  We consider pure-jump transaction-level models for asset prices in continuous
time, driven by point processes. In a bivariate model that admits
cointegration, we allow for time deformations to account for such effects as
intraday seasonal patterns in volatility, and non-trading periods that may be
different for the two assets. We also allow for asymmetries (leverage effects).
We obtain the asymptotic distribution of the log-price process. We also obtain
the asymptotic distribution of the ordinary least-squares estimator of the
cointegrating parameter based on data sampled from an equally-spaced
discretization of calendar time, in the case of weak fractional cointegration.
For this same case, we obtain the asymptotic distribution for a tapered
estimator under more
