scholarly journals Exit problems for general draw-down times of spectrally negative Lévy processes

2019 ◽  
Vol 56 (2) ◽  
pp. 441-457 ◽  
Author(s):  
Bo Li ◽  
Nhat Linh Vu ◽  
Xiaowen Zhou
Keyword(s):  
Lévy Processes ◽  
Exit Time ◽  
Levy Processes ◽  
Laplace Transforms ◽  
Scale Functions ◽  
Dynamic Level ◽  
Running Maximum ◽  
Potential Measure ◽  
Exit Problems ◽  
Spectrally Negative Lévy Processes

AbstractFor spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Lévy processes.

scholarly journals On the last exit times for spectrally negative Lévy processes

2017 ◽  
Vol 54 (2) ◽  
pp. 474-489 ◽  
Author(s):  
Yingqiu Li ◽  
Chuancun Yin ◽  
Xiaowen Zhou
Keyword(s):  
Lévy Processes ◽  
Exit Time ◽  
Levy Processes ◽  
Laplace Transforms ◽  
Occupation Time ◽  
Exit Times ◽  
New Approach ◽  
Last Exit Time ◽  
Spectrally Negative Lévy Processes

Abstract Using a new approach, for spectrally negative Lévy processes we find joint Laplace transforms involving the last exit time (from a semiinfinite interval), the value of the process at the last exit time, and the associated occupation time, which generalize some previous results.

scholarly journals The W, Z scale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems

2020 ◽  
Vol 24 ◽  
pp. 454-525 ◽  
Author(s):  
Florin Avram ◽  
Danijel Grahovac ◽  
Ceren Vardar-Acar
Keyword(s):  
Lévy Processes ◽  
Mathematical Finance ◽  
Levy Processes ◽  
Small Sample ◽  
First Passage ◽  
Scale Functions ◽  
Exit Problems ◽  
Finance Risk ◽  
Storage Theory ◽  
Spectrally Negative Lévy Processes

In the last years there appeared a great variety of identities for first passage problems of spectrally negative Lévy processes, which can all be expressed in terms of two “q-harmonic functions” (or scale functions) W and Z. The reason behind that is that there are two ways of exiting an interval, and thus two fundamental “two-sided exit” problems from an interval (TSE). Since many other problems can be reduced to TSE, researchers developed in the last years a kit of formulas expressed in terms of the “W, Z alphabet”. It is important to note – as is currently being shown – that these identities apply equally to other spectrally negative Markov processes, where however the W, Z functions are typically much harder to compute. We collect below our favorite recipes from the “W, Z kit”, drawing from various applications in mathematical finance, risk, queueing, and inventory/storage theory. A small sample of applications concerning extensions of the classic de Finetti dividend problem is offered. An interesting use of the kit is for recognizing relationships between problems involving behaviors apparently unrelated at first sight (like reflection, absorption, etc.). Another is expressing results in a standardized form, improving thus the possibility to check when a formula is already known.

Scale functions of Lévy processes and busy periods of finite-capacity M/GI/1 queues

2004 ◽  
Vol 41 (4) ◽  
pp. 1145-1156 ◽  
Author(s):  
Parijat Dube ◽  
Fabrice Guillemin ◽  
Ravi R. Mazumdar
Keyword(s):  
Closed Form ◽  
Lévy Processes ◽  
Busy Period ◽  
Exit Time ◽  
Levy Processes ◽  
Finite Capacity ◽  
Scale Functions ◽  
Period Distribution ◽  
Finite Capacity Queues ◽  
Time Theory

In this paper we use the exit time theory for Lévy processes to derive new closed-form results for the busy period distribution of finite-capacity fluid M/G/1 queues. Based on this result, we then obtain the busy period distribution for finite-capacity queues with on–off inputs when the off times are exponentially distributed.

Scale functions of Lévy processes and busy periods of finite-capacity M/GI/1 queues

2004 ◽  
Vol 41 (04) ◽  
pp. 1145-1156 ◽  
Author(s):  
Parijat Dube ◽  
Fabrice Guillemin ◽  
Ravi R. Mazumdar
Keyword(s):  
Closed Form ◽  
Lévy Processes ◽  
Busy Period ◽  
Exit Time ◽  
Levy Processes ◽  
Finite Capacity ◽  
Scale Functions ◽  
Period Distribution ◽  
Finite Capacity Queues ◽  
Time Theory

In this paper we use the exit time theory for Lévy processes to derive new closed-form results for the busy period distribution of finite-capacity fluid M/G/1 queues. Based on this result, we then obtain the busy period distribution for finite-capacity queues with on–off inputs when the off times are exponentially distributed.

scholarly journals Exit Problems for Spectrally Negative Lévy Processes Reflected at Either the Supremum or the Infimum

2007 ◽  
Vol 44 (4) ◽  
pp. 1012-1030 ◽  
Author(s):  
Xiaowen Zhou
Keyword(s):  
Lévy Processes ◽  
Stable Process ◽  
Levy Processes ◽  
Exit Problem ◽  
Brownian Motion With Drift ◽  
Spectrally Negative Lévy Process ◽  
The Times ◽  
Exit Problems ◽  
Supremum Process ◽  
Spectrally Negative Lévy Processes

For a spectrally negative Lévy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let τ(a) and κ(a) denote the times when the reflected processes Ŷ := S − X and Y := X − I first exit level a, respectively; let τ−(a) and κ−(a) denote the times when X first reaches Sτ(a) and Iκ(a), respectively. The main results of this paper concern the distributions of (τ(a), Sτ(a), τ−(a), Ŷτ(a)) and of (κ(a), Iκ(a), κ−(a)). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.

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