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Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 4 / August 2021
- Published online by Cambridge University Press:
- 02 September 2020, pp. 1278-1304
- Print publication:
- August 2021
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- Article
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In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter
$ H \gt \frac{1}{2}$. The drift term of the equation is locally Lipschitz and unbounded in the neighbourhood of the origin. We show the existence, uniqueness and positivity of the solutions. The estimates of moments, including the negative power moments, are given. We also develop the implicit Euler scheme, proved that the scheme is positivity preserving and strong convergent, and obtain rate of convergence. Furthermore, by using Lamperti transformation, we show that our results can be applied to stochastic interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear Aït-Sahalia type model.
