Mean-Square and Asymptotic Stability of the Stochastic Theta Method

By Desmond John Higham. SIAM Journal on Numerical Analysis, 2000

377 citations recorded by the source corpus.

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Citations in the corpus

Exponential stability in p -th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations (2005, background): “Higham’s results [16] in the SODE case).”
Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations (2007, background): “In particular, the “if and only if” results for linear SDEs can be founded in [13,31] while for general nonlinear SDEs in [14].”
A-Stability and Stochastic Mean-Square Stability (2000, methodology): “We mention that removing the last term in (1.2) lowers the strong order of the method and produces the semi-implicit Euler scheme, or Stochastic Theta Method [ 4 , 9]. This method was analyzed in [4], and was shown to inherit the mean-square extension of deterministic A-stability for all 1 2 ≤ θ ≤ 1”
Characterization of bistability for stochastic multistep methods (2010, unknown): “For stochastic ordinary differential equations this kind of stability is studied in [4, 5, 14, 15].”
A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems (2012, methodology): “Preprint submitted to Applied Numerical Mathematics February 20, 2012”
An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations (2001, background): “The material on linear stability in section 7 is based on [2] and [12].”
Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations (2002, methodology): “2] or the stochastic theta method class [10] and is sometimes called the semi-implicit Euler method [2].”
Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations (2006, methodology): “We also give the stability regions for the stochastic θ-Maruyama methods, based on the results in [12].”
CONVERGENCE AND STABILITY OF IMPLICIT METHODS FOR JUMP-DIFFUSION SYSTEMS (2005, methodology): “for introducing and studying (2) is that for the ODE and SDE cases, it has been found that the class offers good linear stability properties [8, 9, 23] and excellent potential for capturing long time dynamics [20, 25].”