In this paper, we study the existence of $n$-dimensional linear stochastic differential equations (SDEs) such that the sign of Lyapunov exponents is changed under an exponentially decaying perturbation. First, we show that the equation with all positive Lyapunov exponents will have $n-1$ linearly independent solutions with negative Lyapunov exponents under the perturbation. Meanwhile, we prove that the equation with all negative Lyapunov exponents will also have solutions with positive Lyapunov exponents under another similar perturbation. Finally, we show that other three kinds of perturbations which appear at different positions of the equation will change the sign of Lyapunov exponents.

在本文中，我们研究了存在$n$维线性随机微分方程 (SDE)，使得 Lyapunov 指数的符号在指数衰减扰动下发生变化。首先，我们证明具有所有正李雅普诺夫指数的方程将有$n-\mathrm{1个}$在扰动下具有负 Lyapunov 指数的线性独立解。同时，我们证明了所有负李雅普诺夫指数的方程在另一个类似的摄动下也将有正李雅普诺夫指数的解。最后证明了出现在方程不同位置的其他三种扰动会改变Lyapunov指数的符号。