分数布朗运动驱动的随机微分方程的增域填充渐近性
Increasing Domain Infill Asymptotics for Stochastic Differential Equations Driven by Fractional Brownian Motion
AI总结：
针对分数布朗运动驱动的SDE推断难题，本文通过离散化建立增域填充渐近理论，证明乘性参数的经典与贝叶斯估计的一致性和渐近正态性，通过模拟和真实收盘价数据验证了贝叶斯分数SDE的优越性。
摘要：
尽管由维纳过程驱动的随机微分方程（SDEs）的统计推断已在文献中得到广泛关注，但相比之下，由分数布朗运动驱动的SDE的推断研究进展却少得多，尽管它们在长程相关性建模中具有重要意义。本文研究了这类基于分数布朗运动的SDE的经典推断与贝叶斯推断，具体考虑了当时域趋于无穷时，两个参数的渐近推断：与漂移函数相关的乘性参数，以及分数布朗运动的所谓“Hurst参数”。由于Hurst参数未知时，似然函数无法采用常用的Girsanov形式，导致常规的渐近推导难以进行。为此，我们通过对SDE进行离散化，建立了增域填充渐近理论。在该框架下，我们证明了最大似然估计量的一致性和渐近正态性，以及贝叶斯后验分布的一致性和渐近正态性，但无法建立Hurst参数对应的经典或贝叶斯渐近正态性。我们通过非渐近设置下的模拟研究补充了理论分析，给出了适用于分数布朗运动驱动SDE的经典分析与贝叶斯分析方法。本文还将方法应用于真实的收盘价数据，并与由维纳过程驱动的标准SDE进行了对比。结果符合预期，我们提出的贝叶斯分数SDE在模拟数据和真实数据应用中均优于其他模型和方法。
英文摘要：
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.
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