Stochastic differential equations driven by fractional Brownian motion with locally Lipschitiz drift and their Euler approximation

Abstract

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.