Abstract
In this chapter we shall apply explicit one-step methods such as strong Taylor schemes and explicit schemes, as well as multi-step and implicit schemes, to approximate the solutions of stochastic differential equations with respect to the strong convergence criterion. Numerical experiments will be used to investigate the convergence of these schemes.
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Literature for Chapter 4
First results on strong convergence of the Euler approximation can be found in Maruyama (1955) and Gikhman & Skorokhod (1979). Higher ord er discrete time strong approximations have been proposed, for example, by Milstein (1974) , Mc Shane (1974), Rao, Borwankar & Ramakrishna (1974), Kloed en & Pearson (1977) , Wagner & Platen, (1978), Clark (1978), Platen (1980, 1981), Clark & Cameron (1980) , Talay (1982), Riimelin (1982), Chang (1987), Milstein (1988a,b) , Nakazawa (1990) , Kloeden & Platen (1992a ,b). Implicit strong approximations were consid ered in Talay (1982), Klauder & Petersen (1985), Milstein (1988a), Smith & Gardiner (1988) , Drummond & Mortimer (1991) , Hernandez & Spigler (1991) , Petersen (1990), Kloeden & Platen (1992a ,b) or Saito & Mitsui (1992). Simulation studies can be found , for example, in Pardoux & Talay (1985) (see (4.5)) , Liske & Platen (1987), Newton (1991), Klauder & Petersen (1985), Kloeden, Platen & Schurz (1993). More references on st rong approximations are given in Kloeden & Platen (1992a).
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© 1994 Springer-Verlag Berlin Heidelberg
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Kloeden, P.E., Platen, E., Schurz, H. (1994). Strong Approximations. In: Numerical Solution of SDE Through Computer Experiments. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57913-4_4
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DOI: https://doi.org/10.1007/978-3-642-57913-4_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57074-5
Online ISBN: 978-3-642-57913-4
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