The problem of integrated volatility estimation for the solution X of a stochastic differential equation with L{\'e}vy-type jumps is considered under discrete high-frequency observations in both short and long time horizon. We provide an asymptotic expansion for the integrated volatility that gives us, in detail, the contribution deriving from the jump part. The knowledge of such a contribution allows us to build an unbiased version of the truncated quadratic variation, in which the bias is visibly reduced. In earlier results the condition $\beta$ > 1 2(2--$\alpha$) on $\beta$ (that is such that (1/n) $\beta$ is the threshold of the truncated quadratic variation) and on the degree of jump activity $\alpha$ was needed to have the original truncated realized volatility well-performed (see [22], [13]). In this paper we theoretically relax this condition and we show that our unbiased estimator achieves excellent numerical results for any couple ($\alpha$, $\beta$). L{\'e}vy-driven SDE, integrated variance, threshold estimator, convergence speed, high frequency data.

中文翻译：

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跳跃扩散过程中波动率估计的无偏截断二次变异

在短期和长期的离散高频观测下，考虑了具有 L{\'e}vy 型跳跃的随机微分方程的解 X 的综合波动率估计问题。我们为综合波动率提供了渐近扩展，详细地给出了从跳跃部分得出的贡献。这种贡献的知识使我们能够构建截断二次变化的无偏版本，其中偏差明显减少。在早期的结果中，$\beta$ 上的条件 $\beta$ > 1 2(2--$\alpha$)（即 (1/n) $\beta$ 是截断二次变化的阈值）和在跳跃活动的程度上，需要 $\alpha$ 才能使原始截断的已实现波动率表现良好（参见 [22]、[13]）。在本文中，我们从理论上放宽了这个条件，并表明我们的无偏估计器对任何一对（$\alpha$、$\beta$）都取得了出色的数值结果。L{\'e}vy-driven SDE，积分方差，阈值估计，收敛速度，高频数据。