Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation - Chenggui Yuan (Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2020)
In this paper, a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1 2$ is studied. The drift term of the equation is locally Lipschitz and unbounded in the neighborhood of the origin. The existence, uniqueness and positivity of the solutions are proved. We estimate moments including the negative power moments. We also develop the implicit Euler scheme, proved that the scheme is positivity preserving and strong convergent, and obtain rate of convergence. Furthermore, we show that our results can be applied to stochastic interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear A\"it-Sahalia type model by using Lamperti transformation