Related papers: On the density of the supremum of a stable process

An absolutely convergent double series representation for the density of the supremum of $\alpha$-stable Levy process is given in [3, Theorem 2] for almost all irrational $\alpha$. This result cannot be made stronger in the following sense:…

We study the density of the supremum of a strictly stable L\'evy process. We prove that for almost all values of the index $\alpha$ -- except for a dense set of Lebesgue measure zero -- the asymptotic series which were obtained in A.…

We study the distribution and various properties of exponential functionals of hypergeometric Levy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic…

We derive explicit asymptotic expansions of the density of the supremum of a strictly stable process when the index $\alpha$ is not rational. In the case when parameters $\alpha$ and $\rho=\p(X_1>0)$ satisfy $\rho+k=l/\alpha$ for some…

If $X$ is a stable process of index $\alpha\in(0,2)$ whose L\'{e}vy measure has density $cx^{-\alpha-1}$ on $(0,\infty)$, and $S_1=\sup_{0<t\leq1}X_t$, it is known that $P(S_1>x)\backsim A\alpha ^{-1}x^{-\alpha}$ as $x\to\infty$ and…

Let $X=(X_t)_{t\ge0}$ be a stable L\'{e}vy process of index $\alpha \in(1,2)$ with no negative jumps and let $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t>0$. We show that the density function $f_t$ of $S_t$ can be…

If $X$ is a spectrally positive stable process of index $\alpha\in(1,2)$ whose L\'{e}vy measure has density $cx^{-\alpha-1}$ on $(0,\infty),$ and $S_1=\sup_{0<t\leq1}X_t,$ it is known that $P(S_1>x)\backsim c\alpha^{-1}x^{-\alpha}$ as…

We consider a process $Z$ on the real line composed from a L\'evy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$…

In this article, the infinite series form of the probability densities of tempered stable and inverse tempered stable subordinators are obtained using Mellin transform. Further, the densities of the products and quotients of stable and…

In this paper we derive a technique of obtaining limit theorems for suprema of L\'evy processes from their random walk counterparts. For each $a>0$, let $\{Y^{(a)}_n:n\ge 1\}$ be a sequence of independent and identically distributed random…

In this paper, we study the purely discontinuous Girsanov transforms which were discussed in Chen and Song \cite{CS2} and Song \cite{S3}. We show that the transition density of any purely discontinuous Girsanov transform of a…

This article uses a combination of three ideas from simulation to establish a nearly optimal polynomial upper bound for the joint density of the stable process and its associated supremum at a fixed time on the entire support of the joint…

We study the Wiener--Hopf factorization and the distribution of extrema for general stable processes. By connecting the Wiener--Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such…

Let us consider a real L\'evy process X whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of X before any deterministic time t is absolutely continuous on (0,\infty). We…

Stable distributions are a celebrated class of probability laws used in various fields. The $\alpha$-stable process, and its exponentially tempered counterpart, the Classical Tempered Stable (CTS) process, are also prominent examples of…

It is, perhaps, surprising that the location of the unique supremum of a stationary process on an interval can fail to be uniformly distributed over that interval. We show that this distribution is absolutely continuous in the interior of…

In this article, we propose a Lyapunov stability approach to analyze the convergence of the density operator of a quantum system. In analog to the classical probability measure for Markovian processes, we show that the set of invariant…

In spite of the Lebesgue density theorem, there is a positive $\delta$ such that, for every non-trivial measurable set $S$ of real numbers, there is a point at which both the lower densities of $S$ and of the complement of $S$ are at least…

In this paper we study the supremum functional $M_t=\sup_{0\le s\le t}X_s$, where $X_t$, $t\ge0$, is a one-dimensional L\'{e}vy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution…

In this article we get simple explicit formulas for $\Exp\sup_{s\leq t}X(s)$ where $X$ is a spectrally positive or negative L\'evy process with infinite variation. As a consequence we derive a generalization of the well-known formula for…