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 $alphain(frac{1}{2},1)$ 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 $t^{-frac{1-alpha}{2alpha}-eps}$ as $t to infty$ for any $eps>0$. 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.