Related papers: On the Lamperti stable processes

We prove that the upward ladder height subordinator $H$ associated to a real valued L\'{e}vy process $\xi$ has Laplace exponent $\phi$ that varies regularly at $\infty$ (resp. at 0) if and only if the underlying L\'{e}vy process $\xi$…

The Lamperti transform offers a powerful bridge between self-similar processes and stationary dynamics, making it especially useful for analyzing anomalous diffusion models that lack stationary increments. In this paper we examine the…

For a L\'evy process $\xi=(\xi_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows \[{\rm{I}}_{\xi}=\int_0^{\infty}e^{\xi_t} dt.\] Under mild conditions on $\xi$, we show that the following…

We give a series representation of the logarithm of the bivariate Laplace exponent $\kappa$ of $\alpha$-stable processes for almost all $\alpha \in (0,2]$.

This paper considers a simulation-based estimator for a general class of Markovian processes and explores some strong consistency properties of the estimator. The estimation problem is defined over a continuum of invariant distributions…

Fixed-time stable dynamical systems are capable of achieving exact convergence to an equilibrium point within a fixed time that is independent of the initial conditions of the system. This property makes them highly appealing for designing…

Recent fluctuation identities for $\alpha$-stable L\'evy processes have decomposed paths using generalised spherical polar coordinates revealing an underlying Markov Additive Process (MAP) for which a more advanced form of excursion theory…

The concept of a L\'evy subordinator is generalized to a family of non-decreasing stochastic processes, which are parameterized in terms of two Bernstein functions. Whereas the independent increments property is only maintained in the…

We investigate exponential stock models driven by tempered stable processes, which constitute a rich family of purely discontinuous L\'{e}vy processes. With a view of option pricing, we provide a systematic analysis of the existence of…

An $\mathbb{R}^d$-valued Markov process $X^{(x)}_t=(X^{1,x_1}_t,\dots,X^{d,x_d}_t)$, $t\ge0,x\in\mathbb{R}^d$ is said to be multi-self-similar with index $(\alpha_1,\dots,\alpha_d)\in[0,\infty)^d$ if the identity in law…

Suppose we have a high-frequency sample from the L\'{e}vy process of the form $X_t^\theta=\beta t+\gamma Z_t+U_t$, where $Z$ is a possibly asymmetric locally $\alpha$-stable L\'{e}vy process, and $U$ is a nuisance L\'{e}vy process less…

Using complex analysis techniques we obtain precise asymptotic approximations for the kernels corresponding to the symmetric $\alpha$-stable processes and their fractional derivatives. We apply our method to general L\'evy processes whose…

Hyper-Positive Real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of Hyper-positive functions, related through nested inclusions, are introduced. Structurally, this…

This paper surveys some of our recent progress on Hardy-type inequa\-lities which consist of a well-known topic in Harmonic Analysis. In the first section, we recall the original probabilistic motivation dealing with the stability speed in…

Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the…

We find an expression for the joint Laplace transform of the law of $(T_{[x,+\infty[},X_{T_{[x,+\infty[}})$ for a L\'evy process $X$, where $T_{[x,+\infty[}$ is the first hitting time of $[x,+\infty[$ by $X$. When $X$ is an $\alpha$-stable…

In this paper, we shall introduce the Tanaka formula from viewpoint of the Doob-Meyer decomposition. For symmetric L\'evy processes, if the local time exists, Salminen and Yor (2007) obtained the Tanaka formula by using the potential…

In this note, we present few examples of Piecewise Deterministic Markov Processes and their long time behavior. They share two important features: they are related to concrete models (in biology, networks, chemistry,. . .) and they are…

This article study the class of distributions obtained by subordinating L\'evy processes and L\'evy bases. To do this we derive properties of a suitable mapping obtained via L\'evy mixing. We show that our results can be used to solve the…

We study a new class of so-called rational-infinitely (or quasi-infinitely) divisible probability laws on the real line. The characteristic functions of these distributions are ratios of the characteristic functions of classical infinitely…