Weak exponential schemes for stochastic differential equations with additive noise

IMA Journal of Numerical Analysis

Volumn 25, Issue 3, 2005, Pages 486-506

  Mora, Carlos M ^a  

^a UNIVERSITY OF CONCEPCIÓN   (Chile)

Author keywords

Exponential schemes; Local Linearization Method; Rate of convergence; Stochastic differential equations; Weak numerical solution

Indexed keywords

ADDITIVE NOISE; ADDITIVES; APPROXIMATION THEORY; DIFFERENTIAL EQUATIONS; ITERATIVE METHODS; STOCHASTIC SYSTEMS;

EXPONENTIAL SCHEMES; LOCAL LINEARIZATION METHOD; NUMERICAL SOLUTION; RATE OF CONVERGENCE; STOCHASTIC DIFFERENTIAL EQUATIONS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 26644435080     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/dri001     Document Type: Article

References (35)

1. ARNOLD, L. (1974) Stochastic Differential Equations: Theory and Applications. New York: John Wiley & Sons.
2. BISCAY, R., JIMENEZ, J. C., RIERA, J. & VALDES, P. A. (1996) Local linearization method for the numerical solution of stochastic differential equations. Ann. Inst. Stat. Math., 48, 631-644.
3. DELLACHERIE, C. & MEYER, P.-A. (1978) Probabilities and Potential. Paris: Hermann
4. (Translation of Probabilities et potentiel. Hermann, 1975).
5. ETHIER, S. N. & KURTZ, T. G. (1986) Markov Processes. New York: John Wiley & Sons.
6. HIGHAM, D. J. (2001) An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 43, 525-546.
7. HOCHBRUCK, M. & LUBICH, CH. (1997) On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal., 34, 1911-1925.
8. HOCHBRUCK, M. & LUBICH, CH. (1999) Exponential integrators for quantum-classical molecular dynamics. BIT, 39, 620-645.
9. HOCHBRUCK, M., LUBICH, CH. & SELHOFER, H. (1998) Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19, 1552-1574.
10. JIMÉNEZ, J. C. & BISCAY, R. J. (2002) Approximation of continuous time stochastic processes by the local linearization method revisited. Stochastic Anal. Appl., 20, 105-121.
11. JIMÉNEZ, J. C., SHOJI, I. & OZAKI, T. (1999) Simulation of stochastic differential equations through the local linearization method. A comparative study. J. Stat. Phys., 94, 587-602.
12. JIMÉNEZ, J. C., BISCAY, R., MORA, C. & RODRIGUEZ, L. M. (2002) Dynamic properties of the local linearization method for initial-value problems. Appl. Math. Comput., 126, 63-81.
13. KLOEDEN, P. E. & PLATEN, E. (1992) Numerical Solution of Stochastic Differential Equations. Berlin: Springer.
14. MATTINGLY, J. C., STUART, A. M. & HIGHAM, D. J. (2002) Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl., 101, 185-232.
15. MILSHTEIN, G. N. (1985) Weak approximation of solutions of systems of stochastic differential equations. Theor. Prob. Appl., 30, 750-766.
16. MILSTEIN, G. N. (1995) The Numerical Integration of Stochastic Differentials Equations. Dordrecht: Kluwer (English translation from Russian 1988).
17. MILSTEIN, G. N. & TRETYAKOV, M. V. (2003) Quasi-symplectic methods for Langevin type equations. IMA J. Numer. Anal., 23, 593-626.
18. MILSTEIN, G. N., PLATEN, E. & SCHURZ, H. (1998) Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Anal., 35, 1010-1019.
19. MILSTEIN, G. N., REPIN, Yu. M. & TRETYAKOV, M. V. (2002) Symplectic integration of Hamiltonian systems with additive noise. SIAM J. Numer. Anal., 39, 2066-2088.
20. MOLER, C. & VAN LOAN, C. (1978) Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev., 20, 801-836.
21. MORA, C. M. (2002) Solución numérica de ecuaciones diferenciales estocásticas mediante métodos exponenciales. Ph.D. Thesis, Pontificia Universidad Católica de Chile.
22. MORA, C. M. (2004a) Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups. Math. Comput., 247, 1393-1415.
23. MORA, C. M. (2004b) Numerical Solution of Conservative Finite-dimensional Stochastic Schrödinger Equations. Preprint.
24. OZAKI, T. (1985a) Nonlinear Time Series Models and Dynamical Systems. Handbook of Statistics 5, Amsterdam: North Holland, 25-83.
25. OZAKI, T. (1985b) Statistical identification of storage models with application to stochastic hydrology. Water Resour. Bull., 21, 663-675.
26. PERCIVAL, I. (1998) Quantum State Diffusion. Cambridge University Press.
27. PROTTER, P. (1990) Stochastic Integration and Differential equations. A New Approach. Berlin: Springer.
28. SHOJI, I. (1998) Approximation of continuous time stochastic processes by a local linearization method. Math. Comput., 67, 287-298.
29. SHOJI, I. & OZAKI, T. (1998) Estimation for nonlinear stochastic differential equations by a local linearization method. Stochastic Anal. Appl., 16, 733-752.
30. TALAY, D. (1984) Efficient Numerical Schemes for the Approximation of Expectations of Functionals of S.D.E. Filtering and Control of Random Processes (J. Szpirglas, H. Korezlioglu & G. Mazziotto eds). Lecture Notes in Control and Information Science 61. Berlin: Springer, 294-313.
31. TALAY, D. (1986) Discrétisation d'une E.D.S. et calcul approché d'espérances de fonctionnelles de la solution. Math. Model. Numer. Anal., 20, 141-179.
32. TALAY, D. (1995) Simulation and numerical analysis of stochastic differential systems: a review. Probabilistic Methods in Applied Physics. Berlin: Springer, pp. 54-96.
33. TALAY, D. (1996) Probabilistic Numerical Methods for Partial Differential Equations: Elements of Analysis. Probabilistic Models for Nonlinear Partial Differential Equations (D. Talay and L. Tubaro eds). Lecture Notes in Math. 1627. Berlin: Springer, 148-196.
34. TALAY, D. (2002) Stochastic Hamiltonian dissipative systems with non globally Lipschitz coefficients: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Related Fields, 163-198.
35. TALAY, D. & TUBARO, L. (1990) Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl., 8, 483-509.