Talks' abstracts > Jaramillo Arturo

We study how to recover the integrated scale of a stochastic integral from high-frequency observations. The signal is driven by a Gaussian noise with a time-dependent positive scale, and it may also contain a low-activity perturbation, such as jumps or other irregular terms. The goal is to estimate the accumulated scale directly from the observed increments, without first detecting or removing those perturbations.

The method works block by block. On each short block, the scale is treated as essentially constant, while the remaining terms are absorbed as an additive error. We estimate the local scale by matching the order statistics of the observed increments with Gaussian quantiles. This gives a robust local estimator, based on the ordered data rather than on a quadratic average of all increments.

Finally, the local estimates are averaged to recover the integrated scale. Under natural assumptions on the perturbation, the local approximation of the integral, and the dependence of the Gaussian noise, we prove convergence of the estimator. Numerical examples for Brownian and fractional Brownian models illustrate the method, which can be seen as an order-statistic alternative to quadratic variation in the presence of jumps, rough noise, or non-semimartingale effects.