Source: Ann. Probab.
Volume 34, Number 3
We consider the problem of absolute continuity for the one-dimensional SDE
where Z is a real Lévy process without Brownian part and a a function of class with bounded derivative. Using an elementary stratification method, we show that if the drift a is monotonous at the initial point x, then Xt is absolutely continuous for every t>0 if and only if Z jumps infinitely often. This means that the drift term has a regularizing effect, since Zt itself may not have a density. We also prove that when Zt is absolutely continuous, then the same holds for Xt, in full generality on a and at every fixed time t. These results are then extended to a larger class of elliptic jump processes, yielding an optimal criterion on the driving Poisson measure for their absolute continuity.
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