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（SDE）的统计推断在文献中已获得广泛关注，但相比之下，分数布朗运动驱动的随机微分方程的推断研究发展明显不足，尽管后者在建模长程依赖性方面具有重要价值。本文在此类基于分数布朗运动的SDE中同时考虑了经典推断与贝叶斯推断。具体而言，我们针对两个参数研究了其时域趋于无穷时的渐近推断问题：一个与漂移函数相关的乘积参数，以及分数布朗运动的所谓"Hurst参数"。当Hurst参数未知时，似然函数无法转化为常用的Girsanov形式，这使得传统的渐近理论发展面临困难。为此，我们通过对SDE进行离散化，建立了增长域填充渐近理论。在此框架下，我们证明了最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，关于Hurst参数的经典或贝叶斯渐近正态性未能得到证明。我们通过非渐近设定下的模拟研究补充理论分析，为分数布朗运动驱动的SDE提供了适用的经典与贝叶斯分析方法。研究还包含对实际收盘价数据的应用，并与维纳过程驱动的标准SDE进行了比较。正如预期，我们的贝叶斯分数SDE模型在模拟和实际数据应用中均优于其他模型与方法。