Stochastic ordinary differential equations in applied and computational mathematics

IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)

Volumn 76, Issue 3, 2011, Pages 449-474

  Higham, Desmond J ^a  

^a UNIVERSITY OF STRATHCLYDE   (United Kingdom)

Author keywords

finance; modelling; stochastic computation; systems biology

Indexed keywords

BACKGROUND KNOWLEDGE; CHEMICAL KINETICS; COMPUTATIONAL MATHEMATICS; DIFFUSION LIMITS; KEY TOPICS; MODEL COMPARISON; MODELLING; MONTE CARLO SIMULATION; MULTI-SCALE MODELLING; MULTI-SCALE SIMULATION; NON-LINEAR MODEL; PHYSICAL MODEL; STOCHASTIC COMPUTATION; STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS; STOCHASTIC PROCESS; SYSTEMS BIOLOGY; UNCERTAINTY QUANTIFICATIONS; WEAK AND STRONG CONVERGENCE;

COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL TECHNIQUES; MODELS; MONTE CARLO METHODS; ORDINARY DIFFERENTIAL EQUATIONS; PARAMETER ESTIMATION; RANDOM PROCESSES; STOCHASTIC SYSTEMS;

STOCHASTIC MODELS;

EID: 79957937424     PISSN: 02724960     EISSN: 14643634     Source Type: Journal    
DOI: 10.1093/imamat/hxr016     Document Type: Article

References (57)

