Small deviations of general Lévy processes

Аурзада, Франк; Dereich, Steffen

doi:10.1214/09-aop457

Cited by 18 publications

(31 citation statements)

References 19 publications

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“…The main motivation for this paper originates from the recent work Aurzada and Dereich [2], where a framework for obtaining the small deviation rate (1.2) for general Lévy processes (but fixed t) is provided. The difficulty in passing over from the small deviation estimate to the respective LIL concerns circumventing the independence assumption of the Borel-Cantelli lemma.…”

Section: Introductionmentioning

confidence: 99%

Small time Chung-type LIL for Lévy processes

Аурзада¹,

Doering²,

Savov³

2013

Bernoulli

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Section: Introductionmentioning

confidence: 99%

Small time Chung-type LIL for Lévy processes

Аурзада¹,

Doering²,

Savov³

2013

Bernoulli

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“…We will need the following system of SDEs which is nothing but the Galerkin approximation of (1): (35) where the sequence { k ; k ∈ N} is defined by (2)…”

Section: Proof Of Theorem 37: Exponential Mixing By Coupling Methodsmentioning

confidence: 99%

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

Fernando

Hausenblas

Razafimandimby

2016

Commun. Math. Phys.

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Abstract:In this paper, we derive several results related to the long-time behavior of a class of stochastic semilinear evolution equations in a separable Hilbert space H:Here A is a positive self-adjoint operator and B is a bilinear map, and the driving noise L is basically a D(A −1/2 )-valued Lévy process satisfying several technical assumptions. By using a density transformation theorem type for Lévy measure, we first prove a support theorem and an irreducibility property of the Ornstein-Uhlenbeck processes associated to the nonlinear stochastic problem. Second, by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain γUsing a coupling argument, the exponential ergodicity is also proved under the stronger assumption that B is continuous on H × H. While the latter condition is only satisfied by the nonlinearities of GOY and Sabra shell models, the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2D Navier-Stokes, Magnetohydrodynamics equations, the 3D Leray-α model, the GOY and Sabra shell models.

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“…However, one can get this asymptotic behaviour for Gaussian processes (see, e.g., [12] and [11]) or real-valued Lévy processes (see [2]). There is a large amount of literature on small ball probabilities in the Gaussian setting and one can consult the survey article [12].…”

Section: Small Ball Problem and Chung's Law Of Iterated Logarithmmentioning

confidence: 99%

On certain integral functionals of squared Bessel processes

Çetın

2015

Stochastics

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Abstract. For a squared Bessel process, X, the Laplace transforms of joint laws of (U, Ry 0 X p s ds) are studied where Ry is the first hitting time of y by X and U is a random variable measurable with respect to the history of X until Ry. A subset of these results are then used to solve the associated small ball problems for Ry 0 X p s ds and determine a Chung's law of iterated logarithm.Ry 0 X p s ds is also considered as a purely discontinuous increasing Markov process and its infinitesimal generator is found. The findings are then used to price a class of exotic derivatives on interest rates and determine the asymptotics for the prices of some put options that are only slightly in-the-money. . IntroductionLet X be a squared Bessel process which is the unique strong solution towhere ν ≥ −1 is a real constant and B is a standard Brownian motion. Letting δ = 2(ν + 1), X is called a δ-dimensional squared Bessel process. We will denote such a process with X 0 = z by BESQ δ (z) and δ and ν will be related by δ = 2(ν + 1) throughout the text. In this paper we are interested in the integral functional where p > −1 and R y := inf{t ≥ 0 : X t = y} for y ∈ [0, ∞) and X is BESQ δ (z). (In the sequel, we will write R δ y only if we need to specify the dimension to avoid ambiguity.) Squared Bessel processes have found wide applications especially in Finance Theory, see Chapter 6 in [8] for a recent account. They can, e.g., be used to model interest rates in a Cox-Ingersoll-Ross framework. In the above setting, if X p models the spot interest rates, then exp Σ δ p,z,y refers to the cumulative interest until the spot rate hits the barrier y p . As such, this random variable is related to certain exotic options on interest rates (see [5] for some formulae regarding barrier options in a similar framework). Bessel processes also appear often in the study of financial bubbles since 1/ √ X is the prime example of a continuous (strict) local martingale when X is a BESQ 3 (see, e.g., [14], [17] and [18] for how strict local martingales, and in particular Bessel processes, appear in mathematical studies of bubbles).In Section 2 we will determine the joint law of (U, Σ δ p,z,y ) by martingale methods, where U is a random variable measurable with respect to the evolution of X until R y . In particular we will obtain the joint distributions of (R y , Σ δ p,z,y ) and (max t≤Ry X t , Σ δ p,z,y ). As a by-product of our findings, if |ν| p+1 = 1 2 , we have a remarkable characterisation of the conditional law of Σ δ p,z,y given that the maximum (resp. minimum) of X at R y is below (resp. above) a fixed level in terms of the first hitting time distributions of a 3-dimensional Bessel process when z ≥ y (resp. z ≤ y).Date: March 9, 2015.

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Small deviations of general Lévy processes

Cited by 18 publications

References 19 publications

Small time Chung-type LIL for Lévy processes

Small time Chung-type LIL for Lévy processes

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

On certain integral functionals of squared Bessel processes

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