Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise

Stochastic Analysis and Applications

Volumn 14, Issue 3, 1996, Pages 313-353

  Schurz, Henri ^a  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Numerical methods; Stability; Stochastic differential equations

Indexed keywords

EID: 21344458305     PISSN: 07362994     EISSN: None     Source Type: Journal    
DOI: 10.1080/07362999608809442     Document Type: Article

References (86)

1. S. Ariaratnam and P. Graefe, Linear systems with stochastic coefficients. I, II, III, Int. J. Control 1 (1965), 239-248 (No. 1), 161-169 (No. 2), 205-211 (No. 2).
2. L. Arnold, Stochastic Differential Equations, Wiley, New York (1974).
3. L. Arnold, A formula connecting sample and moment stability of linear stochastic systems, SIAM J. Appl. Math. 44 (1984), 793-802 (No. 4).
4. L. Arnold, E. Oeljeklaus and E. Pardoux, Almost sure and moment stability for linear Itô equations, in Lyapunov Exponents, Proc. Workshop Bremen 1984, ed. by L. Arnold and V. Wihstutz, Lecture Notes in Mathematics 1186 (1986), Springer, Berlin, 129-159.
5. S.S. Artemiev, Certain aspects of application of numerical methods for solving SDE systems, Bull. Nov. Comp. Center, Num. Anal. 1 (1993), NCC Publisher, Novosibirsk, 1-16.
6. S.S. Artemiev, The stability of numerical methods for solving stochastic differential equations, Research lecture at IAAS, Berlin (1993).
7. P.H. Baxendale, Moment stability and large deviations for linear stochastic differential equations, Taniguchi Symp. on Prob. Meth. in Math. Physics, Katata and Kyoto 1985, ed. by N. Ikeda, (1987), Kinokuniya, Tokyo, 31-54.
8. P. Bougerol and J. Lacroix, Products of random matrices with applications to Schrödinger operators, Birkhauser, Basel (1985).
9. H. Bunke, Über das asymptotische Verhalten von Lösungen linearer stochastischer Differentialgleichungssysteme, ZAMM 45 (1965), 1-9 (No. 1).
10. H. Bunke, Zum asymptotischen Verhalten der Lösungen im quadratischen Mittel von linearen stochastischen Differentialgleichungssystemen, ZAMM 48 (1968), 70-71.
11. H. Bunke, Gewöhnliche Differentialgleichungen mit zufälligen Parametern, Akademie Verlag, Berlin (1972).
12. E.F. Car'kov, Asymptotic exponential stability in the mean square of the trivial solution of stochastic functional differential equations (in Russian), Teor. Verojatnost. i Primenen. 21 (1976), 871-875.
13. C.C. Chang, Numerical solution of stochastic differential equations with constant diffusion coefficients, Math. Comp. 49 (1987), 523-542 (No. 180).
14. J.M.C. Clark and R.J. Cameron, The maximum rate of convergence of discrete approximations for stochastic differential equations, in B. Grigelionis (ed.), Stochastic Differential Systems, Lecture Notes in Control and Information Systems 25 (1980), Springer, Berlin, 162-171.
15. J.E. Cohen and C.M. Newman, The stability of large random matrices and their products, Ann. Prob. 12 (1984), 283-310 (No. 2).
16. G. Dahlquist, A special stability problem for multi-step methods, BIT 3 (1963), 27-43.
17. P.D. Drummond and I.K. Mortimer, Computer simulation of multiplicative stochastic differential equations, J. Comput. Phys. 93 (1991), 144-170 (N1).
18. M. Ehrhardt, Invariant probabilities for systems in a random environment - with applications to the Brusselator, Bull. Math. Biol. 45 (1983), 579-590.
19. J.F. Feng, G.Y. Lei and M.P. Qian, Second order methods for solving stochastic differential equations, J. Comp. Math. 10 (1992), 376-387 (No. 4).
20. A. Friedman, Stochastic Differential Equations and Applications, Academic Press, London (1975).
21. H. Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963), 377-428.
22. T.C. Gard, Introduction to Stochastic Differential Equations, Marcel Dekker, New York (1988).
23. I.I. Gikhman and A.V. Skorokhod, Stochastic Differential Equations, Springer, New York (1972).
24. A. Greiner, W. Strittmatter and J. Honerkamp, Numerical integration of stochastic differential equations, J. Stat. Phys. 51 (1988), 95-108 (Nos. I/2).
25. E. Hairer and G. Wanner, Solving Differential Equations II: Stiff Problems, Springer Series Comp. Math. 8, Springer, Berlin (1991).
26. E. Helfand, Numerical integration of stochastic differential equations, Bell Syst. Techn. J. 58 (1979), 2289-2299.
