 have the same relatively higher numerical precision as the classical Runge-Kutta methods (eg. 4 stage explicit (M3) and 2 stage diagonal implicit (DIM1)).
本文首先介绍了随机微分方程的背景知识及其理论解的重要性质。其中通过随机积分导出了Ito型和Stratonovich型两种重要形式的随机微分方程，并给出了计算随机积分期望的相关引理；介绍了随机微分方程强解的存在唯一性定理，对于线性随机微分方程，给出了解的解析表达式；推导了解的随机Taylor展开式。
由于随机系统的复杂性，一般情况很难得到方程理论解的解析表达式。这样一来，数值方法的构造显得尤为重要。现在对随机微分方程数值解的研究还处在初级阶段。为了构造有效的数值方法，首先要考虑到数值方法的收敛性和稳定性。本文介绍了随机微分方程理论解的随机渐进稳定性和均方(MS)稳定性，同时介绍了数值解的MS-稳定性和T-稳定性。
在主体部分，本文分别通过直接截断随机Taylor展开式和比较理论解与随机Runge-Kutta格式的Taylor展开式的方法分别得到了数值求解随机微分方程的Taylor方法和Runge-Kutta方法，并对具体方法进行了MS-稳定性分析，对实际算例进行了数值模拟。
其中显式Euler-Mayaruma方法和Milstein方法是求解Ito型随机微分方程的基本方法。本文在此基础上介绍了相应的半隐式Euler-Mayaruma方法、Milstein方法和隐式Euler-Taylor方法、Milstein方法，并通过截断随机Taylor展开式的方式推导了1.5阶Taylor方法。
在推导具体的Runge-Kutta方法时，本文首先介绍了Runge-Kutta方法在常微分方程中的应用，形式上类比得到了随机Runge-Kutta方法。通过应用有根树理论简化了Runge-Kutta格式的Taylor展开式，应用阶条件构造了3级显式(M2)和3级半隐式(SIM1)两个具体的Runge-Kutta格式。
稳定性分析表明各种数值方法的隐式格式稳定性优于相应的显式格式和半隐式格式。数值模拟表明新格式M2和SIM1与经典的Runge-Kutta格式(如4级显式(M3)和2级对角隐式(DIM1))一样具有较高的数值精度。
The theory of stochastic differential equation (SDE) was widely applied in the fields of economy, biology, physics and automatization. However, during quite a long period of time, due to the lack of efficient numerical methods for solving stochastic systems and computers with sufficient power, many complicated mathematical models that attempt to represent physical phenomena, such as SDE(s), had been put aside or simplified when applied in practical problems by omitting stochastic factors. Thus these models were just beautiful in form and never fully utilized. Fortunately, in the past decade or so numerical methods for SDE(s) have made some cheering achievements, which predicate some mathematical models represented by SDE(s) are being researched with computers.
First, the background of SDE and the importance of its theoretical solution are introduced. Two of the very important forms of SDE, Ito SDE and Stratonovich SDE, are deduced by stochastic integrals and several lemmas about the moments of stochastic integrals are also given in the paper. In addition, I mention the theorem giving necessary and sufficient conditions for the existence and uniqueness of a solution to SDE and I give representation formulae of solutions of linear SDEs. And the stochastic Taylor series of solution are deduced.
For the complexity of stochastic systems, it's very difficult to calculate the representation formulae of solutions of generic SDE. Thus constructing numeric methods is paramount. Nowadays, the research of numerical solution of SDE is still in its nascent state. Convergence and stability need to be considered before developing efficient numerical methods. Stochastic asymptotical stability and that in mean-square sense (MS-stability) of the theoretical solution is introduced in the paper, as well as MS-stability and T-stability.
In the body of the paper, both direct truncation of stochastic Taylor series and a comparison of the Taylor series of the theoretical solution and its corresponding Runge-Kutta form are considered, which lead to Taylor methods and Runge-Kutta methods. For Taylor methods, explicit Euler-Mayaruma method and Milstein method are basic for solving Ito SDE(s), on which basis Semi-implicit Euler-Mayaruma method, Semi-implicit Milstein method, implicit Euler-Taylor method and implicit Milstein method are introduced and order 1.5 Taylor method are obtain in the similar way. For Runge-Kutta methods, their application to ordinary differential equation are mentioned at first and the stochastic settings are constructed by comparison. Rooted tree theory simplifies the form of Runge-Kutta methods and two new Runge-Kutta methods of 3 stage explicit (M2) and 3 stage semi-implicit (SIM1) are designed.
