Topic
Subordinator
About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.
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TL;DR: In this article, it was shown that the density profile evolves as the solution of the random equation ∂t ρ = LW ρ, where LW is the generalized second-order differential operator d du d dW in which W is a double-sided α-stable subordinator.
Abstract: Considerasystemofparticlesperformingnearestneighborrandomwalks on the lattice Z under hard-core interaction. The rate for a jump over a given bond is direction-independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an α-stable law, 0 <α <1. This exclusion process models conduction in strongly disordered 1D media. We prove that, when varying over the disorder and for a suitable slowly varying function L, under the super-diffusive time scaling N 1+1/α L(N ), the density profile evolves as the solution of the random equation ∂t ρ = LW ρ, where LW is the generalized second-order differential operator d du d dW in which W is a double-sided α-stable subordinator. This resultfollowsfromaquenchedhydrodynamiclimitinthecasethatthei.i.d.jumprates are replaced by a suitable array {ξN ,x : x ∈ Z} having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.
30 citations
01 Jan 1997
TL;DR: In this paper, the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting point is analyzed.
Abstract: Results are obtained regarding the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting point. Various martingales are described in terms of the Levy measure of the Poisson point process of interval lengths on the local time scale. The martingales derived from the zero set of a one-dimensional diffusion are related to martingales studied by Azema and Rainer. Formulae are obtained which show how the distribution of interval lengths is affected when the underlying process is subjected to a Girsanov transoformation. In particular, results for the zero set of an Ornstein-Uhlenbeck process or a Cox-Ingersoll-Ross process are derived from results for a Brownian motion or recurrent Bessel process, when the zero set is the range of a stable subordinator.
29 citations
TL;DR: In this paper, the authors constructed a martingale which has the same marginals as the arithmetic average of geometric Brownian motion in the convex order, and showed that the average is increasing with respect to the number of vertices.
Abstract: We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion.This provides a short proof of the recent result due to P. Carr et al that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion is replaced by a stable subordinator.
29 citations
TL;DR: In this paper, the authors proposed an alternative approach based on the combination of the popular Brownian motion with drift and tempered stable subordinator, which can be described by a Fokker-Planck type equation.
Abstract: In the classical analysis many models used to real data description are based on the standard Brownian diffusion-type processes. However, some real data exhibit characteristic periods of constant values. In such cases the popular systems seem not to be applicable. Therefore we propose an alternative approach, based on the combination of the popular Brownian motion with drift (called also the arithmetic Brownian motion) and tempered stable subordinator. The probability density function of the proposed model can be described by a Fokker-Planck type equation and therefore it has many similar properties as the popular Brownian motion with drift. In this paper we propose the estimation procedure for the considered tempered stable subdiffusive arithmetic Brownian motion and calibrate the analyzed process to the real financial data.
29 citations
TL;DR: Meerschaert et al. as discussed by the authors used a connection between X ( E ( t ) ) and the stable subordinator of index β / α to derive information on the path behavior of X (E ( t ), where β = 1 − 1/α for some constant ρ > 0.
29 citations