Numerical methods for strong solutions of stochastic differential equations: An overview

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 460, Issue 2041, 2004, Pages 373-402

  Burrage, K ^a   Burrage, P M ^a   Tian, T ^a  

^a UNIVERSITY OF QUEENSLAND   (Australia)

Author keywords

Numerical methods; Stochastic differential equations; Strong solutions

Indexed keywords

AB INITIO CALCULATION; ANALYTIC METHOD; CONFERENCE PAPER; MATHEMATICAL ANALYSIS; MATHEMATICAL MODEL; SIMULATION; STOCHASTIC MODEL; THEORETICAL MODEL;

EID: 1542512456     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2003.1247     Document Type: Conference Paper

References (83)

1. Alcock, J. & Burrage, K. 2004 A note on the balanced method. (Submitted.)
2. Allison, S. A. 1986 Brownian dynamics simulation of wormlike chains. Macromolecules 19, 1184-1204.
3. Arnold, L. 1974 Stochastic differential equations: theory and applications. Wiley.
4. Baxendale, P. 1986 Asymptotic behaviour of stochastic flow of diffeomorphisms. Springer Lecture Notes in Mathematics, vol. 1203. Springer.
5. Ben-Arous, G. 1989 Flots et series de Taylor stochastiques. Prob. Theory Relat. Fields 81, 29-77.
6. Brugnano, L., Burrage, K. & Burrage, P. M. 2000 Adams-type methods for the numerical solution of stochastic ordinary differential equations. BIT 40, 451-470.
7. Burrage, K. 1995 Parallel and sequential methods for ordinary differential equations. Oxford University Press.
8. Burrage, P. M. 1999 Runge-Kutta methods for stochastic differential equations. PhD thesis, University of Queensland, Australia.
9. Burrage, K. & Burrage, P. M. 1999 High strong order methods for non-commutative stochastic ordinary differential equation system and the Magnus formula. Physica D 133, 34-48.
10. Burrage, K. & Burrage, P. M. 2000 Order conditions of stochastic Runge-Kutta methods by B-series. SIAM J. Numer. Analysis 38, 1626-1646.
11. Burrage, P. M. & Burrage, K. 2002 A variable stepsize implementation for stochastic differential equations. SIAM J. Sci. Comput. 24, 848-864.
12. Burrage, K. & Tian, T. H. 2000 A note on the stability properties of the Euler methods for solving stochastic differential equations. NZ J. Math. 29, 115-127.
13. Burrage, K. & Tian, T. H. 2002 Predictor-corrector methods of Runge-Kutta type for stochastic differential equations. SIAM J. Numer. Analysis 40, 1516-1537.
14. Burrage, K., Burrage, P. M. & Tian, T. H. 2001 Numerical methods for solving stochastic differential equations on parallel computers. In Proc. 5th Int. Conf. on High-Performance Computing in the Asia-Pacific Region.
15. Butcher, J. C. 1963 Coefficients for the study of Runge-Kutta integration processes. J. Aust. Math. Soc. 3, 185-201.
16. Butcher, J. C. 1987 The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods. Wiley.
17. Caverhill, A., Chappel, M. & Elworthy, K. D. 1986 Characteristic exponents for stochastic flow. Springer Lecture Notes in Mathematics, vol. 1158. Springer.
18. Chang, C. C. 1987 Numerical solution of stochastic differential equations with constant diffusion coefficients. Math. Comp. 49, 523-542.
19. Denk, G. & Schäffler, S. 1997 Adams methods for the efficient solution of stochastic differential equations with additive noise. Computing 59, 153-161.
20. Ermak, D. L. & McCammon, J. A. 1978 Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69, 1352-1360.
21. Fisher, P. & Platen, E. 1999 Applications of the balanced method to stochastic differential equations in filtering. Monte Carlo Meth. Applic. 5, 19-38.
22. Fogelson, A. L. 1984 A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting. J. Computat. Phys. 50, 111-134.
23. Gaines, J. G. & Lyons, T. J. 1997 Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math. 57, 1455-1484.
