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On Paszkiewicz-type criterion for a.e. continuity of processes in L p -spaces

Jakub Olejnik (2010)

Banach Center Publications

In this paper we consider processes Xₜ with values in L p , p ≥ 1 on subsets T of a unit cube in ℝⁿ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing f: ℝ₊ → ℝ₊ ||Xₜ-Xₛ||ₚ ≤ f(||t-s||), s,t ∈ T. We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set T. This criterion turns out to be necessary for a wide class of functions f. We use a geometrical Paszkiewicz-type...

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier (2013)

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

We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order α with drift and diffusion coefficients b , σ . When α ∈ ( 1 , 2 ) , we investigate pathwise uniqueness for this equation. When α ∈ ( 0 , 1 ) , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether α ∈ ( 0 , 1 ) or α ∈ ( 1 , 2 ) and on whether the driving stable process is symmetric or not. Our assumptions...

On Poisson-Dirichlet problems with polynomial data

Henryk Gzyl (2002)

Publicacions Matemàtiques

In this note we provide a probabilistic proof that Poisson and/or Dirichlet problems in an ellipsoid in Rd, that have polynomial data, also have polynomial solutions. Our proofs use basic stochastic calculus. The existing proofs are based on famous lemma by E. Fisher which we do not use, and present a simple martingale proof of it as well.

On preservation under univariate weighted distributions

Salman Izadkhah, Mohammad Amini, Gholam Reza Mohtashami Borzadaran (2015)

Applications of Mathematics

We derive some new results for preservation of various stochastic orders and aging classes under weighted distributions. The corresponding reversed preservation properties as straightforward conclusions of the obtained results for the direct preservation properties, are developed. Damage model of Rao, residual lifetime distribution, proportional hazards and proportional reversed hazards models are discussed as special weighted distributions to try some of our results.

On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, φ | ( - ∞ , 0 ] = 0 , φ | [ 1 , ∞ ) = 1 , = φ: ℝ → ℝ|φ is continuous, φ | ( - ∞ , 0 ] = 0 , φ | [ 1 , ∞ ) = 1 . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection...

Currently displaying 441 – 460 of 1208