27. D.B. Hernandez and R. Spigler, A-stability of implicit Runge-Kutta methods for systems with additive noise, BIT 32 (1992), 620-633.
28. D.B. Hernandez and R. Spigler, Convergence and stability of implicit Runge-Kutta methods for systems with multiplicative noise, BIT 33 (1993), 654-669.
29. C.C. Heyde, Quasi-likelihood and optimality for estimating functions: some current unifying themes, Bull. Int. Stat. Inst. 53 (1989), 19-29 (Book 1).
30. N. Hofmann and E. Platen, Stability of weak numerical schemes for stochastic differential equations, submitted to Comp. Math. Appl. (1993).
31. J. Honerkamp, Stochastic Dynamical Systems (in German), VCH, New York (1990).
32. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam (1981).
33. I.Ja. Kac and N.N. Krasovskij, On stability of systems with random parameters, J. Appl. Math. Mech. 24 (1960), 1225-1246.
34. L.W. Kantorowitsch and G.P. Akilow, Functional Analysis in Normalized Spaces (in German), Akademie Verlag, Berlin (1978).
35. Karmeshu and N.K. Bansal, Stability of moments in a simple neutronic system with stochastic parameters, Nucl. Sc. Eng. 58 (1975), 321-327.
36. Karmeshu and H. Schurz, Moment evolution of the outflow-rate from nonlinear conceptual reservoirs, Preprint 84, IAAS, Berlin (1994), to appear in Proc. Int. Conf. Hydrol. Wat. Res. 1993, ed. by V.P. Singh, New Delhi (1993).
37. Karmeshu and H. Schurz, Effects of distributed delays on the stability of structures under seismic excitation and multiplicative noise, Preprint 100, IAAS, Berlin (1994), submitted to SADHANA, ed. by R.N. Iyengar (Indian Int. J. Eng. Sci.), Roorkee (1994).
38. R.Z. Khas'minskij, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Alphen aan den Rijn (1980).
39. J.R. Klauder and W.P. Petersen, Numerical integration of multiplicative-noise stochastic differential equations, SIAM J. Num. Anal. 22 (1985), 1153-1166.
40. P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Appl. of Math. 23, Springer, New York (1992).
41. P.E. Kloeden and E. Platen, Higher-order implicit strong numerical schemes for stochastic differential equations, J. Statist. Phys. 66 (1992), 283-314.
42. P.E. Kloeden, E. Platen and H. Schurz, Higher order approximate Markov chain filters, in Stochastic Processes - A Festschrift in Honour of Gopinath Kallianpur, ed. by S. Cambanis et al., (1993), Springer, New York, 181-190.
43. P.E. Kloeden, E. Platen, H. Schurz and M. Sørensen, On effects of discretization of estimators of diffusions, submitted to Theory Prob. Appl. (1993), Preprint 249, Univ. Aarhus, Aarhus (1992).
44. P.E. Kloeden, E. Platen and H. Schurz, Numerical Solution of Stochastic Differential Equations through Computer Experiments, Universitext, Springer, New York (1994).
45. F. Kozin, A survey of stability of stochastic systems, Automatica 5 (1969), Pergamon Press, Oxford, 95-112.
46. H.J. Kushner, Stochastic Stability and Control, Academic Press, London (1967).
47. H.J. Kushner and P.G. Dupuis, Numerical methods for stochastic control problems in continuous time, Appl. of Math. 24. Springer, New York (1992).
48. U. Küchler and M. Sørensen, Exponential families of stochastic processes: An unifying semimartingale approach, Int. Stat. Rev. 57 (1989), 123-147.
49. G.S. Ladde and V. Lakshmikantham, Random differential inequalities, Math. Sci. Eng. 150, Academic Press, London (1980).
50. G.S. Ladde, V. Lakshmikantham and M. Sanbandham, Comparison theorem and error estimates of stochastic differential systems, Stoch. Anal. Appl. 3, 23-62 (1985).
51. G.S. Ladde and D.D. Siljak, Convergence and stability of distributed stochastic iterative processes, in Lyapunov functions method and applications, ed. by P. Borne and V. Matrosov, IMACS Annals on Computing and Applied Mathematics 8, (1990), J.C. Baltzer A.G., Basel, (211-218).
52. M.V. Levit and V.A. Jakubovič, Algebraic criteria for stochastic stability of linear systems with parametric control of the type of white noise (in Russian), Prikl. Mat. Meh. 36 (1972), 142-148.
53. G.N. Mil'shtein and U.M. Repin, On mean square stability of systems of stochastic differential equations (in Russian), Prikl. Mat. Meh. 3 (1967), 509-510.
54. G.N. Mil'shtein, Numerical Integration of Stochastic Differential Equations (in Russian), Uralski University Press, Ekaterinburg (1988).
