Related papers: Transition probabilities of L\'evy-type processes:…
We review some of the theory relevant to passage times of one-dimensional L\'evy processes out of bounded regions, highlighting results that are useful in physical phenomena modelled by heavy-tailed L\'evy flights. The process is…
Various recent results on quantum L\'evy processes are presented. The first part provides an introduction to the theory of L\'evy processes on involutive bialgebras. The notion of independence used for these processes is tensor…
We describe basic motivations behind quantum or noncommutative probability, introduce quantum L\'evy processes on compact quantum groups, and discuss several aspects of the study of the latter in the example of quantum permutation groups.…
In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical L\'evy process, and show that these conditions are also necessary if the…
In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) L\'evy processes on half spaces for all $t>0$. These L\'evy processes may…
We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall's identity for spectrally positive L\'evy processes. We obtain the later result as a direct…
In this paper we present some new limit theorems for power variations of stationary increment L\'{e}vy driven moving average processes. Recently, such asymptotic results have been investigated in [Ann. Probab. 45(6B) (2017), 4477--4528,…
We consider solutions of L\'evy-driven stochastic differential equations of the form $\mathrm{d} X_t=\sigma(X_{t-})\mathrm{d} L_t$, $X_0=x$ where the function $\sigma$ is twice continuously differentiable and maximal of linear growth and…
Motivated by the recent results of Nualart and Xu \cite{Nualart} concerning limits laws for occupation times of one dimensional symmetric stable processes, this paper proves a decomposition for functionals of one dimensional symmetric…
We give two-term small-time approximation for the trace of the Dirichlet heat kernel of bounded smooth domain for unimodal L\'evy processes satisfying the weak scaling conditions.
We derive general sufficient conditions for the existence of c\`adl\`ag and continuous modifications of L\'evy-driven mixed moving average processes. The conditions are explicit and easy to verify and applied to supOU, well-balanced supOU,…
We consider the problem of determining the L\'evy exponent in a L\'evy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure $\mathbb P$, consists of a pricing kernel…
Let $X$ be a L\'evy process with absolutely continuous L\'evy measure $\nu$. Small time polynomial expansions of order $n$ in $t$ are obtained for the tails $P(X_{t}\geq{}y)$ of the process, assuming smoothness conditions on the L\'evy…
We construct intrinsic on-and off-diagonal upper and lower estimates for the transition probability density of a L\'evy process in small time. By intrinsic we mean that such estimates reflect the structure of the characteristic exponent of…
This article deals with adaptive nonparametric estimation for L\'evy processes observed at low frequency. For general linear functionals of the L\'evy measure, we construct kernel estimators, provide upper risk bounds and derive rates of…
Let $X$ be a real L\'evy process and let $\Xpos $ be the process conditioned to stay positive. We assume that $ 0 $ is regular for $(-\infty, 0)$ and $(0, +\infty) $ with respect to $X$. Using elementary excursion theory arguments, we…
We give conditions under which the tail probability of the supremum over unit interval of a Levy process with light tail is equivalent to the tail of the value of the process at the right endpoint.
We consider Kallenberg's hypothesis on the characteristic function of a L\'{e}vy process and show that it allows the construction of weakly continuous bridges of the L\'{e}vy process conditioned to stay positive. We therefore provide a…
In our previous publications (IJTAF 2019, Math. Finance 2020), we introduced a general class of SINH-regular processes and demonstrated that efficient numerical methods for the evaluation of the Wiener-Hopf factors and various probability…
In this paper we develop an $L_2$-theory for stochastic partial differential equations driven by L\'evy processes. The coefficients of the equations are random functions depending on time and space variables, and no smoothness assumption of…