In quantitative finance, we often fit a parametric semimartingale model to asset prices. To ensure our model is correct, we must then perform goodness-of-fit tests. In this paper, we give a new goodness-of-fit test for volatility-like processes, which is easily applied to a variety of semimartingale models. In each case, we reduce the problem to the detection of a semimartingale observed under noise. In this setting, we then describe a wavelet-thresholding test, which obtains adaptive and near-optimal detection rates.
"Semimartingale detection and goodness-of-fit tests." Ann. Statist. 45 (3) 1254 - 1283, June 2017. https://doi.org/10.1214/16-AOS1484