Fluctuation identities for lévy processes and splitting at the maximum

1980 ◽  
Vol 12 (04) ◽  
pp. 893-902 ◽  
Author(s):  
Priscilla Greenwood ◽  
Jim Pitman
Keyword(s):  
Point Process ◽  
Lévy Processes ◽  
Poisson Point Process ◽  
Levy Processes ◽  
Fluctuation Theory ◽  
Unified Approach ◽  
Path Decomposition

Itô's notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of Lévy processes, including identities of Pecherskii, Rogozin and Fristedt, and Millar's path decomposition at the maximum.

Fluctuation identities for lévy processes and splitting at the maximum

1980 ◽  
Vol 12 (4) ◽  
pp. 893-902 ◽  
Author(s):  
Priscilla Greenwood ◽  
Jim Pitman
Keyword(s):  
Point Process ◽  
Lévy Processes ◽  
Poisson Point Process ◽  
Levy Processes ◽  
Fluctuation Theory ◽  
Unified Approach ◽  
Path Decomposition

Itô's notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of Lévy processes, including identities of Pecherskii, Rogozin and Fristedt, and Millar's path decomposition at the maximum.

scholarly journals $${{\varvec{L}}}^{{\varvec{p}}}$$-Solutions and Comparison Results for Lévy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting

2020 ◽  
Author(s):  
Stefan Kremsner ◽  
Alexander Steinicke
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Backward Stochastic Differential Equations ◽  
Time Dependent ◽  
Unified Approach ◽  
General Growth ◽  
Comparison Results ◽  
Osgood Condition

AbstractWe present a unified approach to $$L^p$$ L p -solutions ($$p > 1$$ p > 1 ) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the y-variable and have general growth in y. Within this setting, the results generalize those of Royer, Yin and Mao, Yao, Kruse and Popier, and Geiss and Steinicke.

scholarly journals Concave Majorants of Random Walks and Related Poisson Processes

2011 ◽  
Vol 20 (5) ◽  
pp. 651-682
Author(s):  
JOSH ABRAMSON ◽  
JIM PITMAN
Keyword(s):  
Random Walks ◽  
Point Process ◽  
Finite Length ◽  
Poisson Point Process ◽  
Complete Information ◽  
Arithmetic Mean ◽  
Poisson Processes ◽  
Infinite Length ◽  
Unified Approach ◽  
Geometric Length

We offer a unified approach to the theory of concave majorants of random walks, by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant of a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested compositions that naturally arise from our construction of the concave majorant.

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