Abstract
In this chapter we shall describe several methods which allow a reduction in the variance of functionals of weak approximations of Ito diffusions. One method changes the underlying probability measure by means of a Girsanov transformation, another uses general principles of Monte-Carlo integration. Unbiased estimators are also constructed.
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Bibliographical Notes
Hammersley, J. M. and D. C. Handscomb (1964). Monte Carlo Methods. Methuen, London.
Karlin, S. and H. M. Taylor (1975). A First Course in Stochastic Processes ( 2nd ed. ). Academic Press, New York.
Kloeden, P. E. and R. A. Pearson (1977). The numerical solution of stochastic differential equations. J. Austral. Math. Soc. Ser. B 20, 8–12.
Lanska, V. (1979). Minimum contrast estimation in diffusion processes. J. AppL Probab. 16, 65–75.
Le Gland, F. (1981). Estimation de paramétres dans les processus stochastiques en observation incompléte: Application d un probléme de radio-astronomie. Ph. D. thesis, Dr. Ing. Thesis, Univ. Paris IX (Dauphine).
Liske, H. (1982). Distribution of a functional of a Wiener process. Theory of Random Processes 10, 50–54. (in Russian).
Ripley, B. D. (1983b). Stochastic Simulation. Wiley, New York.
Schöner, G., H. Haken, and J. A. S. Kelso (1986). A stochastic theory of phase transitions in human hand movement. Biol. Cybern. 53, 247–257.
Bratley, P., B. L. Fox, and L. Schrage (1987). A Guide to Simulation (2nd ed.). Springer.
Chang, C. C. (1987). Numerical solution of stochastic differential equations with constant diffusion coefficients. Math. Comp. 49, 523–542.
Wagner, W. (1987a). Unbiased Monte-Carlo evaluation of certain functional integrals. J. Comput. Phys. 71 (1), 21–33.
Wagner, W. (1988a). Monte-Carlo evaluation of functionals of solutions of stochastic differential equations. Variance reduction and numerical examples. Stochastic Anal. Appl. 6, 447–468.
Milstein, G. N. (1988b). A theorem of the order of convergence of mean square approximations of systems of stochastic differential equations. Theory Probab. AppL 32, 738–741.
Wagner, W. (1989a). Stochastische numerische Verfahren zur Berechnung von Funktionalintegralen. (in German), Habilitation, Report 02/89, IMATH, Berlin.
Wagner, W. (1989b). Unbiased Monte-Carlo estimators for functionals of weak solutions of stochastic differential equations. Stochastics Stochastics Rep. 28(1), 1–20.
Wozniakowski, H. (1991). Average case complexity of multivariate integration. Bull. Austral. Math. Soc. 24 (1), 185–194.
Yen, V. V. (1992). A stochastic Taylor formula for two-parameter stochastic processes. Probab. Math. Statist. 13 (1), 149–155.
Chorin, A. J. (1971). Hermite expansions in Monte-Carlo computation. J. Comput. Phys. 8, 472–482.
Chorin, A. J. (1973a). Accurate evaluation of Wiener integrals. Math. Comp. 27, 1–15.
Chorin, A. J. (1973b). Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–786.
Maltz, F. H. and D. L. Hitzl (1979). Variance reduction in Monte-Carlo computations using multi-dimensional Hermite polynomials. J. Comput. Phys. 32, 345–376.
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© 1992 Springer-Verlag Berlin Heidelberg
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Kloeden, P.E., Platen, E. (1992). Variance Reduction Methods. In: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12616-5_16
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DOI: https://doi.org/10.1007/978-3-662-12616-5_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08107-1
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