In the end, stability analyses under mean-square sense are performed on concrete methods and numerical simulations are implemented, which illustrate implicit form out-performs semi-implicit, and semi-implicit is better than explicit in stability for every method, and new methods M2, SIM1 have the same relatively higher numerical precision as the classical Runge-Kutta methods (eg. 4 stage explicit (M3) and 2 stage diagonal implicit (DIM1)).
引文
[1] 张柄根，赵玉芝．科学与工程中的随机微分方程．海洋出版社，1980．
[2] 泽夫司曲斯．随机微分方程理论及其应用．上海科学技术文献出版社，1984．
[3] A．弗里德曼．随机微分方程及其应用．科学出版社，1983．
[4] 龚光鲁．随机微分方程引论．北京大学出版社，1995．
[5] Burrage P M. Runge-Kutta methods for stochastic differential equations. Ph.D Thesis, Department of Mathematics, the University of Queensland, Australia, 1998.
[6] Kleoden P E, Platen E, Schurz H. Numerical solution of stochastic differential equations. Berlin, Spring Verlag, 1992.
[7] Xuerong Mao. Stochastic differential equations and their applications. University of Strathclyde, Glasgow, Scotland, 1997.
[8] Christopher T.H.Baker, Buchwar E. Introduction to the numerical analysis of stochastic delay differential equations. Numerical Analysis Report NO.345, ISSN, 1360～1725
[9] Tian Tianhai, Burrage K. Implicit Taylor methods for stiff stochastic differential equations. Applied Numerical Mathematics, 2001, 38(3):167～185
[10] Burrage K, Tianhai Tian. A note on the stability properties of the Euler methods for solving stochastic differential equations. New Zealand J. of Maths., 2000, 29:115～127
[11] Bryden A, Desmond J. Higham. On the Boundedness of asymptotic stability regions for the stochastic theta method. BIT Numerical Mathematics, 2003, 43:1～7
[12] Saito Y, Mitsui T. T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Appl. Anal., 1993, 2:333～344
[13] Saito Y, Mitsui T. Stability analysis of numerical schemes for stochastic differential equations. SIAM. J. Numer. Anal., 1996, 33(6):2254～2267
[14] Rumelin W. Numerical treatment of stochastic differential equations. SIAM. J.Numer.Anal., 1982, 19(3):604～613
[15] Kleoden P E, Platen E. Higher-order implicit strong numerical schemes for stochastic differential equations. J. of Statistical Physics, 1992, 66, Nos. 1/2:283～300
[16] Mao Xuerong, Marion G, Renshaw E. Convergence of the Euler scheme for a lass of stochastic differential equations. International Mathematical Journal, 2002, 1(1): 9～22
[17] Burrage K, Burrage P M. Order conditions of stochastic Runge-Kutta methods by B-series. SIAM. J. Numer. Anal., 2000, 38(5):1626～1646
[18] Desmond J.Higham, Mao Xuerong, Stuart. A Strong convergence of numerical methods for nonlinear stochastic differential equations. SIAM. J. Numer. Anal., 2002, 40:1041-1063
[19] Milstein G N, Platen E, Schurz. Balanced implicit methods for stiff stochastic systems. SIAM. J. Numer. Anal., 1998,35(3): 1010-1019
[20] Burrage K, Tian Tianhai. The composite Euler method for solving stochastic differential equations. J. of Comput. and Appl.Maths., 2001,131:407-426
[21] Una-May O'Reilly, Tina Yu, Rick Riolo, Bill Worzel. Genetic Programming and practice II. Springer 2005
[22] Randy L. Haupt, Sue Ellen Haupt. Practical Genetic Algorithm 2nd Edition. 2004
[23] Nicola Bruti-Liberati, Eckhard Platen. On the Approximation of Jump Diffusion Process. April 15~(th), 2005
[24] Y. d'Halluin, P.A.Forsyth, K.R.Vetzal. Robust Numerical Methods for Contingent Claims under Jump Diffusion Processes. January 2, 2004
[25] Bakshi, G, C. Cao, and Z. Chen (1997). Empirical performance of alternative option pricing models. Journal of Finance 52, 2003-2049.
[26] Milstein, GN. (1974): Approximate integration of stochastic differential equations. Theor. Prob.Appl. 19,557-562.
