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

2018/12/29 by Shao‐Qin Zhang, Chenggui Yuan, Zhang, Shao-Qin +1
Economics, Econometrics and Finance · #60H10 #60H35 #Complex Systems and Time Series Analysis #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1812.11382

openalex publication_date 2018/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Aït-Sahalia type model.

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