scholarly journals Large deviations for the Ornstein-Uhlenbeck process with shift

2015 ◽  
Vol 47 (03) ◽  
pp. 880-901 ◽  
Author(s):  
Bernard Bercu ◽  
Adrien Richou
Keyword(s):  
Maximum Likelihood ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Maximum Likelihood Estimates ◽  
Shift Parameter ◽  
Uhlenbeck Process ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
Ornstein Uhlenbeck Process

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical nonsteepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the estimator previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

scholarly journals Large deviations for the Ornstein-Uhlenbeck process with shift

2015 ◽  
Vol 47 (3) ◽  
pp. 880-901 ◽  
Author(s):  
Bernard Bercu ◽  
Adrien Richou
Keyword(s):  
Maximum Likelihood ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Maximum Likelihood Estimates ◽  
Shift Parameter ◽  
Uhlenbeck Process ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
Ornstein Uhlenbeck Process

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical nonsteepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the estimator previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

scholarly journals The Large Deviations of Estimating Rate Functions

2005 ◽  
Vol 42 (01) ◽  
pp. 267-274 ◽  
Author(s):  
Ken Duffy ◽  
Anthony P. Metcalfe
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Length Distribution ◽  
Rate Function ◽  
Partial Sums ◽  
Single Server ◽  
Mixing Condition ◽  
Deviation Principle ◽  
Queue Length Distribution ◽  
Empirical Estimates

Given a sequence of bounded random variables that satisfies a well-known mixing condition, it is shown that empirical estimates of the rate function for the partial sums process satisfy the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue length distribution at a single-server queue with infinite waiting space is proved.

scholarly journals The Large Deviation Principle for the On-Off Weibull Sojourn Process

2008 ◽  
Vol 45 (01) ◽  
pp. 107-117 ◽  
Author(s):  
Ken R. Duffy ◽  
Artem Sapozhnikov
Keyword(s):  
Renewal Process ◽  
Time Scale ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Compact Set ◽  
Natural Processes ◽  
Deviation Principle ◽  
Rate Functions ◽  
Nonlinear Scale

This article proves that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale. Unusually, its rate function is not convex. Apart from on a compact set, the rate function is infinite, which enables us to construct natural processes that satisfy the large deviation principle with nontrivial rate functions on more than one time scale.

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 THE MATHER MEASURE AND A LARGE DEVIATION PRINCIPLE FOR THE ENTROPY PENALIZED METHOD

2011 ◽  
Vol 13 (02) ◽  
pp. 235-268 ◽  
Author(s):  
D. A. GOMES ◽  
A. O. LOPES ◽  
J. MOHR
Keyword(s):  
Jacobi Equation ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Deviation Function ◽  
Deviation Principle ◽  
Probability Densities ◽  
Aubry Set ◽  
Dynamical Properties ◽  
Hamilton Jacobi Equation

We present the rate function and a large deviation principle for the entropy penalized Mather problem when the Lagrangian is generic (it is known that in this case the Mather measure μ is unique and the support of μ is the Aubry set). We assume the Lagrangian L(x, v), with x in the torus 𝕋N and v∈ℝN, satisfies certain natural hypotheses, such as superlinearity and convexity in v, as well as some technical estimates. Consider, for each value of ϵ and h, the entropy penalized Mather problem [Formula: see text] where the entropy S is given by [Formula: see text], and the minimization is performed over the space of probability densities μ(x, v) on 𝕋N×ℝN that satisfy the discrete holonomy constraint ∫𝕋N×ℝN φ(x + hv) - φ(x) dμ = 0. It is known [17] that there exists a unique minimizing measure μϵ, h which converges to a Mather measure μ, as ϵ, h→0. In the case in which the Mather measure μ is unique we prove a Large Deviation Principle for the limit lim ϵ, h→0ϵ ln μϵ, h(A), where A ⊂ 𝕋N×ℝN. In particular, we prove that the deviation function I can be written as [Formula: see text], where ϕ0 is the unique viscosity solution of the Hamilton – Jacobi equation, [Formula: see text]. We also prove a large deviation principle for the limit ϵ→ 0 with fixed h. Finally, in the last section, we study some dynamical properties of the discrete time Aubry–Mather problem, and present a proof of the existence of a separating subaction.

