We derive a higher-order asymptotic expansion of the conditional characteristic function of the increment of an It\^o semimartingale over a shrinking time interval. The spot characteristics of the It\^o semimartingale are allowed to have dynamics of general form. In particular, their paths can be rough, that is, exhibit local behavior like that of a fractional Brownian motion, while at the same time have jumps with arbitrary degree of activity. The expansion result shows the distinct roles played by the different features of the spot characteristics dynamics. As an application of our result, we construct a nonparametric estimator of the Hurst parameter of the diffusive volatility process from portfolios of short-dated options written on an underlying asset.

中文翻译：

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具有应用的粗波动设置中特征函数的短时扩展

我们推导出 It\^o 半鞅增量的条件特征函数在缩小的时间间隔上的高阶渐近展开。It\^o semimartingale 的点特征被允许具有一般形式的动力学。特别是，它们的路径可能是粗糙的，即表现出类似于分数布朗运动的局部行为，同时具有任意活动程度的跳跃。展开结果显示了光斑特征动态的不同特征所起的不同作用。作为我们结果的应用，我们从标的资产上写的短期期权组合构建了扩散波动率过程的 Hurst 参数的非参数估计量。