Local time of additive Levy process

2000 ◽  
Vol 5 (1) ◽  
pp. 007-012
Author(s):  
Zhong Yu-quan ◽  
Hu Di-he
Keyword(s):  
Local Time ◽  
Lévy Process ◽  
Levy Process

Self-intersection local time of Additive levy process

2002 ◽  
Vol 22 (2) ◽  
pp. 261-268 ◽  
Author(s):  
Yuquan Zhong ◽  
Dike Hu
Keyword(s):  
Local Time ◽  
Lévy Process ◽  
Levy Process ◽  
Intersection Local Time

scholarly journals Analysis of stochastic fluid queues driven by local-time processes

2008 ◽  
Vol 40 (04) ◽  
pp. 1072-1103 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Andreas E. Kyprianou ◽  
Paavo Salminen ◽  
Marina Sirviö
Keyword(s):  
Local Time ◽  
Lévy Process ◽  
Point Processes ◽  
Levy Process ◽  
Fluid Queue ◽  
Idle Period ◽  
Priority Service ◽  
Stationary Version ◽  
Palm Calculus ◽  
Stochastic Fluid

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

scholarly journals Analysis of stochastic fluid queues driven by local-time processes

2008 ◽  
Vol 40 (4) ◽  
pp. 1072-1103 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Andreas E. Kyprianou ◽  
Paavo Salminen ◽  
Marina Sirviö
Keyword(s):  
Local Time ◽  
Lévy Process ◽  
Point Processes ◽  
Levy Process ◽  
Fluid Queue ◽  
Idle Period ◽  
Priority Service ◽  
Stationary Version ◽  
Palm Calculus ◽  
Stochastic Fluid

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

scholarly journals Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations

2021 ◽  
Author(s):  
Fabian A. Harang ◽  
Chengcheng Ling
Keyword(s):  
Differential Equations ◽  
Local Time ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Local Times ◽  
Volterra Kernel ◽  
Spatial Regularity ◽  
Regularizing Effects

AbstractWe investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by $$\alpha $$ α -stable processes for $$\alpha \in (0,2]$$ α ∈ ( 0 , 2 ] . We show that the spatial regularity of the local time for Volterra–Lévy process is $${\mathbb {P}}$$ P -a.s. inverse proportional to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbation of ordinary differential equations by a Volterra–Lévy process which has sufficiently regular local time. Following along the lines of Harang and Perkowski (2020), we show existence, uniqueness and differentiability of the flow associated with such equations.

scholarly journals LAN property for a simple Lévy process

2014 ◽  
Vol 352 (10) ◽  
pp. 859-864 ◽  
Author(s):  
Arturo Kohatsu-Higa ◽  
Eulalia Nualart ◽  
Ngoc Khue Tran
Keyword(s):  
Lévy Process ◽  
Levy Process
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