24. Gard, T. C. 1988 Introduction to stochastic differential equations. New York: Dekker.
25. Geman, S. & Hwang, C. 1986 Diffusions for global optimization. SIAM J. Control Optimiz. 24, 1031-1043.
26. Gillespie, D. T. 1992 Markov processes: an introduction for physical scientists. Academic.
27. d. Adv. Appl. Math. 9, 35-39.
28. Gustafsson, K. 1991 Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods. ACM Trans. Math. Software 17, 533-554.
29. Hairer, E. & Wanner, G. 1991 Solving ordinary differential equations. II. Stiff and differential-algebraic systems. Springer.
30. Hernandez, D. B. & Spigler, R. 1992 A-stability of Runge-Kutta methods for systems with additive noise. BIT 32, 620-633.
31. Higham, D. J. 2000 Mean-square and asymptotic stability of numerical methods for stochastic ordinary differential equations. SIAM J. Numer. Analysis 38, 753-769.
32. Higham, D. J., Mao, X. & Stuart, A. 2002 Strong convergence of Euler-type methods for non-linear stochastic differential equations. SIAM J. Numer. Analysis 40, 1041-1063.
33. Hofmann, N. 1995 Stability of weak numerical schemes for stochastic differential equations. Math. Comput. Simul. 38, 63-68.
34. Hofmann, N. & Müller-Gronbach, T. 2004 On the global error of Itô-Taylor schemes for strong approximation of scalar stochastic differential equations. SIAM J. Sci. Comput. (In the press.)
35. Hofmann, N. & Platen, E. 1994 Stability of weak numerical schemes for stochastic differential equations. Comput. Math. Applic. 28, 45-47.
36. Hofmann, N. & Platen, E. 1996 Stability of superimplicit numerical methods for stochastic differential equations. Fields Inst. Commun. 9, 93-104.
37. Hofmann, N., Müller-Gronbach, T. & Ritter, K. 2000 Step-size control for the uniform approximation of systems of stochastic differential equations with additive noise. Ann. Appl. Prob. 10, 613-633.
38. Hofmann, N., Müller-Gronbach, T. & Ritter, K. 2001 The optimal discretization of stochastic differential equations. J. Complexity 17, 117-153.
39. Hume, D. 2000 Probability in transcriptional regulation and its implications for leukocyte differentiation and inducible gene expression. Blood 96, 2323-2328.
40. Iserles, A., Marthinsen, A. & Nørsett, S. P. 1999 On the implementation of the method of Magnus series for linear differential equations. BIT 39, 281-304.
41. Itô, K. 1984 Introduction to probability theory. Cambridge University Press.
42. Kloeden, P. E. & Pearson, R. A. 1977 The numerical solution of stochastic differential equations. J. Aust. Math. Soc. B 20, 8-12.
43. Kloeden, P. E. & Platen, E. 1991 Stratonovich and Itô stochastic Taylor expansions. Math. Nachr. 151, 33-50.
44. Kloeden, P. E. & Platen, E. 1992 Numerical solution of stochastic differential equations. Springer.
45. Komori, Y., Saito, Y. & Mitsui, T. 1994 Some issues in discrete approximate solution for stochastic differential equations. Comput. Math. Applic. 28, 10-12, 269-278.
46. Komori, Y., Mitsui, T. & Sugiura, H. 1997 Rooted tree analysis of the order conditions of ROW-type scheme for stochastic differential equations. BIT 37, 43-66.
47. Kunita, H. 1990 Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, vol. 24. Cambridge University Press.
48. Kusle, R. 2000 Gradient particle solutions of Fokker-Planck equations for noisy delay bifurcations. SIAM J. Sci. Comput. 22, 351-367.
49. Lehn, J., Rössler, A. & Schein, O. 2002 Adaptive schemes for the numerical solution of SDEs - a comparison. J. Comput. Appl. Math. 138, 297-308.
50. Lelarasmee, E., Ruehli, A. & Sangiovanni-Vincentelli, A. 1982 The waveform relaxation method for time-domain analysis of large-scale integrated circuits. IEEE Trans. CAD Integr. Circuits Syst. 1, 131-145.
51. Lépingle, D. & Ribémont, B. 1991 A multistep approximation scheme for the Langevin equation. Stoch. Process. Applic. 37, 61-69.
