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An Improved Test for Continuous Local Martingales

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  • David A. Rolls
  • Owen D. Jones

Abstract

We present a new test for the “continuous martingale hypothesis”. That is, a test for the hypothesis that observed data are from a process which is a continuous local martingale. The basis of the test is an embedded random walk at first passage times, obtained from the well-known representation of a continuous local martingale as a continuous time-change of Brownian motion. With a variety of simulated diffusion processes our new test shows higher power than existing tests using either the crossing tree or the quadratic variation, including the situation where non-negligible drift is present. The power of the test in the presence of jumps is also explored with a variety of simulated jump diffusion processes. The test is also applied to two sequences of high-frequency foreign exchange trade-by-trade data. In both cases the continuous martingale hypothesis is rejected at times less than hourly and we identify significant dependence in price movements at these small scales.

Suggested Citation

  • David A. Rolls & Owen D. Jones, 2015. "An Improved Test for Continuous Local Martingales," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(13), pages 2674-2688, July.
    • Handle: RePEc:taf:lstaxx:v:44:y:2015:i:13:p:2674-2688
    DOI: 10.1080/03610926.2013.788709
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