Abstract
Abstract This paper is devoted to the study of reproduction of asymptotic boundedness in the second moment and small moments of stochastic differential equations by the stochastic theta method. In addition, we illustrate that the asymptotic moment boundedness of the numerical solution stand-alone plays a key role in the study of numerical stationary distribution.
Highlights
The numerical reproduction of asymptotic properties of stochastic differential equations (SDEs) studies that given the underlying SDE has certain asymptotic property how one chooses a proper numerical method such that the corresponding discrete numerical solution can reproduce the same property
Many papers are devoted to the numerical reproduction of stability of SDEs in different senses, such as mean square stability [ – ], almost sure stability [ – ], stability in small moment [ ], we just mention some of them here
The asymptotic boundedness plays an important role in the study of stationary distribution of SDEs
Summary
The numerical reproduction of asymptotic properties of stochastic differential equations (SDEs) studies that given the underlying SDE has certain asymptotic property how one chooses a proper numerical method such that the corresponding discrete numerical solution can reproduce the same property. Since the STM is semi-implicit when θ = , to ensure that this method is well defined, let us impose the one-sided Lipschitz condition on the drift coefficient f : there exists a constant b such that for any x, y ∈ Rn, x – y, f (x) – f (y) ≤ b|x – y|. 3.1 The second moment First we discuss the situation for θ ∈ [ , / ), which has the linear growth condition on both drift and diffusion coefficients. ) is asymptotically bounded in the second moment lim sup E x(t) ≤ a + a , ∀x( ) ∈ Rn. we consider reproducing this boundedness property by the STM. Without the linear growth condition on the drift coefficient, this theorem shows that the STM can still reproduce the boundedness property of true solution
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