Seminar Ausgewählte Kapitel der Statistik und Stochastik

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Abstract: We propose methods to infer jumps of a semi-martingale in a stochastic boundary model with one-sided microstructure noise for high-frequency limit order prices. In the process, we consider estimation of the spot volatility and establish stable limit theorems and uniform consistency. The asymptotic analysis builds upon an expansion of tail probabilities for local minima based on a generalized arcsine law. To use the involved distribution of local minima for a bias correction, a numerical algorithm is presented. We develop methods to estimate, locate and test for jumps using local minima. We provide a local test and show that we can consistently estimate price jumps. The main contribution is a global test for jumps. We establish the asymptotic properties of this test which is based on a maximum statistic. For this purpose, we derive the asymptotic distribution of this statistic under the null hypothesis of no jumps based on extreme value theory and convolution tails. We prove consistency under the alternative hypothesis. The rate of convergence for local alternatives is determined and shown to be faster than optimal rates for the classical noise model with centered market microstructure. This allows the identification of smaller jumps which is also demonstrated in simulations.