Related papers: Levy processes with summable Levy measures, long t…

It is well known that, in the description of quantum observables, positive operator valued measures (POVMs) generalize projection valued measures (PVMs) and they also turn out be more optimal in many tasks. We show that a commutative POVM…

We obtain series expansions of the $q$-scale functions of arbitrary spectrally negative L\'evy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit $q$-scale functions.…

Suppose Xt is either a regular exponential type Levy process or a Levy process with a bounded variation jumps measure. The distribution of the extrema of Xt play a crucial role in many financial and actuarial problems. This article employs…

Path decomposition is performed to characterize the law of the pre/post-supremum, post-infimum and the intermediate processes of a spectrally negative Levy process taken up to an independent exponential time T: As a result, mainly the…

We study stationary max-stable processes $\{\eta(t)\colon t\in\mathbb R\}$ admitting a representation of the form $\eta(t)=\max_{i\in\mathbb N}(U_i+ Y_i(t))$, where $\sum_{i=1}^{\infty} \delta_{U_i}$ is a Poisson point process on $\mathbb…

We derive characteristic function identities for conditional distributions of an r-trimmed Levy process given its r largest jumps up to a designated time t. Assuming the underlying Levy process is in the domain of attraction of a stable…

In this paper we analyze the quasi-stationary workload of a L\'evy-driven storage system. More precisely, assuming the system is in stationarity, we study its behavior conditional on the event that the busy period $T$ in which time 0 is…

Generalizing the concept of quantiles to the jump measure of a L\'evy process, the generalized quantiles $q_{\tau}^{\pm}>0$, for $\tau>0$, are given by the smallest values such that a jump larger than $q_{\tau}^{+}$ or a negative jump…

We consider a class of L\'evy-type processes derived via a Doob-transform from L\'evy processes conditioned by a control function called potential. These processes have position-dependent and generally unbounded components, with stationary…

Upper estimates of densities of convolution semigroups of probability measures are given under explicit assumptions on the corresponding L\'evy measure and the L\'evy--Khinchin exponent.

In this paper, we study the compressibility of random processes and fields, called generalized L\'evy processes, that are solutions of stochastic differential equations driven by $d$-dimensional periodic L\'evy white noises. Our results are…

We study class of L\'{e}vy processes having distributions being indentifiable by moments. We define system of polynomial martingales \newline $\left\{ M_{n}(X_{t},t),\mathcal{F}_{\leq t}\right\} _{n\geq 1},$ where $% \mathcal{F}_{\leq t}$…

We present a new class of prior measures in connection to $\ell_p$ regularization techniques when $p \in(0,1)$ which is based on the generalized Gamma distribution. We show that the resulting prior measure is heavy-tailed, non-convex and…

Multivariate subordinated L\'evy processes are widely employed in finance for modeling multivariate asset returns. We propose to exploit non-linear dependence among financial assets through multivariate cumulants of these processes, for…

We study the density of the supremum of a strictly stable L\'evy process. We prove that for almost all values of the index $\alpha$ -- except for a dense set of Lebesgue measure zero -- the asymptotic series which were obtained in A.…

We establish optimal logarithmic rates of convergence in the strong invariance principle for multivariate cumulative processes in the Smith's sense. Exponential probabilistic inequalities of Koml\'{o}s-Major-Tusn\'{a}dy type are obtained.…

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…

Quasi-power law ensembles are discussed from the perspective of nonextensive Tsallis distributions characterized by a nonextensive parameter $q$. A number of possible sources of such distributions are presented in more detail. It is further…

We extract the long-distance asymptotic behaviour of two-point correlation functions in massless quantum integrable models containing multi-species excitations. For such a purpose, we extend to these models the method of a large-distance…

Let $\mu$ be a probability measure (or corresponding random variable) such that all moments $\mu_n$ exist. Knowledge of the moments is not sufficient to determine infinite divisibility of the measure; we show also that infinitely divisible,…