Large deviations in estimation of an Ornstein-Uhlenbeck model

1999 ◽  
Vol 36 (01) ◽  
pp. 60-77 ◽  
Author(s):  
Danielle Florens-Landais ◽  
Huyên Pham
Keyword(s):  
Large Deviation ◽  
Score Function ◽  
Rate Function ◽  
Estimating Functions ◽  
Uhlenbeck Process ◽  
Large Deviation Principles ◽  
Dependent Change ◽  
Parameter Dependent ◽  
Explicit Rate ◽  
Drift Parameter

A large deviation principle (LDP) with an explicit rate function is proved for the estimation of drift parameter of the Ornstein-Uhlenbeck process. We establish an LDP for two estimating functions, one of them being the score function. The first one is derived by applying the Gärtner–Ellis theorem. But this theorem is not suitable for the LDP on the score function and we circumvent this key point by using a parameter-dependent change of measure. We then state large deviation principles for the maximum likelihood estimator and another consistent drift estimator.

scholarly journals The Large Deviation Principle for the On-Off Weibull Sojourn Process

2008 ◽  
Vol 45 (1) ◽  
pp. 107-117 ◽  
Author(s):  
Ken R. Duffy ◽  
Artem Sapozhnikov
Keyword(s):  
Renewal Process ◽  
Time Scale ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Rate Function ◽  
Compact Set ◽  
Natural Processes ◽  
Deviation Principle ◽  
Rate Functions ◽  
Nonlinear Scale

This article proves that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale. Unusually, its rate function is not convex. Apart from on a compact set, the rate function is infinite, which enables us to construct natural processes that satisfy the large deviation principle with nontrivial rate functions on more than one time scale.

scholarly journals The Large Deviations of Estimating Rate Functions

2005 ◽  
Vol 42 (1) ◽  
pp. 267-274 ◽  
Author(s):  
Ken Duffy ◽  
Anthony P. Metcalfe
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Length Distribution ◽  
Rate Function ◽  
Partial Sums ◽  
Single Server ◽  
Mixing Condition ◽  
Deviation Principle ◽  
Queue Length Distribution ◽  
Empirical Estimates

Given a sequence of bounded random variables that satisfies a well-known mixing condition, it is shown that empirical estimates of the rate function for the partial sums process satisfy the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue length distribution at a single-server queue with infinite waiting space is proved.

scholarly journals Large deviation principles of one-dimensional maps for Hölder continuous potentials

2014 ◽  
Vol 36 (1) ◽  
pp. 127-141 ◽  
Author(s):  
HUAIBIN LI
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Weak Form ◽  
Periodic Points ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
One Dimensional Maps ◽  
Level 2

We show some level-2 large deviation principles for real and complex one-dimensional maps satisfying a weak form of hyperbolicity. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages.

THE LARGE DEVIATION PRINCIPLE FOR MEASURES WITH RANDOM WEIGHTS

1993 ◽  
Vol 05 (04) ◽  
pp. 659-692 ◽  
Author(s):  
R. S. ELLIS ◽  
J. GOUGH ◽  
J. V. PULÉ
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Spherical Model ◽  
Fermi Gas ◽  
Rate Function ◽  
Bose Statistics ◽  
Deviation Principle ◽  
Random Weights ◽  
Abstract Setting ◽  
Special Case

In this paper, we study the problem of large deviations for measures with random weights. We are motivated by previous work dealing with the special case occuring in the statistical mechanics of the Bose gas. We study the problem in an abstract setting, isolating what is general from what is dependent on Bose statistics. We succeed in proving the large deviation principle for a large class of measures with random weights and obtaining the corresponding rate function in an explicit form. In particular, our results are applicable to the Fermi gas and the spherical model.

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