Rate-optimal Tests for Jumps in Diffusion Processes

Abstract

Suppose one has given a sample of high-frequency intra-day discrete observations of a continuous-time random process (e.g. stock market data) and wants to test for the presence of jumps. We show that the power of any test of this hypothesis depends on the frequency of observation. In particular, we show that if the process is observed at intervals of length 1=n and the instantaneous volatility of the process is given by t, at best one can detect jumps of height no smaller than ...... We construct a test which achieves this rate in the case for di¤usion-type processes.