Almost sure asymptotic stability and convergence of stochastic Theta methods applied to systems of linear SDEs in
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Bibliographic record
Abstract
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 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 non-anticipative 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 2 -martingales (i.e. martingale differences, not necessarily Gaussian) and can be interpreted as nonautonomous discretizations (e.g. with variable step sizes or dependence on time).
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