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

Shao-Qin Zhang, Chenggui Yuan Orcid Logo

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume: 151, Issue: 4, Pages: 1278 - 1304

Swansea University Author: Chenggui Yuan Orcid Logo

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DOI (Published version): 10.1017/prm.2020.60

Abstract

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 positivi...

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Published in: Proceedings of the Royal Society of Edinburgh: Section A Mathematics
ISSN: 0308-2105 1473-7124
Published: Cambridge University Press (CUP) 2020
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa57824
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Abstract: 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
Keywords: locally Lipschitz drift; fractional Brownian motion; implicit Euler scheme; optimal strong convergence rate; interest rate models
College: Faculty of Science and Engineering
Issue: 4
Start Page: 1278
End Page: 1304

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