Related papers: Two continua of embedded regenerative sets

In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise,…

Let $^{(r,s)}X_t$ be the L\'evy process $X_t$ with the $r$ largest jumps and $s$ smallest jumps up till time $t$ deleted and let $^{(r)}\tilde X_t$ be $X_t$ with the $r$ largest jumps in modulus up till time $t$ deleted. We show that…

For a (killed) spectrally negative L\'evy process we provide an analytic expression for the distribution of its overshoot over a fixed level in terms of the infinitesimal generator and the scale function of the process. Our identity…

We present a class of L\'evy processes for modelling financial market fluctuations: Bilateral Gamma processes. Our starting point is to explore the properties of bilateral Gamma distributions, and then we turn to their associated L\'evy…

We study the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, where $Z_t = (Z_t^{(1)},\ldots,Z_t^{(d)})^T$ and $Z_t^{(1)}, \ldots, Z_t^{(d)}$ are independent one-dimensional L{\'e}vy processes with characteristic…

A L\'evy process on a *-bialgebra is given by its generator, a conditionally positive hermitian linear functional vanishing at the unit element. A *-algebra homomorphism k from a *-bialgebra C to a *-bialgebra B with the property that k…

We consider the stochastic differential equations of the form \begin{equation*} \begin{cases} dX^ x(t) = \sigma(X(t-)) dL(t) \\ X^ x(0)=x,\quad x\in\mathbb{R}^ d, \end{cases} \end{equation*} where $\sigma:\mathbb{R}^ d\to \mathbb{R}^ d$ is…

We provide analytical tools for pricing power options with exotic features (capped or log payoffs, gap options ...) in the framework of exponential L\'evy models driven by one-sided stable or tempered stable processes. Pricing formulas take…

We consider renewal stochastic processes generated by non-independent events from the perspective that their basic distribution and associated generating functions obey the statistical-mechanical structure of systems with interacting…

We develop a general construction for nonlinear L\'evy processes with given characteristics. More precisely, given a set $\Theta$ of L\'evy triplets, we construct a sublinear expectation on Skorohod space under which the canonical process…

A step reinforced random walk is a discrete time process with memory such that at each time step, with fixed probability $p \in (0,1)$, it repeats a previously performed step chosen uniformly at random while with complementary probability…

In the present work, we consider spectrally positive L\'evy processes $(X_t,t\geq0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process…

For a L\'evy process $\xi=(\xi_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows \[{\rm{I}}_{\xi}=\int_0^{\infty}e^{\xi_t} dt.\] Under mild conditions on $\xi$, we show that the following…

The class of Levy processes for which overshoots are almost surely constant quantities is precisely characterized.

We study subexponential tail asymptotics for the distribution of the maximum $M_t:=\sup_{u\in[0,t]}X_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and…

Motivated by a recent paper of Budd, where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of L\'evy processes, called the double hypergeometric class, whose…

Let $(\xi,\eta)$ be a bivariate L\'evy process such that the integral $\int\_0^\infty e^{-\xi\_{t-}} d\eta\_t$ converges almost surely. We characterise, in terms of their \LL measures, those L\'evy processes for which (the distribution of)…

In this paper, we establish the monotone convex order between two $\mathbb{R}$-valued McKean-Vlasov processes $X=(X_t)_{t\in [0, T]}$ and $Y=(Y_t)_{t\in [0, T]}$ defined on a filtered probability space $(\Omega, \mathcal{F},…

We develop criteria for recurrence and transience of one-dimensional Markov processes which have jumps and oscillate between $+\infty$ and $-\infty$. The conditions are based on a Markov chain which only consists of jumps (overshoots) of…

For a stochastic process $(X_t)_{t\geq 0}$ we establish conditions under which the inverse first-passage time problem has a solution for any random variable $\xi >0$. For Markov processes we give additional conditions under which the…