Although statistical inference in stochastic differential equations (SDEs) driven by Wiener process has received significant attention in the literature, inference in those driven by fractional Brownian motion seem to have seen much less development in comparison, despite their importance in modeling long range dependence. In this article, we consider both classical and Bayesian inference in such fractional Brownian motion based SDEs. In particular, we consider asymptotic inference for two parameters in this regard; a multiplicative parameter associated with the drift function, and the so-called "Hurst parameter" of the fractional Brownian motion, when the time domain tends to infinity. For unknown Hurst parameter, the likelihood does not lend itself amenable to the popular Girsanov form, rendering usual asymptotic development difficult. As such, we develop increasing domain infill asymptotic theory, by discretizing the SDE. In this setup, we establish consistency and asymptotic normality of the maximum likelihood estimators, as well as consistency and asymptotic normality of the Bayesian posterior distributions. However, classical or Bayesian asymptotic normality with respect to the Hurst parameter could not be established. We supplement our theoretical investigations with simulation studies in a non-asymptotic setup, prescribing suitable methodologies for classical and Bayesian analyses of SDEs driven by fractional Brownian motion. Applications to a real, close price data, along with comparison with standard SDE driven by Wiener process, is also considered. As expected, it turned out that our Bayesian fractional SDE triumphed over the other model and methods, in both simulated and real data applications.

翻译：尽管维纳过程驱动的随机微分方程(SDEs)的统计推断在文献中已得到大量关注，但分数布朗运动驱动的SDEs的推断研究相比之下进展甚微，尽管此类方程在建模长程依赖性方面具有重要价值。本文同时考虑此类基于分数布朗运动的SDEs的经典推断与贝叶斯推断。具体而言，我们关注时间域趋于无穷时两个参数的渐近推断：漂移函数相关的乘性参数，以及分数布朗运动的所谓"赫斯特参数"。对于未知的赫斯特参数，似然函数无法适用流行的Girsanov形式，导致常规渐近分析难以开展。为此，我们通过对SDE进行离散化处理，发展了域内填充渐近理论。在该框架下，我们建立了极大似然估计量的一致性与渐近正态性，以及贝叶斯后验分布的一致性与渐近正态性。然而，针对赫斯特参数的经典或贝叶斯渐近正态性未能建立。我们在非渐近框架下通过模拟研究补充理论分析，为分数布朗运动驱动的SDEs的经典与贝叶斯分析提供了适当的方法论。同时，本文将该方法应用于实际收盘价数据，并与标准维纳过程驱动的SDE进行对比。正如预期，我们的贝叶斯分数SDE在模拟数据与实际数据应用中均优于其他模型与方法。