  In this paper, we study a class of one-dimensional stochastic differential
equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1
2$. The drift term of the equation is locally Lipschitz and unbounded in the
neighborhood of $0$. We show the existence, uniqueness and positivity of the
solutions. The estimations of moments, including the negative power moments,
are given. Based on these estimations, strong convergence of the positivity
preserving drift-implicit Euler-type scheme is proved, and optimal convergence
rate is obtained. By using Lamperti transformation, we show that our results
can be applied to interest rate models such as mean-reverting stochastic
volatility model and strongly nonlinear A\"it-Sahalia type model.
