A remark on the rates of convergence for integrated volatility estimation in the presence of jumps

By Jean Jacod, Markus Reiß. 2012

50 citations recorded by the source corpus.

The optimal rate of convergence of estimators of the integrated volatility, for a discontinuous It\^{o} semimartingale sampled at regularly spaced times and over a fixed time interval, has been a long-standing problem, at least when the jumps are not summable. In this paper, we study this optimal rate, in the minimax sense and for appropriate "bounded" nonparametric classes of semimartingales. We show that, if the $r$th powers of the jumps are summable for some $r\in[0,2)$, the minimax rate is equal to $\min(\sqrt{n},(n\log n)^{(2-r)/2})$, where $n$ is the number of observations.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/51522605decc10085c14a101f6d5ef29312a9f79

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Citations in the corpus

Efficient estimation of integrated volatility in presence of infinite variation jumps (2014, background): “Of course, given the result in [9], discussed in point (2) above, this is only possible under some additional assumption, namely that the “small” jumps behave like those of a stable process, or of the integral with respect to a stable-like process, with some index β ∈ (1,2) [recall that in this case”
Quarticity and other functionals of volatility: Efficient estimation (2012, unknown): “Indeed, it is shown in [6] that if r > 1, the optimal rate in the minimax sense is ( √ ∆n log(1/ √ ∆n)) −(2−r)/2.”
Econometrics of co-jumps in high-frequency data with noise (2015, unknown): “For a discussion on optimality for this setup we refer to Jacod and Reiß (2012).”