On the infimum attained by a reflected Lévy process

On the infimum attained by the reflected fractional Brownian motion

Let $\{B_H(t):t\ge 0\}$ be a fractional Brownian motion with Hurst parameter $H\in(\frac{1}{2},1)$. For the storage process $Q_{B_H}(t)=\sup_{-\infty\le s\le t} \left(B_H(t)-B_H(s)-c(t-s)\right)$ we show that, for any $T(u)>0$ such that $T(u)=o(u^\frac{2H-1}{H})$, \[\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_H}(s)>u)\sim\mathbb P(Q_{B_H}(0)>u),\quad\text{as}\quad u\to\infty.\] This finding, known in the literature as the strong Piterbarg property, is in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.

Control of a finite dam when the input process is either spectrally positive Levy or spectrally positive Levy reflected at its infimum

We consider the control of a finite dam when the input process is either spectrally positive Levy or spectrally positive Levy reflected at its infimum, using P(M,Lambda,tau) control policies. Our results extend and unify the results Bea et al (2003), Lam and Lou (1987), and Attia (1987). The techniques used by the above authors involve solving systems of differential or integral equations. Our methods of proof are based on the theory and methods of scale functions of Levy processes.

The excursion measure away from zero for spectrally negative Lévy processes

We provide a description of the excursion measure from a point for a spectrally negative Lévy process. The description is based in two main ingredients. The first is building a spectrally negative Lévy process conditioned to avoid zero and the study of its entrance law at zero. The latter is connected with both the excursion measure from zero of the process reflected in its infimum and reflected in its supremum. This lead us to establish a connection with the excursion measure from the state zero with the excursion measure from zero for the process reflected at the infimum and reflected at the…

谱负Lévy过程下确界和最大回撤的条件路径分解

Conditional Path Decomposition at the Infimum and Maximum Drawdowns for Spectrally Negative Lévy Processes

针对谱负Lévy过程，研究在极端事件条件下最大回撤的分布，通过两种路径分解（下确界处分解和先下确界后上确界分解）给出尺度函数刻画，并得到条件分布的Doob h-变换表示。

Asymptotic behaviour of first passage time distributions for Lévy processes

Let $X$ be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function $c.$ As an analogue of the random walk results in \cite{vw} and \cite{rad} we study the local behaviour of the distribution of the lifetime $ζ$ under the characteristic measure $\underline{n}$ of excursions away from 0 of the process $X$ reflected in its past infimum, and of the first passage time of $X$ below $0,$ $T_{0}=\inf \{t>0:X_{t}<0\},$ under $\mathbb{P}_{x}(\cdot),$ for $x>0,$ in two different regimes for $x,$ viz. $x=o(c(\cdot))$ and $x>D c(\cdot),$ for so…

反射莱维过程的大偏移：渐进行为

Large Excursions of Reflected Lévy Processes: Asymptotic Shapes

本文研究了反射莱维过程偏移的渐进行为，揭示了长寿命与大高度在不同条件下的等价性及收敛性质。