55. G.N. Mil'shtein, E. Platen and H. Schurz, Balanced implicit methods for stiff stochastic systems: An introduction and numerical experiments, submitted to SIAM J. Num. Anal. (1994), Preprint 33, IAAS, Berlin (1992).
56. T. Mitsui and G. Saito, Stability analysis of numerical schemes for stochastic differential equations, Report 9202. School of Eng., Univ. Nagoya, Nagoya (1992).
57. T. Morozan, Stability of stochastic discrete systems, J. Math. Anal. Appl. 23 (1968), 1-9 (MR 37 # 4864).
58. N.J. Newton, Asymptotically efficient Runge-Kutta methods for a class of Itô and Stratonovich equations, SIAM J. Appl. Math. 51 (1991), 542-567.
59. B. Øksendal, Stochastic Differential Equations, Universitext (3rd edition), Springer, Berlin (1992).
60. E. Pardoux and D. Talay, Discretization and simulation of stochastic differential equations, Acta Appl. Math. 3 (1985), 23-47.
61. E. Pardoux and D. Talay, Stability of linear differential systems with parametric excitation, in Nonlin. Stoch. Dyn. Eng. Sys., ed. by F. Ziegler and G.I. Schueller, Proc. IUTAM Symp. Innsbruck (1988), 153-168.
62. W.P. Petersen, Stability and accuracy of simulations for stochastic differential equations, IPS Report 90-02, ETH Zürich, Zürich (1990).
63. N.J. Rao, J.D. Borwankar and D. Ramkrishna, Numerical solution of Itô integral equations, SIAM J. Control 12, (1974), 124-139 (No. 1).
64. W. Rümelin, Numerical treatment of stochastic differential equations, SIAM J. Num. Anal. 19 (1982), 604-613.
65. T. Sasagawa, On exponential stability and instability of linear stochastic systems, Int. J. Control 33 (1981), 363-373 (No. 2).
66. T. Sasagawa and J.L. Willems, P-th mean stability of linear systems with multiplicative noise, Report 247, Univ. Bremen, Bremen (1991).
67. H. Schurz, Asymptotical mean stability of numerical solutions with multiplicative noise, Preprint 47, IAAS, Berlin (1993).
68. H. Schurz, Mean square stability for discrete linear stochastic systems, Preprint 72, IAAS, Berlin (1993).
69. H. Schurz, A note on pathwise approximation of stationary Ornstein-Uhlenbeck processes with diagonalizable drift, Preprint 112, WIAS, Berlin (1994).
70. H. Schurz, Construction of nonnegative numerical solutions for SDEs: Explosions, life time, nonnegativity and uniform boundedness, Discussion paper 40, SFB 373, Humboldt University, Berlin (1994).
71. Z. Schuss, Theory and Application of Stochastic Differential Equations, Wiley, New York (1980).
72. I.O. Shkurko, Numerical solution of linear systems of stochastic differential equations (in Russian), in Numerical Methods for Statistics and Modeling. Collected Scientific Works, (1987), Novosibirsk, 101-109.
73. A.M. Smith and C.W. Gardiner, Simulation of nonlinear quantum damping using the positive representation, Phys. Rev. 39 (1989), 3511-3524.
74. K. Sobczyk, Stochastic differential equations, Kluwer, Dordrecht (1991).
75. T.T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York (1973).
76. D.W. Stroock and S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer, New York (1982).
77. D. Talay, Analyse Numérique del Equations Différentielles Stochastiques, Thèse Sème Cycle, Universite de Provence, Centre Saint Charles (1982).
78. D. Talay, Simulation and numerical analysis of stochastic differential systems: A review, Rapports de Recherche 1313, INRIA, Rocquencourt (1990).
79. D. Talay, Approximation of upper Lyapunov exponents of bilinear stochastic differential equations, SIAM J. Num. Anal. 28 (1991), 1141-1164.
80. C. Törok, Numerical solution of linear stochastic differential equations, Comp. Math. Appl. 27 (1994), 1-10 (No. 4).
81. R.A. Usmani, Applied Linear Algebra, Marcel Dekker Inc., New York (1987).
82. W. Wagner and E. Platen, Approximation of Itô integral equations, Preprint ZIMM, Acad. Sc. GDR, Berlin (1978).
83. W. Wedig, Stochastische Schwingungen - Simulation, Schätzung und Stabilität, ZAMM 67 (1987), T34-T42 (No. 4).
84. J.L. Willems, Mean square stability criteria for linear white noise stochastic systems, Probl. Cont. Inf. Theory 2 (1973), 199-217 (No. 3-4).
85. E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci. 3 (1965), 213-229.
86. W.M. Wonham, Optimal stationary control of a linear system with state-dependent noise, SIAM J. Control 5 (1967), 486-500.