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Denseness of certain smooth Lévy functionals in D1;2

100%

Geiss C. , Laukkarinen E.

Probability and Mathematical Statistics

2011

tom Vol. 31, Fasc. 1

1--15

The Malliavin derivative for a Lévy process (Xt) can be defined on the space D1;2 using a chaos expansion or in the case of a pure jump process also via an increment quotient operator. In this paper we define the Malliavin derivative operator D on the class S of smooth random variables f(Xt1 ; : : : ;Xtn); where f is a smooth function with compact support. We show that the closure of L2(P) ⊇ S D→ L2(m⊗P) yields to the space D1;2: As an application we conclude that Lipschitz functions operate on D1;2:

Rosiński measures for tempered stable and related Ornstein-Uhlenbeck processes

80%

Terdik G. , Woyczyński W. A.

Probability and Mathematical Statistics

2006

tom Vol. 26, Fasc. 2

213--243

Several concrete parametric classes of tempered stable distributions are discussed in terms of explicit calculations of their Rosiński measures. The hope is that they will provide a family of concrete models useful in applied areas and for which the fitting can be done by parametric methods. Related Ornstein-Uhlenbeck processes are analyzed. The emphasis throughout the paper is on obtaining exact analytic formulas.

On Lamperti stable processes

80%

Caballero M. E. , Pardo J. C. , Pérez J. L.

Probability and Mathematical Statistics

2010

tom Vol. 30, Fasc. 1

1--28

In this paper, we consider a new family of Rd-valued Lévy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitly. In the one-dimensional case we provide an explicit form for the characteristic exponent and other several useful properties of the class. This family of processes shares many tractable properties with the tempered stable and the layered stable processes, defined by Rosiński [33] and Houdré and Kawai [16], respectively. We also find a series representation which is used for sample path simulation, illustrated in the case d = 1. Finally, we provide many examples, some of which appear in recent literature.

On harmonic measure for Lévy processes

80%

Sztonyk P.

Probability and Mathematical Statistics

2000

tom Vol. 20, Fasc. 2

383--390

Let {Xt} be a Lévy process in Rd, d ≥ 2, with infinite Lévy measure. If {Xt} has no Gaussian component, then the process does not hit the boundary of Lipschitz domain S ⊂ Rd at the first exit time of S under mild conditions on {Xt}. The conditions are met, e.g., if {Xt} is rotation invariant.

Generalized backward doubly stochastic differential equations driven by Lévy processes with non-Lipschitz coefficients

70%

Aman A. , Owo J.-M.

Probability and Mathematical Statistics

2010

tom Vol. 30, Fasc. 2

259--272

We prove an existence and uniqueness result for generalized backward doubly stochastic differential equations driven by Lévy processes with non-Lipschitz assumptions.

On Adjoint Additive Processes

70%

Evans K. P. , Jacob N.

Probability and Mathematical Statistics

2020

tom Vol. 40, Fasc. 2

205--223

Starting with an additive process (Yt)t≥0, it is in certain cases possible to construct an adjoint process (Xt)t≥0 which is itself additive. Moreover, assuming that the transition densities of (Yt)t≥0 are controlled by a natural pair of metrics dψ;t and δψ;t, we can prove that the transition densities of (Xt)t≥0 are controlled by the metrics δ ψ,1/treplacing dψ,t and dψ,/treplacing δψ,t.

Path-wise versus kinetic modeling for equilibrating non-Langevin jump-type processes

60%

Żaba M. , Garbaczewski P. , Stephanovich V.

Open Physics

2014

tom 12

nr 3

175-184

We discuss two independent methods of solution of a master equation whose biased jump transition rates account for long jumps of Lévy-stable type and admit a Boltzmannian (thermal) equilibrium to arise in the large time asymptotics of a probability density function ρ(x, t). Our main goal is to demonstrate a compatibility of a direct solution method (an explicit, albeit numerically assisted, integration of the master equation) with an indirect pathwise procedure, recently proposed in [Physica A 392, 3485, (2013)] as a valid tool for a dynamical analysis of non-Langevin jump-type processes. The path-wise method heavily relies on an accumulation of large sample path data, that are generated by means of a properly tailored Gillespie’s algorithm. Their statistical analysis in turn allows to infer the dynamics of ρ(x, t). However, no consistency check has been completed so far to demonstrate that both methods are fully compatible and indeed provide a solution of the same dynamical problem. Presently we remove this gap, with a focus on potential deficiencies (various cutoffs, including those upon the jump size) of approximations involved in simulation routines and solutions protocols.