Almost sure asymptotic stability and convergence of stochastic Theta methods applied to systems of linear SDEs in

Abstract

Almost sure asymptotic stability of trivial solution and almost sure convergence of stochastic Theta methods applied to bilinear systems of ordinary stochastic differential equations (SDEs) of Itô-type in R d are proven. For this purpose, we prove and exploit a convergence theorem for non-negative semi-martingale decompositions, and verify a practical criteria based on the uniform boundedness of nonrandom eigenvalues related to certain matrix systems in any dimension d . We do not assume commutativity or simultaneous diagonalizability of drift and diffusion parts as many other authors, neither we restrict our analysis and applicability to only 2D or 3D cases nor to uniform step sizes (since the problem of adequate stochastic test equations cannot be solved within nonanticipative Itô calculus). However, an example of 2D diagonal-noised systems illustrates our approach. The discrete time systems of stochastic Theta methods are driven by L 2martingales (i.e. martingale differences, not necessarily Gaussian) and can be interpreted as nonautonomous discretizations (e.g. with variable step sizes or dependence on time).

A Martingale Approach to Asymptotic Stability Of Nonlinear Stochastic Difference Equations With Bounded Noise in R1

“…Remark 7.5. There are possible extensions of martingale approach to systems of stochastic difference equations (see [45]) and or systems of stochastic numerical methods (for an application to drift-implicit stochastic Theta methods, see [44]) in any dimension. Even systems of delay equations with memory effects can be studied by our approach (cf.…”

“…also [40], [41], [42], [43]. For a.s. asymptotic stability of linear systems of drift-implicit stochastic Theta methods in R d , see [44].…”

The Semimartingale Approach to Almost Sure Stability Analysis of a Two-Stage Numerical Method for Stochastic Delay Differential Equation

“…Theorem 4 (see [8,9]). Let = ( ) ∈ be an almost sure nonnegative stochastic sequence of (F , B)-measurable random variables on probability space (Ω, F, (F ) ∈ , P).…”

“…The mean-square stability analysis of numerical methods for SDDE has received a great deal of attention (see, e.g., [2,3] and the references therein). Recently, the almost sure (a.s.) stability (or the trajectory stability) is becoming prevalent in the science literature [4][5][6][7][8][9][10][11]. However, the prior works concerned with SDDE are [7,8,10].…”

Almost sure exponential stability of the Euler–Maruyama approximations for stochastic functional differential equations

“…On the other hand, by the discrete semimartingale convergence theorem (cf. [26,33]), [25][26][27] investigated the stability of stochastic difference equations and [31] examined almost sure stability of the stochastic theta methods of linear SDEs. Noting that there are similar expressions for the continuous and discrete semimartingale convergence theorems, [26] obtained the sufficient conditions for almost surely asymptotic stability of both exact and numerical solutions of linear stochastic differential equations.…”