Abstract
The aim of this chapter is to introduce weak approximations such as weak Taylor schemes, explicit and implicit weak schemes and their extrapolations to approximate functionals of the solutions of stochastic differential equations. Such functionals include, for example, moments of the solution at a given time instant as well as expectations of integrals of functions of such solutions over a given time interval. Simulation studies will provide an indication of the numerical efficiency of the schemes. Finally, some variance reduction methods will be discussed.
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Literature for Chapter 5
The weak Euler scheme and an order 2.0 weak Taylor scheme appeared in Milstein (1978). Order 2.0 weak convergence is shown for a class of schemes in Talay (1984) . General proofs on the convergence of weak Taylor schemes are given in Platen (1984), Milstein (1985, 1988a) and Kloeden & Platen (1992a), and for Ito processes with jump component in Mikulevicius & Platen (1991). Weak convergence for the Euler scheme under Holder continuous coefficients follows from a result in Mikulevicius & Platen (1991).
Completely derivative free explicit order 2.0 weak schemes were first proposed in Platen (1984). Other Runge-Kutta type methods can also be found in Talay (1984), Milstein (1978, 1988a) and Kloeden & Platen (1992a). Recent results on extrapolation methods for deterministic ODEs are summarized in Deuflhard (1985). Extrapolations for stochastic differential equations were first developed hy Talay & Tubaro (1990). Further results on extrapolation methods can be found in Kloeden & Platen (1992a) or Kloeden, Platen & Hofmann (1993). Milstein proposed some implicit weak schemes for additive noise in (1985, 1988a). Further general implicit and also predictor-corrector methods are given in Kloeden & Platen (1992a).
There are only a few simulation studies for weak approximations, for example, in Pardoux & Talay (1985), Klauder & Petersen (1985) and Liske & Platen (1987). The measure transformation method to reduce the variance is due to Milstein (1988a). A recent paper by Newton (1992) discusses also the control variate method for variance reduction. Other Monte-Carlo variance reduction methods can be found in Rubinstein (1981), Kalos & Whitlock (1986) and Mikhailov (1992). The results on unbiased and variance reducing estimators are due to Wagner (1988a,b, 1989a,b). Variance reduction using Hermite polynomials was proposed in Chang (1987). Further references on weak convergence can be found 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). Weak Approximations. In: Numerical Solution of SDE Through Computer Experiments. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57913-4_5
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DOI: https://doi.org/10.1007/978-3-642-57913-4_5
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