Joint asymptotic distribution of certain path functionals of the reflected process
TLDR
In this article, the authors investigated the joint asymptotic distribution of the reflected process of a Levy process and the path functionals of the path functions of the Levy process for a certain non-linear curve.Abstract:
Let $\tau (x)$ be the first time that the reflected process $Y$ of a Levy 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 Levy processes $X$ satisfying Cramer’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 Cramer 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 $(Y_\infty , Z_\infty )$.read more
Citations
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Convergence of probability measures
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Buffer-overflows: Joint limit laws of undershoots and overshoots of reflected processes
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Renewal theory and level passage by subordinators
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References
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Book
Modelling Extremal Events: for Insurance and Finance
TL;DR: In this article, an approach to Extremes via Point Processes is presented, and statistical methods for Extremal Events are presented. But the approach is limited to time series analysis for heavy-tailed processes.
Book ChapterDOI
Convergence of probability measures
TL;DR: Weakconvergence methods in metric spaces were studied in this article, with applications sufficient to show their power and utility, and the results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables.
Journal ArticleDOI
Modelling Extremal Events for Insurance and Finance
TL;DR: In this article, Modelling Extremal Events for Insurance and Finance is discussed. But the authors focus on the modeling of extreme events for insurance and finance, and do not consider the effects of cyber-attacks.