A note on the Balanced method

Abstract

Recently the Balanced method was introduced as a class of quasi-implicit methods for solving stiff stochastic differential equations. We examine asymptotic and meansquare stability for several implementations of the Balanced method and give a generalized result for the mean-square stability region of any Balanced method. We also investigate the optimal implementation of the Balanced method with respect to strong convergence. (2000): 65C30, 65L07. AMS subject classification

Strong Predictor–corrector Euler Methods for Stochastic Differential Equations

“…As already emphasized, there exists a wide range of numerical stability concepts. For instance, it has been common to use second moments for identifying some type of mean-square stability, see Saito & Mitsui (1996), Higham (2000), Higham & Kloeden (2005) and Alcock & Burrage (2006). This may lead for some schemes to mathematically convenient characterizations of the resulting regions of meansquare stability.…”

“…We may refer for this approach, for instance, to papers by Talay (1982), Klauder & Petersen (1985), Milstein (1988), Hernandez & Spigler (1992, 1993, Saito & Mitsui (1993a, 1993b, Kloeden & Platen (1992, 1999, Milstein, Platen & Schurz (1998), Higham (2000) and Alcock & Burrage (2006). For general SDEs it is not straightforward to introduce implicitness into the approximation of the diffusion terms since ad hoc attempts lead typically to terms involving the inverse of Gaussian random variables, which can lead to explosions and, thus, makes numerically no sense.…”

“…In some applications, as given by the SDE (1.1), there is no point in making some drift terms in a scheme implicit since it would not change anything. For these reasons, balanced implicit methods have been developed in Milstein, Platen & Schurz (1998) and Alcock & Burrage (2006). These methods introduce some implicitness into the approximation of the diffusion term, which can significantly improve the numerical stability of the scheme.…”

Numerical Homogenization

“…We mention for example -methods that supplement oscillatory functions to a coarse FE space, pioneered by Babuška & Osborn [8], generalized through the so-called multiscale finite-element method (MsFEM) [9], developed since then by many authors (MsFEM using harmonic coordinates [10,11], see [12] for a survey and additional references), -methods based on the variational multiscale method (VMM) introduced in [13] and the residual free bubble (RFB) method [14] that are closely related to MsFEM-type strategy for homogenization problems [15], -methods based on the two-scale convergence theory and its generalization [3,16] as proposed in [17] and developed in [18] using sparse tensor product FEM, -projection-based numerical homogenization method based on projecting a fine-scale discretized problem into a low-dimensional space and eliminating successively the fine-scale components [19,20], and -numerical homogenization methods that supplement effective data for coarse FE computation and approximate the fine-scale solution via reconstruction such as the heterogeneous multiscale method (HMM) [21,22] or related micro-macro methods [23][24][25][26].…”

Quasi-exact approximation of hidden Markov chain filters

“…In particular, implicit or predictor-corrector methods are used to control the propagation of errors. We refer for this approach to papers by [1], [2], [7], [8], [11], [12], [15], [16], [20], [21] and [22]. There are various numerical schemes that perform well on some SDEs for certain parameter ranges and sufficiently small step sizes.…”