scholarly journals The two-sided exit problem for spectrally positive Lévy processes

1990 ◽  
Vol 22 (2) ◽  
pp. 486-487 ◽  
Author(s):  
L. C. G. Rogers
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Exit Problem

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.

Some fluctuation identities for Lévy processes with jumps of the same sign

2004 ◽  
Vol 41 (04) ◽  
pp. 1191-1198 ◽  
Author(s):  
Xiaowen Zhou
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Joint Distribution ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Exit Problem ◽  
The Laplace Transform ◽  
The One

We consider a two-sided exit problem for a Lévy process with no positive jumps. The Laplace transform of the time when the process first exits an interval from above is obtained. It is expressed in terms of another Laplace transform for the one-sided exit problem. Applications of this result are discussed. In particular, a new expression for the solution to the two-sided exit problem is obtained. The joint distribution of the minimum and the maximum values of such a Lévy process is also studied.

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

2007 ◽  
Vol 44 (04) ◽  
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 S τ(a) and I κ(a), respectively. The main results of this paper concern the distributions of (τ(a), S τ(a), τ−(a), Ŷ τ(a)) and of (κ(a), I κ(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.

Some fluctuation identities for Lévy processes with jumps of the same sign

2004 ◽  
Vol 41 (4) ◽  
pp. 1191-1198 ◽  
Author(s):  
Xiaowen Zhou
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Joint Distribution ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Exit Problem ◽  
The Laplace Transform ◽  
The One

We consider a two-sided exit problem for a Lévy process with no positive jumps. The Laplace transform of the time when the process first exits an interval from above is obtained. It is expressed in terms of another Laplace transform for the one-sided exit problem. Applications of this result are discussed. In particular, a new expression for the solution to the two-sided exit problem is obtained. The joint distribution of the minimum and the maximum values of such a Lévy process is also studied.

On correlating Lévy processes

2010 ◽  
Vol 13 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Ernst Eberlein ◽  
Dilip Madan
Keyword(s):  
Lévy Processes ◽  
Levy Processes
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