scholarly journals Sample path large deviations for Lévy processes and random walks with Weibull increments

2020 ◽  
Vol 30 (6) ◽  
pp. 2695-2739
Author(s):  
Mihail Bazhba ◽  
Jose Blanchet ◽  
Chang-Han Rhee ◽  
Bert Zwart
Keyword(s):  
Large Deviations ◽  
Random Walks ◽  
Lévy Processes ◽  
Sample Path ◽  
Levy Processes ◽  
Sample Path Large Deviations

scholarly journals Lifting Lévy processes to hyperfinite random walks

2006 ◽  
Vol 130 (8) ◽  
pp. 697-706 ◽  
Author(s):  
Sergio Albeverio ◽  
Frederik S. Herzberg
Keyword(s):  
Random Walks ◽  
Lévy Processes ◽  
Levy Processes

scholarly journals Sample Path Large Deviations for Stochastic Evolutionary Game Dynamics

2018 ◽  
Vol 43 (4) ◽  
pp. 1348-1377 ◽  
Author(s):  
William H. Sandholm ◽  
Mathias Staudigl
Keyword(s):  
Large Deviations ◽  
Evolutionary Game ◽  
Sample Path ◽  
Game Dynamics ◽  
Sample Path Large Deviations

scholarly journals Asymptotics for the First Passage Times of Lévy Processes and Random Walks

2013 ◽  
Vol 50 (1) ◽  
pp. 64-84 ◽  
Author(s):  
Denis Denisov ◽  
Vsevolod Shneer
Keyword(s):  
Large Deviations ◽  
Random Walks ◽  
Levy Processes ◽  
Levy Process ◽  
First Passage Times ◽  
Exact Asymptotics ◽  
First Passage ◽  
Heavy Tailed ◽  
First Time ◽  
Passage Times

We study the exact asymptotics for the distribution of the first time, τx, a Lévy process Xt crosses a fixed negative level -x. We prove that ℙ{τx >t} ~V(x) ℙ{Xt≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τx>t} explicitly in both light- and heavy-tailed cases.

scholarly journals Random walks and Lévy processes as rough paths

2017 ◽  
Vol 170 (3-4) ◽  
pp. 891-932 ◽  
Author(s):  
Ilya Chevyrev
Keyword(s):  
Random Walks ◽  
Lévy Processes ◽  
Levy Processes ◽  
Rough Paths

Generalized fractional Lévy processes with fractional Brownian motion limit

2015 ◽  
Vol 47 (04) ◽  
pp. 1108-1131 ◽  
Author(s):  
Claudia Klüppelberg ◽  
Muneya Matsui
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Lévy Processes ◽  
Sample Path ◽  
Levy Processes ◽  
Order Structure ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Volatility Models ◽  
Ornstein Uhlenbeck Process

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

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