[27] Milstein, GN. (1978): A method of second-order accuracy integration of stochastic differential equations. Theor. Prob. Appl. 23, 396-401.
[28] Milstein, GN. and Tret'yakov, M.V. (1994): Numerical solution of differential equations with colored noise. J. Statist. Physics 77, Nos. 3/4, 691-715.
[29] Ottinger, H.C. (1996): Stochastic Processes in Polymeric Fluids. Springer-Verlag, Berlin.
[30] Platen, E. (1984): Zur zeitdiskreten Approximation von Itoprozessen. Diss. B., IMath, Akad. der Wiss. der DDR, Berlin.
[31] Platen, E. and Wagner, W. (1982): On a Taylor formula for a class of Ito processes. Prob. Math. Statist.3,37-51.
[32] Richardson, L.F. (1910): The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Phil. Trans., A, 210, 307-357.
[33] Richardson, L.F. (1927): The deferred approach to the limit. Phil. Trans., A, 226,299-349.
[34] Saito, Y. and Mitsui, T. (1993): T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Appl. Anal. 2, 333-344.
[35] Saito, Y. and Mitsui, T. (1996): Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33, 6, 2254-2267.
[36] Shampine, L.F. (1985): Some practical Runge-Kutta formulas. Math. Comp. 46,135-150.
[37] Shampine, L.F. and Watts, H.A. (1979): The art of writing a Runge-Kutta code II. Appl. Math. Comput. 5, 93-121.
[38] Skorohod, A.V. (1965): Studies in the Theory of Random Processes, Addison-Wesley, Reading. Mass.
[39] Sobczyk, K. (1991): Stochastic Differential Equations, with applications to Physics and Engineering. Kluwer Academic Publishers, Dordrecht.
[40] Tong, H. (1990): Non-linear Time Series. Oxford University Press. Oxford.
[41] van Kampen, N.G (1981): Ito versus Stratonovich. J. Statist. Physics 24,1,175-187.
[42] Verhulst, F (1990): Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag, Berlin.
[43] Veuthey, A.-L. and Stucki, J. (1987): The adenylate kinase reaction acts as a frequency filter towards fluctuations of ATP utilization in the cell. Biophys. Chem. 26,19-28.
[44] Watanabe, S. and Yamada, T. (1971): On the uniqueness of solutions to stochastic differential equations II, J. Math. Kyoto Univ. 11,553-563.
[45] Werner, M.J. and Drummond, P.D. (1997): Robust algorithms for solving stochastic partial differential equations, J. Comput. Phys. 132,312-326.
[2] 泽夫司曲斯．随机微分方程理论及其应用．上海科学技术文献出版社，1984．
[3] A．弗里德曼．随机微分方程及其应用．科学出版社，1983．
[4] 龚光鲁．随机微分方程引论．北京大学出版社，1995．
[5] Burrage P M. Runge-Kutta methods for stochastic differential equations. Ph.D Thesis, Department of Mathematics, the University of Queensland, Australia, 1998.
[6] Kleoden P E, Platen E, Schurz H. Numerical solution of stochastic differential equations. Berlin, Spring Verlag, 1992.
[7] Xuerong Mao. Stochastic differential equations and their applications. University of Strathclyde, Glasgow, Scotland, 1997.
