  In this paper, we study the mean-square stability of the solution and its
stochastic theta scheme for the following stochastic differential equations
drive by fractional Brownian motion with Hurst parameter $H\in (\frac 12,1)$:
$$ dX(t)=f(t,X(t))dt+g(t,X(t))dB^{H}(t). $$ Firstly, we consider the special
case when $f(t,X)=-\lambda\kappa t^{\kappa-1}X$ and $g(t,X)=\mu X$. The
solution is explicit and is mean-square stable when $\kappa\geq 2H$. It is
proved that if the parameter $2H\leq\kappa\le 3/2$ and $\frac{\sqrt{3/2}\cdot
e}{\sqrt{3/2}\cdot e+1} (\approx 0.77)\leq \theta\leq 1$ or $\kappa>3/2$ and
$1/2<\theta\le 1$, the stochastic theta method reproduces the mean-square
stability; and that if $0<\theta<\frac 12$, the numerical method does not
preserve this stability unconditionally. Secondly, we study the stability of
the solution and its stochastic theta scheme for nonlinear equations. Due to
the presence of long memory, even the problem of stability in the mean square
sense of the solution has not been well studied since the conventional
techniques powerful for stochastic differential equations driven by Brownian
motion are no longer applicable. Let alone the stability of numerical schemes.
We need to develop a completely new set of techniques to deal with this
difficulty. Numerical examples are carried out to illustrate our theoretical
results.
