Watanabe proved that if Xt is a point process such that Xt−t is a martingale, then Xt is a Poisson process and this result was generalized by Bremaud for doubly stochastic Poisson processes. Here we define two-parameter point processes and extend this property without needing the strong martingale condition. Using this characterization, we study the problem of transforming a two-parameter point process into a two-parameter Poisson process by means of a family of stopping lines as a time change. Nualart and Sanz gave conditions in order to transform a square integrable strong martingale into a Wiener process. Here, we do the same for the Poisson process by a similar method but under more general conditions.
This is the published version, also available here: http://dx.doi.org/10.1214/aop/1176992378.
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