Abstract

Consider \(d\)-dimensional nonautonomous linear systems of real-valued Stratonovich- or Itô-interpreted stochastic differential equations \[ dX_t = G_0(t) X_t dt + \sum^m_{j=1} G_j(t) X_t \circ dW^j_t \tag{*} \] driven by independent standard Wiener processes \(W^j\), where \(G_j(t)\) \((j=0,1,\dots,m; t \in R^1_{+})\) are continuous-time matrix-valued functions which are supposed to be bounded by a constant \(K\). Let \(\Phi_{s,t}(\omega)\) denote the related two-parameter flow of linear operators acting on \(R^d\) and \(\{\lambda_1(\omega),\lambda_2(\omega),\dots,\lambda_d(\omega)\}\) its Lyapunov spectrum. Then, the SDE \((*)\) (and also \(\Phi\)) is called (forward sample) Lyapunov regular if its coefficient of regularity \(\sigma=\sigma(\omega)\) defined by \[ \sigma := \sum^d_{k=1} \lambda_k - \liminf_{t \to + \infty} \frac{1}{t} \ln | \det(\Phi_{0,t}(\omega))| \] is equal to \(0\) (P-a.s.). This paper studies this concept of sample Lyapunov regularity. Recall that regular ODE-systems have a number of good qualitative properties such as all its solutions have regular growth rate (exact Lyapunov exponents) and its asymptotic stability is robust under small nonlinear perturbations. The author shows that almost all nonautonomous systems of SDEs \((*)\) are (forward sample) Lyapunov regular. This extends classical results of Millionshchikov (1968) and \textit{V. I. Oseledets} [Trans. Mosc. Math. Soc. 19, 197–231 (1968); translation from Tr. Mosk. Mat. Obshch. 19, 179–210 (1968)] for ODEs. For this purpose, the author proves a stochastic version of the Perron theorem on triangularization of linear ODEs adapted to linear SDEs and a theorem on polar decomposition of linear SDEs in order to derive his main result on sample Lyapunov regularity of systems \((*)\).