Differentiability of spectral functions for nearly stable processes and large deviations

2014 ◽  
Vol 17 (03) ◽  
pp. 1450017
Author(s):  
Yoshihiro Tawara ◽  
Kaneharu Tsuchida
Keyword(s):  
Large Deviations ◽  
Spectral Function ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Characteristic Exponent ◽  
Stable Processes ◽  
Slowly Varying Function ◽  
Spectral Functions ◽  
Additive Functionals ◽  
Deviation Principle

We consider the differentiability of a spectral function generated by a Lévy process M with characteristic exponent |ξ|αl(|ξ|2), where l(x) is a slowly varying function at ∞. As an application, we obtain the large deviation principle for positive continuous additive functionals of M. Finally, we show that the exponent l(x) = ( log (1 + x))β/2 (0 < β < 2 - α) is an example for which our theorem is applicable.

ON A SUFFICIENT CONDITION FOR LARGE DEVIATIONS OF ADDITIVE FUNCTIONALS

2011 ◽  
Vol 11 (01) ◽  
pp. 157-181 ◽  
Author(s):  
KANEHARU TSUCHIDA
Keyword(s):  
Brownian Motion ◽  
Large Deviations ◽  
Markov Processes ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Stable Processes ◽  
Sufficient Condition ◽  
Additive Functionals ◽  
Deviation Principle ◽  
Symmetric Markov Processes

We prove the large deviation principle for continuous additive functionals under certain assumptions. The underlying symmetric Markov processes include Brownian motion and symmetric and relativistic α-stable processes.

scholarly journals Large deviations for unbounded additive functionals of a Markov process with discrete time (noncompact case)

1994 ◽  
Vol 7 (3) ◽  
pp. 423-436 ◽  
Author(s):  
O. V. Gulinskii ◽  
Robert S. Lipster ◽  
S. V. Lototskii
Keyword(s):  
Markov Process ◽  
Large Deviations ◽  
Discrete Time ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Occupation Measure ◽  
Additive Functionals ◽  
Deviation Principle ◽  
Exponential Tightness

We combine the Donsker and Varadhan large deviation principle (l.d.p) for the occupation measure of a Markov process with certain results of Deuschel and Stroock, to obtain the l.d.p. for unbounded functionals. Our approach relies on the concept of exponential tightness and on the Puhalskii theorem. Three illustrative examples are considered.

scholarly journals LARGE DEVIATIONS FOR FLOWS OF INTERACTING BROWNIAN MOTIONS

2010 ◽  
Vol 10 (03) ◽  
pp. 315-339 ◽  
Author(s):  
A. A. DOROGOVTSEV ◽  
O. V. OSTAPENKO
Keyword(s):  
Brownian Motion ◽  
Large Deviations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Stochastic Flows ◽  
Brownian Motions ◽  
Deviation Principle ◽  
And Brownian Motion

We establish the large deviation principle (LDP) for stochastic flows of interacting Brownian motions. In particular, we consider smoothly correlated flows, coalescing flows and Brownian motion stopped at a hitting moment.

scholarly journals Large Deviations for Stochastic Differential Equations on Sd Associated with the Critical Sobolev Brownian Vector Fields

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Qinghua Wang
Keyword(s):  
Differential Equations ◽  
Large Deviations ◽  
Stochastic Differential Equations ◽  
Vector Fields ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Deviation Principle

We obtain a large deviation principle for the stochastic differential equations on the sphere Sd associated with the critical Sobolev Brownian vector fields.

