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On Adjoint Additive Processes

Evans Kristian P., Jacob Niels

Probability and Mathematical Statistics

2020

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.

Denseness of certain smooth Lévy functionals in D1;2

Geiss C., Laukkarinen E.

Probability and Mathematical Statistics

2011

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:

On Lamperti stable processes

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

Probability and Mathematical Statistics

2010

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.

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

Aman A., Owo J.-M.

Probability and Mathematical Statistics

2010

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.

Rosiński measures for tempered stable and related Ornstein-Uhlenbeck processes

Terdik G., Woyczyński W. A.

Probability and Mathematical Statistics

2006

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 harmonic measure for Lévy processes

Sztonyk P.

Probability and Mathematical Statistics

2000

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.