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In this paper, we study the purely discontinuous Girsanov transforms which were discussed in Chen and Song cite{CS2} and Song cite{S3}. We show that the transition density of any purely discontinuous Girsanov transform of a $alpha$-stable-like process, which can be nonsymmetric, is comparable to the transition density of the $alpha$-stable-like process.
Suppose that $alpha in (0,2)$ and that $X$ is an $alpha$-stable-like process on $R^d$. Let $F$ be a function on $R^d$ belonging to the class $bf{J_{d,alpha}}$ (see Introduction) and $A_{t}^{F}$ be $sum_{s le t}F(X_{s-},X_{s}), t> 0$, a discontinuous
We consider a new family of $R^d$-valued L{e}vy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance cite{cc}, cite{ckp}, cite{kp} and ci
We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave
We consider a real-valued diffusion process with a linear jump term driven by a Poisson point process and we assume that the jump amplitudes have a centered density with finite moments. We show upper and lower estimates for the density of the solutio
Using a generalization of the skew-product representation of planar Brownian motion and the analogue of Spitzers celebrated asymptotic Theorem for stable processes due to Bertoin and Werner, for which we provide a new easy proof, we obtain some limit