On stochastic integral representation of stable processes with sample paths in Banach spaces (original) (raw)

Continuity of symmetric stable processes

Journal of Multivariate Analysis, 1989

The path continuity of a symmetric p-stable process is examined in terms of any stochastic integral representation for the process. When 0 < p < 1, we give necessary and suflicient conditions for path continuity in terms of any (every) representation. When 1 &p<2, we extend the known sutliciency condition in terms of metric entropy and offer a conjecture for the stable version of the Dudley-Fernique theorem. Finally, necessary and sufficient conditions for path continuity are given in terms of continuity at a point for 0 < p < 2.

Dimension results for sample paths of operator stable Lévy processes

Stochastic Processes and their Applications, 2005

Let X ¼ fX ðtÞ; t 2 R þ g be an operator stable Le´vy process in R d with exponent B, where B is an invertible linear operator on R d : We determine the Hausdorff dimension and the packing dimension of the range X ð½0; 1Þ in terms of the real parts of the eigenvalues of B. r

On sample path properties of semistable processes

Journal of Multivariate Analysis, 1987

This paper contains three main results: In the first result a correspondence principle between semistable measures on L,, 1 <p < co, and Banach space valued semistable processes is established. In the second result it is shown that the paths of a Banach space valued semistable process belong to L, with probability zero or one, and necessary and sufficient conditions for the two alternatives to hold are given. In the third result necessary and sufficient conditions are given for almost sure path absolute continuity for certain Banach space valued semistable processes.

Absolute continuity and singularity of probability measures induced by a purely discontinuous Girsanov transform of a stable process

Transactions of the American Mathematical Society

In this paper we study mutual absolute continuity and singularity of probability measures on the path space which are induced by an isotropic stable Lévy process and the purely discontinuous Girsanov transform of this process. We also look at the problem of finiteness of the relative entropy of these measures. An important tool in the paper is the question under which circumstances the a.s. finiteness of an additive functional at infinity implies the finiteness of its expected value.

Stable Lévy Motion with Values in the Skorokhod Space: Construction and Approximation

Journal of Theoretical Probability, 2019

In this article, we introduce an infinite-dimensional analogue of the α-stable Lévy motion, defined as a Lévy process Z = {Z(t)} t≥0 with values in the space D of càdlàg functions on [0, 1], equipped with Skorokhod's J 1 topology. For each t ≥ 0, Z(t) is an α-stable process with sample paths in D, denoted by {Z(t, s)} s∈[0,1]. Intuitively, Z(t, s) gives the value of the process Z at time t and location s in space. This process is closely related to the concept of regular variation for random elements in D introduced in [9] and [13]. We give a construction of Z based on a Poisson random measure, and we show that Z has a modification whose sample paths are càdlàg functions on [0, ∞) with values in D. Finally, we prove a functional limit theorem which identifies the distribution of this modification as the limit of the partial sum sequence {S n (t) = [nt] i=1 X i } t≥0 , suitably normalized and centered, associated to a sequence (X i) i≥1 of i.i.d. regularly varying elements in D.

The Stability of some stochastic processes

2010

We formulate and prove a new criterion for stability of e-processes. It says that any e-process which is averagely bounded and concentrating is asymptotically stable. In the second part, we show how this general result applies to some shell models (the Goy and the Sabra model). Indeed, we manage to prove that the processes corresponding to these models satisfy the