Search ResultsShowing 1-5 of 5
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arXiv:1603.02506 (Published 2016-03-08)
Joint law of the hitting time, overshoot and undershoot for a Lévy process
Categories: math.PRLet be $(X_t, t\geq 0)$ be a L\'evy process which is the sum of a Brownian motion with drift and a compound Poisson process. We consider the first passage time $\tau_x$ at a fixed level $x>0$ by $(X_t, t\geq 0)$ and $K_x:= X_{\tau_x}-x$ the overshoot and $L_x:= x-X_{\tau_x^-}$ the undershoot. We first study the regularity of the density of the first passage time. Secondly, we calculate the joint law of $(\tau_x, K_x, L_x).$
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arXiv:1307.6947 (Published 2013-07-26)
Buffer-overflows: joint limit laws of undershoots and overshoots of reflected processes
Comments: 11 pages, no figuresCategories: math.PRLet $\tau(x)$ be the epoch of first entry into the interval $(x,\infty)$, $x>0$, of the reflected process $Y$ of a L\'evy process $X$, and define the overshoot $Z(x) = Y(\tau(x))-x$ and undershoot $z(x) = x - Y(\tau(x)-)$ of $Y$ at the first-passage time over the level $x$. In this paper we establish, separately under the Cram\'{e}r and positive drift assumptions, the existence of the weak limit of $(z(x), Z(x))$ as $x$ tends to infinity and provide explicit formulae for their joint CDFs in terms of the L\'{e}vy measure of $X$ and the renewal measure of the dual of $X$. We apply our results to analyse the behaviour of the classical M/G/1 queueing system at the buffer-overflow, both in a stable and unstable case.
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arXiv:math/0603210 (Published 2006-03-09)
Overshoots and undershoots of Lévy processes
Comments: Published at http://dx.doi.org/10.1214/105051605000000647 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)Journal: Annals of Applied Probability 2006, Vol. 16, No. 1, 91-106Categories: math.PRKeywords: lévy processes, undershoot, first passage contribute, concerning asymptotic overshoot distribution, fluctuation identityTags: journal articleWe obtain a new fluctuation identity for a general L\'{e}vy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Kl\"{u}ppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801] concerning asymptotic overshoot distribution of a particular class of L\'{e}vy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying L\'{e}vy process is spectrally one sided.
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arXiv:math/0507193 (Published 2005-07-10)
Levy processes: Hitting time, overshoot and undershoot II - Asymptotic behaviour
Comments: Manuscript P661 submitted to SPA, October 2004. 28 pagesCategories: math.PRLet (X_t, t>=0) be a Levy process started at 0, with Levy measure nu and T_x the first hitting time of level x>0: T_x:=inf{t>=0; X_t>x}. Let $F(theta, mu, rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x) :=E(e^(-theta T_x - mu K_x \rho L_x) 1_(T_x<+infinity)), where theta>=0, mu>=0, rho>=0, x>=0, K_x:=X_(T_x)-x and L_x:=x-X_(T_(x^-)). If we assume that nu has finite exponential moments we exhibit an asymptotic expansion for F(theta,mu,rho,x), as x -> +infinity. A limit theorem involving a normalization of the triplet (T_x,K_x,L_x) as x -> +infinity, may be deduced. At last, if nu_(|_R_+) has finite moment of fixed order, we prove that the ruin probability P(T_x<+infinity) has at most a polynomial decay.
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arXiv:math/0507034 (Published 2005-07-02)
Levy Processes: Hitting time, overshoot and undershoot - part I: Functional equations
Comments: Manuscript P660 submitted to SPA, October 2004. 30 pages, 5 figures, 25 referencesCategories: math.PRLet (X_t, t >=0) be a Levy process started at 0, with Levy measure nu, and T_x the first hitting time of level x>0: T_x := inf{t>=0; X_t>x}. Let F(theta,mu,rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x) := E (e^{-theta T_x - mu K_x - rho L_x} 1_{T_x<+infinity}), where theta>=0, mu>=0, rho>=0, x>0, K_x := X_{T_x} - x and L_x := x - X_{T_{x^-}}. If nu(R) < + \infinity and integral_1^{+\infty} e^{sy} nu (dy) < +infinity for some s>0, then we prove that F(theta,mu,rho,.) is the unique solution of an integral equation and has a subexponential decay at infinity when theta>0 or theta=0 and E(X_1)<0. If nu is not necessarily a finite measure but verifies integral_{-infinity}^{-1} e^{-sy} nu (dy) < +infinity for any s>0, then the x-Laplace transform of F(theta,mu,rho,.) satisfies some kind of integral equation. This allows us to prove that F(theta,mu,rho,.) is a solution to a second integral equation.