Glivenko--Cantelli定理：纯跳跃半鞅中积分波动率泛函及其在加密货币市场的应用
Glivenko--Cantelli Theorems for Integrated Volatility Functionals in Pure-Jump Semimartingales with an Application to Cryptocurrency Markets
查看机构详情
- School of Economics, Singapore Management University（新加坡管理大学经济管理学院）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文提出两步估计法，用于纯跳跃半鞅中积分波动率泛函的一致估计，建立Glivenko--Cantelli型收敛性，并应用于加密货币市场。
中文摘要 AI 辅助
我们开发了一个两步程序，用于估计通过潜在现货波动率过程的占据测度定义的积分波动率泛函，当资产价格是纯跳跃半鞅时。第一步，基于高频增量绝对幂的块估计量一致地近似波动率幂的局部平均值。第二步，将这些估计聚合为经验占据测度。由于在此设定下价格增量具有无限方差，基于局部高斯性的论证不可用，均匀理论转而依赖于针对稳定状态定制的极大不等式。我们建立了对有界单调、参数Lipschitz和局部Hölder检验函数类上的Glivenko--Cantelli型一致收敛性。这些结果提供了波动率占据时间和分位数的一致估计，以及基于非参数恢复潜在过程的$M$-估计量的argmax一致性理论。我们进一步提出了一种基于稳定性的规则来选择波动率估计器的幂指数，该规则在蒙特卡洛实验中紧密跟踪不可行的先验最优选择。对高频加密货币市场的应用说明了该框架在跳跃主导、重尾环境中的有效性。
英文摘要
We develop a two-step procedure for estimating integrated volatility functionals, defined through the occupation measure of the latent spot volatility process, when the asset price is a pure-jump semimartingale. In the first step, block-based estimators formed from absolute powers of high-frequency increments uniformly approximate local averages of powers of volatility. In the second step, these estimates are aggregated into an empirical occupation measure. Since price increments have infinite variance in this setting, arguments based on local Gaussianity are unavailable, and the uniform theory instead rests on maximal inequalities tailored to the stable regime. We establish Glivenko--Cantelli-type uniform consistency over classes of bounded monotone, Lipschitz-in-parameter, and locally Hölder test functions. These results deliver consistent estimation of volatility occupation times and quantiles, together with an argmax-consistency theory for $M$-estimators built on nonparametrically recovered latent processes. We further propose a stability-based rule for selecting the power index of the volatility estimator, which tracks an infeasible ex ante optimal choice closely in Monte Carlo experiments. An application to high-frequency cryptocurrency markets illustrates the framework in a jump-dominated, heavy-tailed environment.
