Asymptotic separation between solutions of Caputo fractional stochastic differential equations

Using a temporally weighted norm we first establish a result on the global existence and uniqueness of solutions for Caputo fractional stochastic differential equations of order $\alpha \in (\frac{1}{2},1)$ whose coefficients satisfy a standard Lipschitz condition. For this class of systems we then show that the asymptotic distance between two distinct solutions is greater than $t^{-\frac{1-\alpha }{2\alpha }-\text{eps}}$ as $t\to \mathrm{\infty }$ for any $\text{eps}>0$. As a consequence, the mean square Lyapunov exponent of an arbitrary non-trivial solution of a bounded linear Caputo fractional stochastic differential equation is always non-negative.