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Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Philip S. Griffin, Ross A. Maller (2013)

Annales de l'I.H.P. Probabilités et statistiques

This paper is concerned with the small time behaviour of a Lévy process X . In particular, we investigate thestabilitiesof the times, T ¯ b ( r ) and T b * ( r ) , at which X , started with X 0 = 0 , first leaves the space-time regions { ( t , y ) 2 : y r t b , t 0 } (one-sided exit), or { ( t , y ) 2 : | y | r t b , t 0 } (two-sided exit), 0 b l t ; 1 , as r 0 . Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in L p . In many instances these are...

Smoothing and occupation measures of stochastic processes

Mario Wschebor (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

This is a review paper about some problems of statistical inference for one-parameter stochastic processes, mainly based upon the observation of a convolution of the path with a non-random kernel. Most of the results are known and presented without proofs. The tools are first and second order approximation theorems of the occupation measure of the path, by means of functionals defined on the smoothed paths. Various classes of stochastic processes are considered starting with the Wiener process,...

Spectral analysis of subordinate Brownian motions on the half-line

Mateusz Kwaśnicki (2011)

Studia Mathematica

We study one-dimensional Lévy processes with Lévy-Khintchine exponent ψ(ξ²), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy processes whose Lévy measure has completely monotone density on (0,∞). Examples include symmetric stable processes and relativistic processes. The main result is a formula for the generalized eigenfunctions of transition...

Stable random fields and geometry

Shigeo Takenaka (2010)

Banach Center Publications

Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(O) = 0. 1. X(a) - X(b) is subject to the Gaussian law of mean 0 and variance d(a,b). He gave an example for M = S m , the m-dimensional sphere. Let Y ( B ) ; B ( S m ) be the Gaussian random measure on S m , that is, 1. Y(B) is a centered Gaussian system, 2. the variance of Y(B) is equal of μ(B), where μ is the uniform measure on S m , 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for B i , i = 1,2,..., B i B j = , i ≠ j, we...

Stationary distributions for jump processes with memory

K. Burdzy, T. Kulczycki, R. L. Schilling (2012)

Annales de l'I.H.P. Probabilités et statistiques

We analyze a jump processes Z with a jump measure determined by a “memory” process S . The state space of ( Z , S ) is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of ( Z , S ) is the product of the uniform probability measure and a Gaussian distribution.

Stochastic flow for SDEs with jumps and irregular drift term

Enrico Priola (2015)

Banach Center Publications

We consider non-degenerate SDEs with a β-Hölder continuous and bounded drift term and driven by a Lévy noise L which is of α-stable type. If β > 1 - α/2 and α ∈ [1,2), we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise L. In our previous paper L was assumed to be non-degenerate, α-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes...

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