在离散观测值下识别扩散部分与协跳之间的协变
Identifying the covariation between the diffusion parts and the co-jumps given discrete observations
摘要：
本文考虑两个由扩散和跳驱动的半鞅。我们允许有限的活动和无限的活动跳跃组件。给定离散观测值，我们将{ \it积分协变}（两个扩散部分之间的协变，用IC表示）与协跳分离。这对预测、期权定价、风险和信用风险管理等多资产价格建模具有重要的应用。通常用于估计IC的方法是取两个过程增量的向量积之和;然而，该估计量在存在跳跃的情况下可能高度有偏，因为它接近二次变异，其也包含共同跳跃。我们的估计IC是基于阈值（或截断）技术。我们证明了在这两种情况下的估计量是一致的，因为观察的数量增加到无穷大。此外，在仅存在有限活动跳跃的情况下1）\hat{IC}也是渐进高斯的; 2）\hat{IC}的联合CLT和单个过程的积分方差的阈值估计量甚至可以达到$β$的一致性和渐进高斯估计量和两个过程的扩散部分之间的相关系数，从而可以更好地测量它们的依赖性; 3）阈值化给出了对观测值的不确定性具有鲁棒性的IC估计。我们进行了模拟研究，以检查我们的技术的应用实际上是信息的值之间的步骤的观察足够大，以避免在数据中的微观结构噪声的存在下出现的典型问题，并评估的门限参数的选择。
英文摘要：
In this paper we consider two semimartingales driven by diffusions and jumps. We allow both for finite activity and for infinite activity jump components. Given discrete observations we disentangle the {\it integrated covariation} (the covariation between the two diffusion parts, indicated by IC) from the co-jumps. This has important applications to multiple assets price modeling for forecasting, option pricing, risk and credit risk management. An approach commonly used to estimate IC is to take the sum of the cross products of the two processes increments; however this estimator can be highly biased in the presence of jumps, since it approaches the quadratic ovariation, which contains also the co-jumps. Our estimator of IC is based on a threshold (or truncation) technique. We prove that the estimator is consistent in both cases as the number of observations increases to infinity. Further, in presence of only finite activity jumps 1) \hat{IC} is also asymptotically Gaussian; 2) a joint CLT for \hat{IC} and threshold estimators of the integrated variances of the single processes allows to reach consistent and asymptotically Gaussian estimators even of the $β$s and of the correlation coefficient among the diffusion parts of the two processes, allowing a better measurement of their dependence; 3) thresholding gives an estimate of IC which is robust to the asynchronicity of the observations. We conduct a simulation study to check that the application of our technique is in fact informative for values of the step between the observations large enough to avoid the typical problems arising in presence of microstructure noises in the data, and to asses the choice of the threshold parameters.
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