Related papers: Zooming in on a L\'evy process at its supremum

For a L\'evy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as…

We consider the problem of finding a stopping time that minimises the $L^1$-distance to $\theta$, the time at which a L\'evy process attains its ultimate supremum. This problem was studied in [12] for a Brownian motion with drift and a…

This paper studies small-time behavior at the supremum of a diffusion process. For a solution to the SDE $\mathrm{d} X_t=\mu(X_t)\mathrm{d} t+\sigma(X_t)\mathrm{d} W_t$ (where $W$ is a standard Brownian motion) we consider…

Let X_t, 0<=t<=T be a one-dimensional stochastic process with independent and stationary increments. This paper considers the problem of stopping the process X_t "as close as possible" to its eventual supremum M_T:=sup{X_t: 0<=t<=T}, when…

This paper provides a multivariate extension of Bertoin's pathwise construction of a L\'evy process conditioned to stay positive/negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original…

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 $\overline Z$…

Distributional identities for a L\'evy process $X_t$, its quadratic variation process $V_t$ and its maximal jump processes, are derived, and used to make "small time" (as $t\downarrow0$) asymptotic comparisons between them. The…

The {\em drawdown} process $Y$ of a completely asymmetric L\'{e}vy process $X$ is equal to $X$ reflected at its running supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the…

An obvious way to simulate a L\'evy process $X$ is to sample its increments over time $1/n$, thus constructing an approximating random walk $X^{(n)}$. This paper considers the error of such approximation after the two-sided reflection map…

Motivated by recent studies of record statistics in relation to strongly correlated time series, we consider explicitly the drawdown time of a Levy process, which is defined as the time since it last achieved its running maximum when…

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…

The first passage time process of a L\'evy subordinator with heavy-tailed L\'evy measure has long-range dependent paths. The random fluctuations that appear under two natural schemes of summation and time scaling of such stochastic…

Last passage times arise in a number of areas of applied probability, including risk theory and degradation models. Such times are obviously not stopping times since they depend on the whole path of the underlying process. We consider the…

We consider a one-dimensional diffusion in a stable L\'evy environment. We show that the normalized local time process refocused at the bottom of the standard valley with height $\log t$, $(L_X(t,\mathfrak m_{\log t}+x)/t,x\in \R)$,…

We prove that when a sequence of L\'evy processes $X^{(n)}$ or a normed sequence of random walks $S^{(n)}$ converges a.s. on the Skorokhod space toward a L\'evy process $X$, the sequence $L^{(n)}$ of local times at the supremum of $X^{(n)}$…

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…

This paper addresses the question of predicting when a positive self-similar Markov process X attains its pathwise global supremum or infimum before hitting zero for the first time (if it does at all). This problem has been studied in…

For an arbitrary diffusion process $X$ with time-homogeneous drift and variance parameters $\mu(x)$ and $\sigma^2(x)$, let $V_\varepsilon$ be $1/\varepsilon$ times the total time $X(t)$ spends in the strip…

This paper considers discretization of the L\'evy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity…

We find an expression for the joint Laplace transform of the law of $(T_{[x,+\infty[},X_{T_{[x,+\infty[}})$ for a L\'evy process $X$, where $T_{[x,+\infty[}$ is the first hitting time of $[x,+\infty[$ by $X$. When $X$ is an $\alpha$-stable…