Related papers: On exponential functionals of Levy processes
In this paper we study the exponential functionals of the processes $X$ with independent increments , namely $$I_t= \int _0^t\exp(-X_s)ds, _,\,\, t\geq 0,$$ and also $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ When $X$ is a…
In the L\'evy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process $W^{N}$ and to define the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends to infinity.…
By killing a stable L\'{e}vy process when it leaves the positive half-line, or by conditioning it to stay positive, or by conditioning it to hit 0 continuously, we obtain three different positive self-similar Markov processes which…
We consider Gaussian subordinated L\'evy fields (GSLFs) that arise by subordinating L\'evy processes with positive transformations of Gaussian random fields on some spatial domain $\mathcal{D}\subset \mathbb{R}^d$, $d\geq 1$. The resulting…
For a bivariate \Levy process $(\xi_t,\eta_t)_{t\ge 0}$ and initial value $V_0$ define the Generalised Ornstein-Uhlenbeck (GOU) process \[ V_t:=e^{\xi_t}\Big(V_0+\int_0^t e^{-\xi_{s-}}\ud \eta_s\Big),\quad t\ge0,\] and the associated…
We derive a criterium for the almost sure finiteness of perpetual integrals of \LL processes for a class of real functions including all continuous functions and for general one-dimensional L\'evy processes that drifts to plus infinity.…
We study natural invariance properties of functionals defined on L\'evy processes and show that they can be described by a simplified structure of the deterministic chaos kernels in It\^o's chaos expansion. These structural properties of…
L\'evy-driven Ornstein-Uhlenbeck (OU) processes represent an intriguing class of stochastic processes that have garnered interest in the energy sector for their ability to capture typical features of market dynamics. However, in the current…
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…
The concept of a L\'evy subordinator is generalized to a family of non-decreasing stochastic processes, which are parameterized in terms of two Bernstein functions. Whereas the independent increments property is only maintained in the…
The infinitesimal generator of a one-dimensional strictly $\alpha$-stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order $\alpha$ and one obtains the fractional Laplacian…
In this article we introduce a theory of integration for deterministic, operator-valued integrands with respect to cylindrical L\'evy processes in separable Banach spaces. Here, a cylindrical L\'evy process is understood in the classical…
It is shown that some convolution semigroups of infinitely divisible measures are invariant under the random integral mappings $I^{h,r}_{(a,b]}$ defined in $(\star)$ below. The converse implication is specified for the semigroups of…
Let $V$ be a projective limit, with respect to the renormalized norm mappings, of the groups of principal units corresponding to a strictly increasing sequence of finite separable totally and tamely ramified Galois extensions of a local…
We derive a necessary and sufficient condition for stochastic processes to have almost periodic finite dimensional distributions; in particular, we obtain characterizations for infinitely divisible processes to be almost periodic in terms…
We continue the investigation of the Levy processes on a q-deformed full Fock space started in a previous paper. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a…
We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric $\alpha $-stable L\'{e}vy processes, whose permanent residence in a finite interval on a line…
When is it possible to interpret a given Markov process as a L\'evy-like process? Since the class of L\'evy processes can be defined by the relation between transition probabilities and convolutions, the answer to this question lies in the…
As in the classical case of L\'evy processes on a group, L\'evy processes on a Voiculescu dual group are constructed from conditionally positive functionals. It is essential for this construction that Schoenberg correspondence holds for…
We investigate the upper tail probabilities of the all-time maximum of a stable L\'evy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with…