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We present some aspects of the so-called additive coalescence, with a focus on its connections with random trees, Brownian excursion, certain bridges with exchangeable increments, L\'evy processes, and sticky particle systems.
Based on the concept of a L\'evy copula to describe the dependence structure of a multivariate L\'evy process we present a new estimation procedure. We consider a parametric model for the marginal L\'evy processes as well as for the L\'evy…
We develop a method of driving a Markov processes through a continuous flow. In particular, at the level of the transition functions we investigate an approach of adding a first order operator to the generator of a Markov process, when the…
By using large deviation theory that deals with the decay of probabilities of rare events on an exponential scale, we study the longtime behaviors and establish action functionals for scaled Brownian motion and L\'evy processes with…
Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional L\'evy processes are defined by integrating…
We present an optimal control approach to the problem of model calibration for L\'evy processes based on a non parametric estimation procedure. The calibration problem is of considerable interest in mathematical finance and beyond.…
We review some developments concerning Markov and Feller processes with jumps in geometric settings. These include stochastic differential equations in Markus canonical form, the Courr\`{e}ge theorem on Lie groups, and invariant Markov…
We are concerned with the absolute continuity of stationary distributions corresponding to some piecewise deterministic Markov process, being typically encountered in biological models. The process under investigation involves a…
The main purpose of this chapter is to present some theoretical aspects of parametric estimation of L\'evy processes based on high-frequency sampling, with a focus on infinite activity pure-jump models. Asymptotics for several classes of…
We characterize the small-time asymptotic behavior of the exit probability of a L\'evy process out of a two-sided interval and of the law of its overshoot, conditionally on the terminal value of the process. The asymptotic expansions are…
For spectrally negative L\'evy processes, adapting an approach from \cite{BoLi:sub1} we identify joint Laplace transforms involving local times evaluated at either the first passage times, or independent exponential times, or inverse local…
In this paper, we study the L\'evy process time-changed by independent L\'evy subordinators, namely, the incomplete gamma subordinator, the $\epsilon$-jumps incomplete gamma subordinator and tempered incomplete gamma subordinator. We derive…
In this paper a Malliavin calculus for L\'evy processes based on a family of true derivative operators is developed. The starting point is an extension to L\'evy processes of the pioneering paper by Carlen and Pardoux [8] for the Poisson…
We elaborate on the theorem saying that as permeability coefficients of snapping-out Brownian motions tend to infinity in such a way that their ratio remains constant, these processes converge to a skew Brownian motion. In particular,…
We study a decomposition of a general Markov process in a manifold invariant under a Lie group action into a radial part (transversal to orbits) and an angular part (along an orbit). We show that given a radial path, the conditioned angular…
We consider a process $Z$ on the real line composed from a L\'evy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$…
Couplings play a central role in contemporary Markov chain Monte Carlo methods and in the analysis of their convergence to stationarity. In most cases, a coupling must induce relatively fast meeting between chains to ensure good…
This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known L\'evy process: The compound Poisson process. The semi-Markov extension of…
We establish exponential ergodicity for the stochastic Hamiltonian system $(X_t, V_t)_{t\ge0}$ on $\mathbb{R}^{2d}$ with L\'evy noises \begin{align*} \begin{cases} \mathrm{d} X_t=\big(a X_t+bV_t\big)\,\mathrm{d} t,\\ \mathrm{d}…
A Markov Additive Process is a bi-variate Markov process $(\xi,J)=\big((\xi_t,J_t),t\geq0\big)$ which should be thought of as a multi-type L\'evy process: the second component $J$ is a Markov chain on a finite space $\{1,\ldots,K\}$, and…