Testing for common arrivals of jumps for discretely observed multidimensional processes

Abstract

We consider a bivariate process Xt = (X 1 t , X 2 t ), which is observed on a finite time interval [0, T ] at discrete times 0, ∆n, 2∆n, . . . . Assuming that its two components X 1 and X 2 have jumps on [0, T ], we derive tests to decide whether they have at least one jump occurring at the same time ("common jumps") or not ("disjoint jumps"). There are two different tests for the two possible null hypotheses (common jumps or disjoint jumps). Those tests have a prescribed asymptotic level, as the mesh ∆n goes to 0. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use for some exchange rates data.

Bipower Variation for Gaussian Processes with Stationary Increments

“…On the other hand, (15) and (20) for any t 1 ≤ t ≤ t 2 and λ > 0, from which we deduce the tightness of the sequence Z n t by Theorem 15.6 of [12]. This completes the proof of Theorem 3.…”

“…Extensions of the theory to Lévy processes and Itô semimartingales have been obtained, particularly by Jacod [18] (cf. also [5]), and applications to finance of such extensions are discussed in [20] and [28].…”

Jumps in High-Frequency Data on the Chinese Stock Market

“…Therefore, the method can be used to test cojumps for financial markets with temporal asynchronicity. Jaeod and Todorov (2009) [2] examine the comovement between individual stocks in the U.S. stock market, proposing the concept of multidimensional realized power variation to estimate the volatility of all subprocesses. Their work provides a basis for analyzing cojumps nonparametrically by using high-frequency data.…”

“…Each semimartingale can be divided into drifts and continuous and discontinuous local martingales. These semimartingale components are ranked according to the level of activity: finite-activity jumps (activity level = 0) such as compound Poisson processes, infinite-activity finite-variation jumps (activity level = [0, 1]), infinite-variation jumps (activity level = [1,2]), and continuous semimartingales (activity level = 2). Moreover, the activity level of a semimartingale is determined by the activity of its most active component.…”

“…On the basis of two-dimensional realized power variation proposed by Jacod and Todorov (2009) [2], a ratio analysis is subsequently conducted on two independent realized power variations to derive a ratio analysis method for cojumps. This method can be employed to determine the level of cojumps between different stochastic processes.…”