Related papers: On exponential functionals of processes with indep…
Extending It\^o's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-It\^o, applies to one dimensional semimartingales and convex functions.…
In [16], under mild conditions, a Wiener-Hopf type factorization is derived for the exponential functional of proper L\'evy processes. In this paper, we extend this factorization by relaxing a finite moment assumption as well as by…
For each $n\geq 1$, let $ {X_{in}, \quad i \geq 1} $ be independent copies of a nonnegative continuous stochastic process $X_{n}=(X_n(t))_{t\in T}$ indexed by a compact metric space $T$. We are interested in the process of partial maxima…
We show that the SDE $dX_t = \sigma(X_{t-}) \, dL_t$, $X_0 \sim \mu$ driven by a one-dimensional symnmetric $\alpha$-stable L\'evy process $(L_t)_{t \geq 0}$, $\alpha \in (0,2]$, has a unique weak solution for any continuous function…
Dilative stability generalizes the property of selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. Inspired by results of Igl\'oi, we will show how dilatively stable…
We examine the Ising chain in a transverse field at zero temperature from the point of view of a family of moment formalisms based upon the cumulant generating function, where we find exact solutions for the generating functions and…
In this paper, we obtain explicit product and moment formulas for products of iterated integrals generated by families of square integrable martingales associated with an arbitrary L\'evy process. We propose a new approach applying the…
In this paper, we establish the existence of moments and moment estimates for L\'evy-type processes. We discuss whether the existence of moments is a time dependent distributional property, give sufficient conditions for the existence of…
Markov-modulated L\'evy processes with two different regimes of restarting are studied. These regimes correspond to the completely renewed process and to the process of Markov modulation, accompanied by jumps. We give explicit expressions…
Let $(X_j)_{j\geq1}$ be a multivariate long-range dependent Gaussian process. We study the asymptotic behavior of the corresponding sequential empirical process indexed by a class of functions. If some entropy condition is satisfied we have…
We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. Other than in the classical L\'evy case studied by Erickson and Maller we have to distinguish between almost sure…
We consider additive functionals of Markov processes in continuous time with general (metric) state spaces. We derive concentration bounds for their exponential moments and moments of finite order. Applications include diffusions,…
This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing…
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…
We study spectral-theoretic properties of non-self-adjoint operators arising in the study of one-dimensional L\'evy processes with completely monotone jumps with a one-sided barrier. With no further assumptions, we provide an integral…
Let $\xi_i$, $i\in \mathbb {N}$, be independent copies of a L\'{e}vy process $\{\xi(t),t\geq0\}$. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process…
We investigate the properties of the moments of the cot function using the central factorial numbers. Using a new integral representation of the central factorial numbers, we find a new way to express these moments in terms of recursive…
We present a systematic study of the reconstruction of a non-negative function via maximum entropy approach utilizing the information contained in a finite number of moments of the function. For testing the efficacy of the approach, we…
Integral representations for expectations of functions of a stable L\'evy process $X$ and its supremum $\bar X$ are derived. As examples, cumulative probability distribution functions (cpdf) of $X_T, \barX_T$, the joint cpdf of $X_T$ and…
We study functionals of the form \[\zeta_{t}=\int_0^{t}...\int_0^{t} | X_1(s_1)+...+ X_p(s_p)|^{-\sigma}ds_1... ds_p\] where $X_1(t),..., X_p(t)$ are i.i.d. $d$-dimensional symmetric stable processes of index $0<\bb\le 2$. We obtain results…