52. Levy, P. 1948 Processus stochastiques et mouvement Brownien. Monographies des Probabilités. Paris: Gauthier-Villars.
53. Lythe, G. 1994 Noise and dynamic transactions. In Stochastic PDEs (ed. A. Etheridge), pp. 181-188. Cambridge University Press.
54. Lythe, G. 1996 Domain formation in transitions with noise and a time-dependent bifurcation parameter. Phys. Rev. E 53, 4271-4274.
55. Magnus, W. 1954 On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649-673.
56. Malliavin, P. 1997 Stochastic analysis. Springer.
57. Maruyama, G. 1955 Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo 4, 48-90.
58. Mauthner, S. 1998 Step-size control in the numerical solution of stochastic differential equations. J. Comput. Appl. Math. 100, 93-109.
59. Merton, R. 1976 Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125-144.
60. Milstein, G. N. 1995 Numerical integration of stochastic differential equations. Dordrecht: Kluwer Academic.
61. Milstein, G. N., Platen, E. & Schurz, H. 1998 Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Analysis 35, 1010-1019.
62. Milstein, G. N., Repin, Y. M. & Tretyakov, M. V. 2002 Numerical methods for stochastic systems preserving symplectic structure. SIAM J. Numer. Analysis 40, 1583-1604.
63. Newton, N. J. 1991 Asymptotically efficient Runge-Kutta methods for a class of Itô and Stratonovich equations. SIAM J. Appl. Math. 51, 542-567.
64. Öttinger, H. C. 1996 Stochastic processes in polymeric fluids. Springer.
65. Platen, E. 1984 Zur zeitdiskreten Approximation von Itoprozessen. Diss. B., IMath, Akademie der Wissenschaften der DDR, Berlin.
66. Platen, E. 1999 An introduction to numerical methods for stochastic differential equations. Acta Numerica 8, 197-246.
67. Platen, E. & Wagner, W. 1982 On a Taylor formula for a class of Itô processes. Prob. Math. Statis. 3, 37-51.
68. Rümelin, W. 1982 Numerical treatment of stochastic differential equations. SIAM J. Numer. Analysis 19, 604-613.
69. Saito, Y. & Mitsui, T. 1993 T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Appl. Analysis 2, 333-344.
70. Saito, Y. & Mitsui, T. 1996 Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Analysis 33, 2254-2267.
71. Sano, Y., Shimada, T., Nakashima, H., Nicholson, R. H., Eliason, J. F., Kocarek, T. A. & Ko, M. S. H. 2001 Random monoallelic expression of three genes clustered within 60 kb of mouse t complex genomic DNA. Genome Res. 11, 1833-1845.
72. Schlick, T. 2002 Molecular modelling and simulation: an interdisciplinary guide. Springer.
73. Schneider, K. R. & Schurz, H. 2000 Waveform relaxation methods for stochastic differential equations. WIAS Report 620.
74. 2([0, T]μ). Report, Mathematics Department, Southern Illinois University.
75. Schuss, Z. 1980 Theory and applications of stochastic differential equations. Wiley.
76. Skorokhod, A. V. 1965 Studies in the theory of random processes. Addison-Wesley.
77. Sobczyk, K. 1991 Stochastic differential equations, with applications to physics and engineering. Dordrecht: Kluwer Academic.
78. Söderlind, G. 1998 The automatic control of numerical integration. CWI Q. 11, 55-74.
79. Tian, T. H. & Burrage, K. 2001 Implicit Taylor methods for stiff stochastic differential equations. Appl. Numer. Math. 38, 167-185.
80. Tian, T. H. & Burrage, K. 2002 Two-stage stochastic Runge-Kutta methods for stochastic differential equations. BIT 42, 625-643.
81. Veuthey, A.-L. & 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.
82. Wagner, W. & Platen, E. 1978 Approximation of Itô integral equations. Preprint ZIMM, Akademie der Wissenschaften der DDR, Berlin.
83. Watanabe, S. & Yamada, T. 1971 On the uniqueness of solutions to stochastic differential equations. II. J. Math. Kyoto Univ. 11, 553-563.