Open Access
2016 Joint asymptotic distribution of certain path functionals of the reflected process
Aleksandar Mijatović, Martijn Pistorius
Electron. Commun. Probab. 21: 1-18 (2016). DOI: 10.1214/16-ECP4359


Let $\tau (x)$ be the first time that the reflected process $Y$ of a Lévy process $X$ crosses $x>0$. The main aim of this paper is to investigate the joint asymptotic distribution of $Y(t)=X(t) - \inf _{0\leq s\leq t}X(s)$ and the path functionals $Z(x)=Y(\tau (x))-x$ and $m(t)=\sup _{0\leq s\leq t}Y(s) - y^*(t)$, for a certain non-linear curve $y^*(t)$. We restrict to Lévy processes $X$ satisfying Cramér’s condition, a non-lattice condition and the moment conditions that $E[|X(1)|]$ and $E[\exp (\gamma X(1))|X(1)|]$ are finite (where $\gamma $ denotes the Cramér coefficient). We prove that $Y(t)$ and $Z(x)$ are asymptotically independent as $\min \{t,x\}\to \infty $ and characterise the law of the limit $(Y_\infty ,Z_\infty )$. Moreover, if $y^*(t) = \gamma ^{-1}\log (t)$ and $\min \{t,x\}\to \infty $ in such a way that $t\exp \{-\gamma x\}\to 0$, then we show that $Y(t)$, $Z(x)$ and $m(t)$ are asymptotically independent and derive the explicit form of the joint weak limit $(Y_\infty , Z_\infty , m_\infty )$. The proof is based on excursion theory, Theorem 1 in [7] and our characterisation of the law $(Y_\infty , Z_\infty )$.


Download Citation

Aleksandar Mijatović. Martijn Pistorius. "Joint asymptotic distribution of certain path functionals of the reflected process." Electron. Commun. Probab. 21 1 - 18, 2016.


Received: 12 June 2015; Accepted: 9 May 2016; Published: 2016
First available in Project Euclid: 23 May 2016

zbMATH: 1346.60062
MathSciNet: MR3510251
Digital Object Identifier: 10.1214/16-ECP4359

Primary: 60F05 , 60G17 , 60G51

Keywords: Asymptotic independence , Cramér condition , limiting overshoot , Reflected Lévy process

Back to Top