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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.
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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.
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).
An aggregated model is proposed, of which the partial-sum process scales to the Karlin stable processes recently investigated in the literature. The limit extremes of the proposed model, when having regularly-varying tails, are characterized by the convergence of the corresponding point processes. The proposed model is an extension of an aggregated model proposed by Enriquez (2004) in order to approximate fractional Brownian motions with Hurst index $Hin(0,1/2)$, and is of a different nature of the other recently investigated Karlin models which are essentially based on infinite urn schemes.
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 $(t,x,y)in (0,infty) times R^d times R^d$.
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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