Abstract
The Local Linearization (LL) approach for the numerical solution of stochastic differential equations (SDEs) is extended to general scalar SDEs, as well as to non-autonomous multidimensional SDEs with additive noise. In case of autonomous SDEs, the derivation of the method introduced gives theoretical support to one of the previously proposed variants of the LL approach. Some numerical examples are given to demonstrate the practical performance of the method.
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References
Arnold, L. (1974). Stochastic Differential Equations: Theory and Applications, Wiley-Interscience Publications, New York.
Gard, T. C. (1988). Introduction to Stochastic Differential Equations, Marcel Dekker, New York.
Golub, G. H. and Van Loan, C. F. (1989). Matrix Computations, 2nd ed., The Johns Hopkins University Press, Maryland.
Henrici, P. (1962). Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York.
Kloeden, P. E. and Platen, E. (1989). A survey of numerical methods for stochastic differential equations, Stochastic Hydrology & Hydraulics, 3, 155–178.
Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer, Berlin.
Kloeden, P. E., Platen, E. and Scurz, H. (1993). Numerical Solution of SDE Through Computer Experiments, Springer, Berlin.
Maruyama, G. (1955). Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo, 4, 48–90.
McShane, E. J. (1974). Stochastic Calculus and Stochastic Models, Academic Press, New York.
Milshtein, G. N. (1974). Approximate integration of stochastic differential equations, Theory Probab. Appl., 19, 557–562.
Newton, N. J. (1991). Asymptotically efficient Runge-Kutta methods for a class of Ito and Stratonovich equations, SIAM J. Appl. Math., 52, 542–567.
Ozaki, T. (1985a). Nonlinear time series models and dynamical systems (eds. E. J. Hannan, P. R. Krishnaiah and M. M. Rao), Handbook of Statistics 5, North Holland, Amsterdam.
Ozaki, T. (1985b). Statistical identification of storage models with application to stochastic hydrology, Water Resource Bull., 21, 663–675.
Ozaki, T. (1992). A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: a local linearization approach, Statistica Sinica, 2, 113–135.
Ozaki, T. (1994). Personal communication.
Saito, Y. and Mitsui, T. (1992). Discrete approximation for stochastic differential equations, Transactions of the Japan Society for Industrial and Applied Mathematics, 2, 1–16 (in Japanese).
Saito, Y. and Mitsui, T. (1993). Simulation of stochastic differential equations, Ann. Inst. Statist. Math., 45, 419–432.
Schuss, Z. (1980). Theory and Applications of Stochastic Differential Equations, Wiley, New York.
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Biscay, R., Jimenez, J.C., Riera, J.J. et al. Local Linearization method for the numerical solution of stochastic differential equations. Ann Inst Stat Math 48, 631–644 (1996). https://doi.org/10.1007/BF00052324
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DOI: https://doi.org/10.1007/BF00052324