LARGE DEVIATIONS FOR SMALL PERTURBATIONS OF SDES WITH NON-MARKOVIAN COEFFICIENTS AND THEIR APPLICATIONS

2006 ◽  
Vol 06 (04) ◽  
pp. 487-520 ◽  
Author(s):  
FUQING GAO ◽  
JICHENG LIU
Keyword(s):  
Large Deviations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Wiener Space ◽  
Small Perturbations ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
Sobolev Norms

We prove large deviation principles for solutions of small perturbations of SDEs in Hölder norms and Sobolev norms, where the SDEs have non-Markovian coefficients. As an application, we obtain a large deviation principle for solutions of anticipating SDEs in terms of (r, p) capacities on the Wiener space.

scholarly journals Large deviations for stochastic integrodifferential equations of the Itô type with multiple randomness

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 473-487 ◽  
Author(s):  
A. Haseena ◽  
M. Suvinthra ◽  
N. Annapoorani
Keyword(s):  
Weak Convergence ◽  
Large Deviations ◽  
Finite Number ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Integrodifferential Equations ◽  
Multiplicative Noises ◽  
Deviation Principle ◽  
Laplace Principle ◽  
Weak Convergence Approach

A Freidlin-Wentzell type large deviation principle is derived for a class of It? type stochastic integrodifferential equations driven by a finite number of multiplicative noises of the Gaussian type. The weak convergence approach is used here to prove the Laplace principle, equivalently large deviation principle.

scholarly journals Higher order asymptotics for large deviations — Part II

2020 ◽  
pp. 2150025
Author(s):  
Kasun Fernando ◽  
Pratima Hebbar
Keyword(s):  
Large Deviations ◽  
Asymptotic Expansions ◽  
Continuous Time ◽  
Compact Manifold ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Hörmander Condition ◽  
Additive Functionals ◽  
Higher Order Asymptotics ◽  
Weakly Dependent

We obtain asymptotic expansions for the large deviation principle (LDP) for continuous time stochastic processes with weakly-dependent increments. As a key example, we show that additive functionals of solutions of stochastic differential equations (SDEs) satisfying Hörmander condition on a [Formula: see text]-dimensional compact manifold admit these asymptotic expansions of all orders.

Large deviations for stochastic differential equations with general delayed generator

2020 ◽  
Vol 28 (3) ◽  
pp. 197-207
Author(s):  
Clément Manga ◽  
Auguste Aman
Keyword(s):  
Differential Equations ◽  
Large Deviations ◽  
Stochastic Differential Equations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Differential Delay ◽  
Deviation Principle ◽  
Nonlinear Anal ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations

AbstractThis paper is devoted to derive a Freidlin–Wentzell type of the large deviation principle for stochastic differential equations with general delayed generator. We improve the result of Chi Mo and Jiaowan Luo [C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal. 80 2013, 202–210].

scholarly journals Weak specification properties and large deviations for non-additive potentials

2013 ◽  
Vol 35 (3) ◽  
pp. 968-993 ◽  
Author(s):  
PAULO VARANDAS ◽  
YUN ZHAO
Keyword(s):  
Dynamical Systems ◽  
Large Deviations ◽  
Lyapunov Exponents ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Deviation Principle ◽  
Hyperbolic Dynamical Systems

AbstractWe obtain large deviation bounds for the measure of deviation sets associated with asymptotically additive and sub-additive potentials under some weak specification properties. In particular, a large deviation principle is obtained in the case of uniformly hyperbolic dynamical systems. Some applications to the study of the convergence of Lyapunov exponents are given.

scholarly journals Sharp large deviations for some hyperbolic systems

2013 ◽  
Vol 35 (1) ◽  
pp. 249-273 ◽  
Author(s):  
VESSELIN PETKOV ◽  
LUCHEZAR STOYANOV
Keyword(s):  
Large Deviations ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Hyperbolic Systems ◽  
Axiom A ◽  
Deviation Principle ◽  
Basic Set ◽  
Sharp Large Deviations ◽  
Additional Regularity ◽  
Sharp Large Deviation

AbstractWe prove a sharp large deviation principle concerning intervals shrinking with sub-exponential speed for certain models involving the Poincaré map related to a Markov family for an Axiom A flow restricted to a basic set $\Lambda $ satisfying some additional regularity assumptions.

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