1. AIT-SAHALIA, Y. (1999) Testing continuous-time models of the spot interest rate. Rev. Financ. Stud., 9, 385-426.
2. ANDERSON, D. F., GANGULY, A. & KURTZ, T. G. (2011) Error analysis of tau-leap simulation methods. Ann. Appl. Probab. (to appear).
3. ANDERSON, D. F. & MATTINGLY, J. C. (2011) A weak trapezoidal method for a class of stochastic differential equations. Commun. Math. Sci., 9, 301-318.
4. AVIKAINEN, R. (2009) Convergence rates for approximations of functionals of SDEs. Finance Stochastics, 13, 381-401.
5. BALL, K., KURTZ, T. G., POPOVIC, L. & REMPALA, G. (2006) Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Probab., 16, 1925-1961.
6. BESKOS, A., ROBERTS, G. O. & STUART, A. M. (2009) Optimal scalings of Metropolis-Hastings algorithms for non-product targets in high dimensions. Ann. Appl. Probab., 19, 863-898.
7. BESKOS, A., ROBERTS, G. O., STUART, A. M. & VOSS, J. (2008) MCMC methods for diffusion bridges. Stochastics Dyn., 8, 319-350.
8. CALDERHEAD, B., GIROLAMI, M. & HIGHAM, D. J. (2011) Is it safe to go out yet? Statistical inference in a zombie outbreak model. Academics on Zombies (Robert Smith? ed.). University of Ottawa Press (to appear).
9. CAO, Y., GILLESPIE, D. T. & PETZOLD, L. (2005) The slow-scale stochastic simulation algorithm. J. Chem. Phys., 122, 014116.
10. COX, D. & MILLER, H. (1965) The Theory of Stochastic Processes. London: Methuen.
11. CYGANOWSKI, S., KLOEDEN, P. & OMBACH, J. (2002) From Elementary Probability to Stochastic Differential Equations with MAPLE. Berlin: Springer.
12. LIU, D. & VANDEN-EIJNDEN, E. (2005) Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Chem. Phys., 123, 194107.
13. ERBAN, R., KEVREKIDIS, I. G., ADALSTEINSSON, D. & ELSTON, T. C. (2006) Gene regulatory networks: A coarse-grained equation-free approach to multiscale computation. J. Chem. Phys., 124, 084106.
14. GADGIL, C., LEE, C. H. & OTHMER, H. G. (2005) A stochastic analysis of first-order reaction networks. Bull. Math. Biol., 67, 901-946. (Pubitemid 40912459)
15. GILES, M. B. (2007) Improved multilevel Monte Carlo convergence using the Milstein scheme. Monte Carlo and Quasi-Monte Carlo Methods. Springer, pp. 343-358. Available at http://www.springerlink.com/content/ x03273ju277j7524/.
16. GILES, M. B. (2008) Multilevel Monte Carlo path simulation. Oper. Res., 56, 607-617.
17. GILES, M. B., HIGHAM, D. J. & MAO, X. (2009) Analysing multi-level Monte Carlo for options with nonglobally Lipschitz payoff. Financ. Stochastics, 13, 403-413.
18. GILES, M. B. & WATERHOUSE, B. J. (2009) Multilevel Quasi-Monte Carlo Path Simulation. Radon Series Comp. Appl. Math., 8, 165-181.
19. GILLESPIE, D. T. (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys., 22, 403-434.
20. GILLESPIE, D. T. (1977) Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81, 2340-2361.
21. GILLESPIE, D. T. (2001) Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys., 115, 1716-1733. (Pubitemid 32715518)
22. GIROLAMI, M. & CALDERHEAD, B. (2011) Riemann manifold Langevin and Hamiltonian Monte Carlo methods (with discussion). J. R. Stat. Soc. B, 73, 1-37.
23. HAIRER, E. & WANNER, G. (1996) Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, 2nd edn. Berlin: Springer.
24. HASTY, J., PRADINES, J., DOLNIK, M. & COLLINS, J. J. (2000) Noise-based switches and amplifiers for gene expression. Proc. Natl. Acad. Sci. U.S.A., 97, 2075-2080. (Pubitemid 30134024)
25. HIGHAM, D. J. (2000) Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal., 38, 753-769. (Pubitemid 33778937)
26. HIGHAM, D. J. (2004) An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation. Cambridge: Cambridge University Press.
27. HIGHAM, D. J., MAO, X. & STUART, A. M. (2002) Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal., 40, 1041-1063.
28. HIGHAM, D. J., MAO, X. & YUAN, C. (2007) Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM J. Numer. Anal., 45, 592-609. (Pubitemid 351599169)
29. HUTZENTHALER, M., JENTZEN, A. & KLOEDEN, P. E. (2011) Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. A Available at http://rspa.royalsocietypublishing.org/content/early/2010/12/08/ rspa.2010.0348.full.
30. IACUS, S. M. (2008) Simulation and Inference for Stochastic Differential Equations: With R Examples. New York: Springer.
31. INTEP, S., HIGHAM, D. J. & MAO, X. (2009) Switching and diffusion models for gene regulation networks. Multiscale Model. Simul., 8, 30-45.
32. JAHNKE, T. (2010) An adaptive wavelet method for the chemical master equation. SIAM J. Sci. Comput., 31, 4373-4394.
33. JAYNES, E. (2003) Probability Theory: The Logic of Science. Cambridge: Cambridge University Press.
34. KLOEDEN, P. & NEUENKIRCH, A. (2007) The pathwise convergence of approximation schemes for stochastic differential equations. Lond. Math. Soc., 10, 235-253.
35. KLOEDEN, P. E. & PLATEN, E. (1999) Numerical Solution of Stochastic Differential Equations, 3rd printing. Berlin: Springer.
36. KURTZ, T. G. (1981) Approximation of Population Processes. Philadelphia, UA: SIAM.
37. LEIER, A., MARQUEZ-LAGO, T. & BURRAGE, K. (2008) Generalized binomial tau-leap method for biochemical processes incorporating both delay and intrinsic noise. J. Chem. Phys., 128, 205107.
38. LEIMKUHLER, B. & REICH, S. (2005) Simulating Hamiltonian Dynamics. Cambridge: Cambridge University Press.
39. MAO, X. (2007) Stochastic Differential Equations and Applications, 2nd edn. Chichester, UK: Horwood.
40. MATTINGLY, J., STUART, A. M. & HIGHAM, D. (2002) Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl., 101, 185-232.
41. MATTINGLY, J., STUART, A. M. & TRETYAKOV, M. V. (2010) Convergence of numerical time-averaging and stationary measures via the poisson equation. SIAM J. Numer. Anal., 48, 552-577.
42. MIKOSCH, T. (1998) Elementary Stochastic Calculus (with Finance in View). Singapore: World Scientific.
43. MILSTEIN, G. N. & TRETYAKOV, M. V. (2004) Stochastic Numerics for Mathematical Physics. Berlin: Springer.
44. MILSTEIN, G. N. & TRETYAKOV, M. V. (2005) Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients. SIAM J. Numer. Anal., 43, 1139-1154. (Pubitemid 43952283)
45. RENSHAW, E. (1991) Modelling Biological Populations in Space and Time. Cambridge: Cambridge University Press.
46. RIPLEY, B. D. (1987) Stochastic Simulation. Chichester: Wiley.
47. ROBERT, C. P. & CASELLA, G. (2004) Monte Carlo Statistical Methods, 2nd edn. Berlin, Germany: Springer.
48. SIVIA, D. & SKILLING, J. (2006) Data Analysis: A Bayesian Tutorial. Oxford: Oxford University Press.
49. STUART, A. M. (2010) Inverse problems: A Bayesian perspective. Acta Numerica, 19, 451-559.
50. SZPRUCH, L. & HIGHAM, D. J. (2010) Comparing hitting time behavior of Markov jump processes and their diffusion approximations. Multiscale Model. Simul., 8, 605-621.
51. SZPRUCH, L., MAO, X., HIGHAM, D. J. & PAN, J. (2011) Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model. BIT Numer. Math. Available at http://www.springerlink.com/content/ 424tt0076k38685u/.
52. TALAY, D. (1999) Approximation of invariant measures of nonlinear Hamiltonian and dissipative stochastic differential equations. Progress in Stochastic Structural Dynamics (R. Bouc & C. Soize eds), vol. 152. Publication du L.M.A.-C.N.R.S., pp. 139-169.
53. TALAY, D. & TUBARO, L. (1990) Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl., 8, 94-120.
54. VYSHEMIRSKY, V. & GIROLAMI, M. (2008) Bayesian ranking of biochemical system models. Bioinformatics, 24, 833-839. (Pubitemid 351386922)
55. WILKIE, J. & WONG, Y. M. (2008) Positivity preserving chemical Langevin equations. Chem. Phys., 353, 132-138.
56. WILKINSON, D. J. (2006) Stochastic Modelling for Systems Biology. Boca Raton, FL: Chapman & Hall/CRC.
57. YOUNG, W. M. & ELCOCK, E. W. (1966) Monte Carlo studies of vacancy migration in binary ordered alloys: I. Proc. Phys. Soc., 89, 735-746.