[8] Christopher T.H.Baker, Buchwar E. Introduction to the numerical analysis of stochastic delay differential equations. Numerical Analysis Report NO.345, ISSN, 1360～1725
[9] Tian Tianhai, Burrage K. Implicit Taylor methods for stiff stochastic differential equations. Applied Numerical Mathematics, 2001, 38(3):167～185
[10] Burrage K, Tianhai Tian. A note on the stability properties of the Euler methods for solving stochastic differential equations. New Zealand J. of Maths., 2000, 29:115～127
[11] Bryden A, Desmond J. Higham. On the Boundedness of asymptotic stability regions for the stochastic theta method. BIT Numerical Mathematics, 2003, 43:1～7
[12] Saito Y, Mitsui T. T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Appl. Anal., 1993, 2:333～344
[13] Saito Y, Mitsui T. Stability analysis of numerical schemes for stochastic differential equations. SIAM. J. Numer. Anal., 1996, 33(6):2254～2267
[14] Rumelin W. Numerical treatment of stochastic differential equations. SIAM. J.Numer.Anal., 1982, 19(3):604～613
[15] Kleoden P E, Platen E. Higher-order implicit strong numerical schemes for stochastic differential equations. J. of Statistical Physics, 1992, 66, Nos. 1/2:283～300
[16] Mao Xuerong, Marion G, Renshaw E. Convergence of the Euler scheme for a lass of stochastic differential equations. International Mathematical Journal, 2002, 1(1): 9～22
[17] Burrage K, Burrage P M. Order conditions of stochastic Runge-Kutta methods by B-series. SIAM. J. Numer. Anal., 2000, 38(5):1626～1646
[18] Desmond J.Higham, Mao Xuerong, Stuart. A Strong convergence of numerical methods for nonlinear stochastic differential equations. SIAM. J. Numer. Anal., 2002, 40:1041-1063
[19] Milstein G N, Platen E, Schurz. Balanced implicit methods for stiff stochastic systems. SIAM. J. Numer. Anal., 1998,35(3): 1010-1019
[20] Burrage K, Tian Tianhai. The composite Euler method for solving stochastic differential equations. J. of Comput. and Appl.Maths., 2001,131:407-426
[21] Una-May O'Reilly, Tina Yu, Rick Riolo, Bill Worzel. Genetic Programming and practice II. Springer 2005
[22] Randy L. Haupt, Sue Ellen Haupt. Practical Genetic Algorithm 2nd Edition. 2004
[23] Nicola Bruti-Liberati, Eckhard Platen. On the Approximation of Jump Diffusion Process. April 15~(th), 2005
[24] Y. d'Halluin, P.A.Forsyth, K.R.Vetzal. Robust Numerical Methods for Contingent Claims under Jump Diffusion Processes. January 2, 2004
[25] Bakshi, G, C. Cao, and Z. Chen (1997). Empirical performance of alternative option pricing models. Journal of Finance 52, 2003-2049.
[26] Milstein, GN. (1974): Approximate integration of stochastic differential equations. Theor. Prob.Appl. 19,557-562.
[27] Milstein, GN. (1978): A method of second-order accuracy integration of stochastic differential equations. Theor. Prob. Appl. 23, 396-401.
[28] Milstein, GN. and Tret'yakov, M.V. (1994): Numerical solution of differential equations with colored noise. J. Statist. Physics 77, Nos. 3/4, 691-715.
[29] Ottinger, H.C. (1996): Stochastic Processes in Polymeric Fluids. Springer-Verlag, Berlin.
[30] Platen, E. (1984): Zur zeitdiskreten Approximation von Itoprozessen. Diss. B., IMath, Akad. der Wiss. der DDR, Berlin.
[31] Platen, E. and Wagner, W. (1982): On a Taylor formula for a class of Ito processes. Prob. Math. Statist.3,37-51.
[32] Richardson, L.F. (1910): The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Phil. Trans., A, 210, 307-357.
[33] Richardson, L.F. (1927): The deferred approach to the limit. Phil. Trans., A, 226,299-349.
[34] Saito, Y. and Mitsui, T. (1993): T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Appl. Anal. 2, 333-344.
[35] Saito, Y. and Mitsui, T. (1996): Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33, 6, 2254-2267.
[36] Shampine, L.F. (1985): Some practical Runge-Kutta formulas. Math. Comp. 46,135-150.
[37] Shampine, L.F. and Watts, H.A. (1979): The art of writing a Runge-Kutta code II. Appl. Math. Comput. 5, 93-121.
[38] Skorohod, A.V. (1965): Studies in the Theory of Random Processes, Addison-Wesley, Reading. Mass.
[39] Sobczyk, K. (1991): Stochastic Differential Equations, with applications to Physics and Engineering. Kluwer Academic Publishers, Dordrecht.
[40] Tong, H. (1990): Non-linear Time Series. Oxford University Press. Oxford.
[41] van Kampen, N.G (1981): Ito versus Stratonovich. J. Statist. Physics 24,1,175-187.
[42] Verhulst, F (1990): Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag, Berlin.
[43] Veuthey, A.-L. and Stucki, J. (1987): The adenylate kinase reaction acts as a frequency filter towards fluctuations of ATP utilization in the cell. Biophys. Chem. 26,19-28.
[44] Watanabe, S. and Yamada, T. (1971): On the uniqueness of solutions to stochastic differential equations II, J. Math. Kyoto Univ. 11,553-563.
[45] Werner, M.J. and Drummond, P.D. (1997): Robust algorithms for solving stochastic partial differential equations, J. Comput. Phys. 132,312-326.