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Register a new userOn the Estimates of the Density of the Purely Discontinuous Girsanov Transforms of $alpha$-Stable-like Processes
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Chunlin Wang
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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.
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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 additive functional of $X$. With neither $F$ nor $X$ being symmetric, under certain conditions, we show that the Feynman-Kac semigroup ${S_{t}^{F}:t ge 0}$ defined by $$ S_{t}^{F}f(x)=mathbb{E}_{x}(e^{-A_{t}^{F}}f(X_{t}))$$ has a density $q$ and that there exist positive constants $C_1,C_2,C_3$ and $C_4$ such that $$C_{1}e^{-C_{2}t}t^{-frac{d}{alpha}}(1 wedge frac{t^{frac{1}{alpha}}}{|x-y|})^{d+alpha} leq q(t,x,y) leq C_{3}e^{C_{4}t}t^{-frac{d}{alpha}}(1 wedge frac{t^{frac{1}{alpha}}}{|x-y|})^{d+alpha}$$ for all $(t,x,y)in (0,infty) times R^d times R^d$.
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 cite{pp}). This family of processes shares many properties with the tempered stable and the layered stable processes, defined in Rosinski cite{ro} and Houdre and Kawai cite{hok} respectively, for instance their short and long time behaviour. Additionally, in the real valued case we find a series representation which is used for sample paths simulation. In this work we find general properties of this class and we also provide many examples, some of which appear in recent literature.
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 majorants of Levy processes) and apply it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse numerically the performance of the algorithm (the code, written in Julia 1.0, is available on GitHub).
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 solution in the case that the jump amplitudes follow a Gaussian or Laplacian law. The proof of the lower bound uses a general expression for the density of the solution in terms of the convolution of the density of the continuous part and the jump amplitude density. The upper bound uses an upper tail estimate in terms of the jump amplitude distribution and techniques of the Malliavin calculus in order to bound the density by the tails of the solution. We also extend the lower bounds to the multidimensional case.
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 Theorems for the exit time from a cone of stable processes of index $alphain(0,2)$. We also study the case $trightarrow0$ and we prove some Laws of the Iterated Logarithm (LIL) for the (well-defined) winding process associated to our planar stable process.
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