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A note on one-dimensional stochastic equations

Hans-Jürgen Engelbert (2001)

Czechoslovak Mathematical Journal

We consider the stochastic equation X t = x 0 + ∫ 0 t b ( u , X u ) d B u , t ≥ 0 , where B is a one-dimensional Brownian motion, x 0 ∈ ℝ is the initial value, and b [ 0 , ∞ ) × ℝ → ℝ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients b , beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on b ensuring the existence as well as the uniqueness in law of the solution.

A note on prediction for discrete time series

Gusztáv Morvai, Benjamin Weiss (2012)

Kybernetika

Let { X n } be a stationary and ergodic time series taking values from a finite or countably infinite set 𝒳 and that f ( X ) is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times λ n along which we will be able to estimate the conditional expectation E ( f ( X λ n + 1 ) | X 0 , ⋯ , X λ n ) from the observations ( X 0 , ⋯ , X λ n ) in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series...

A note on the characterization ofsome minification processes

Wiesław Dziubdziela (1997)

Applicationes Mathematicae

We present a stochastic model which yields a stationary Markov process whose invariant distribution is maximum stable with respect to the geometrically distributed sample size. In particular, we obtain the autoregressive Pareto processes and the autoregressive logistic processes introduced earlier by Yeh et al

A note on the existence of Gibbs marked point processes with applications in stochastic geometry

Martina Petráková (2023)

Kybernetika

This paper generalizes a recent existence result for infinite-volume marked Gibbs point processes. We try to use the existence theorem for two models from stochastic geometry. First, we show the existence of Gibbs facet processes in ℝ d with repulsive interactions. We also prove that the finite-volume Gibbs facet processes with attractive interactions need not exist. Afterwards, we study Gibbs-Laguerre tessellations of ℝ 2 . The mentioned existence result cannot be used, since one of its assumptions...

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