Related papers: Recent developments in exponential functionals of …
In the recent development in a various disciplines of physics, it is noted the need for including the deformed versions of the exponential functions. In this paper, we consider the deformations which have two purposes: to have them like…
Exponential L\'evy processes have been used for modelling financial derivatives because of their ability to exhibit many empirical features of markets. Using their multidimensional analogue, a general analytic pricing formula is obtained,…
The classical notion of L\'evy process is generalized to one that takes as its values probabilities on a first order model equipped with a commutative semigroup. This is achieved by applying a convolution product on definable probabilities…
Monotone L\'evy processes with additive increments are defined and studied. It is shown that these processes have a natural Markov structure and their Markov transition semigroups are characterized using the monotone L\'evy-Khintchine…
During the last decade Levy processes with jumps have received increasing popularity for modelling market behaviour for both derviative pricing and risk management purposes. Chan et al. (2009) introduced the use of empirical likelihood…
For a broad class of the Levy processes the new form (convolution type) of the infinitesimal generators is introduced. It leads to the new notions: a truncated generator, a quasi-potential. The probability of the Levy process remaining…
Exponential L\'evy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such…
In this paper, we study the existence of the density associated to the exponential functional of the L\'evy process $\xi$, \[ I_{\ee_q}:=\int_0^{\ee_q} e^{\xi_s} \, \mathrm{d}s, \] where $\ee_q$ is an independent exponential r.v. with…
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 prove simple general formulas for expectations of functions of a L\'evy process and its running extremum. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener-Hopf factorization, and…
Let $J(\cdot)$ be a compound Poisson process with rate $\lambda>0$ and a jumps distribution $G(\cdot)$ concentrated on $(0,\infty)$. In addition, let $V$ be a random variable which is distributed according to $G(\cdot)$ and independent from…
The likelihood function is central to both frequentist and Bayesian formulations of parametric statistical inference, and large-sample approximations to the sampling distributions of estimators and test statistics, and to posterior…
Exponential convergence rates in the $L^2$-tail norm and entropy are characterized for the second quantization semigroups by using the corresponding base Dirichlet form. This supplements the well known result on the $L^2$-exponential…
This paper provides rate-efficient estimators of the volatility parameter in the presence of L\'{e}vy jumps
In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a L\'evy process. More specifically, we investigate the asymptotic theory for the conditional mean…
We characterize the support of the law of the exponential functional $\int_0^\infty e^{-\xi_{s-}} \, d\eta_s$ of two one-dimensional independent L\'evy processes $\xi$ and $\eta$. Further, we study the range of the mapping $\Phi_\xi$ for a…
In contrast to their seemingly simple and shared structure of independence and stationarity, L\'evy processes exhibit a wide variety of behaviors, from the self-similar Wiener process to piecewise-constant compound Poisson processes.…
These lecture notes are an extended version of my lectures on L\'evy and L\'evy-type (Feller) processes given at the "Second Barcelona Summer School on Stochastic Analysis" 2014 organized by the Centre de Recerca Matemaatica (CRM). The…
Explicit coupling property and gradient estimates are investigated for the linear evolution equations on Hilbert spaces driven by an additive cylindrical L\'evy process. The results are efficiently applied to establish the exponential…
We develop a scale-invariant truncated L\'evy (STL) process to describe physical systems characterized by correlated stochastic variables. The STL process exhibits L\'evy stability for the probability density, and hence shows scaling…