Subordinator

About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.

Reflected stable subordinators for fractional Cauchy problems

Abstract: In a fractional Cauchy problem, the first time derivative is replaced by a Caputo fractional derivative of order less than one If the original Cauchy problem governs a Markov process, a non-Markovian time change yields a stochastic solution to the fractional Cauchy problem, using the first passage time of a stable subordinator This paper proves that a spectrally negative stable process reflected at its infimum has the same one dimensional distributions as the inverse stable subordinator Therefore, this Markov process can also be used as a time change, to produce stochastic solutions to fractional Cauchy problems The proof uses an extension of the D Andr\'e reflection principle The forward equation of the reflected stable process is established, including the appropriate fractional boundary condition, and its transition densities are explicitly computed

Extreme statistics of superdiffusive Levy flights and every other Levy subordinate Brownian motion

Abstract: The search for hidden targets is a fundamental problem in many areas of science, engineering, and other fields. Studies of search processes often adopt a probabilistic framework, in which a searcher randomly explores a spatial domain for a randomly located target. There has been significant interest and controversy regarding optimal search strategies, especially for superdiffusive processes. The optimal search strategy is typically defined as the strategy that minimizes the time it takes a given single searcher to find a target, which is called a first hitting time (FHT). However, many systems involve multiple searchers and the important timescale is the time it takes the fastest searcher to find a target, which is called an extreme FHT. In this paper, we study extreme FHTs for any stochastic process that is a random time change of Brownian motion by a Levy subordinator. This class of stochastic processes includes superdiffusive Levy flights in any space dimension, which are processes described by a Fokker-Planck equation with a fractional Laplacian. We find the short-time distribution of a single FHT for any Levy subordinate Brownian motion and use this to find the full distribution and moments of extreme FHTs as the number of searchers grows. We illustrate these rigorous results in several examples and numerical simulations.

Distribution of suprema for generalized risk processes

Abstract: We study a generalized risk process $X(t)=Y(t)-C(t)$, $t\in[0,\tau]$, where $Y$ is a Levy process, $C$ an independent subordinator and $\tau$ an independent exponential time. Dropping the standard assumptions on the finite expectations of the processes $Y$ and $C$ and the net profit condition, we derive a Pollaczek-Khinchine type formula for the supremum of the dual process $\widehat{X}=-X$ on $[0,\tau]$ which generalizes the results obtained in \cite{HPSV1}. We also discuss which assumptions are necessary for deriving this formula, specially from the point of view of the ladder process.

Self-Similarity in the Wide Sense for Information Flows With a Random Load Free on Distribution

Abstract: For description of dynamics of changes random loads of information flows we examine the stochastic model of Double Stochastic Poisson process which manages points of changes the random loads. A special case of a discrete distribution for the random intensity provides the following covariance property to the corresponding Double Stochastic Poisson subordinator for a sequence of the random loads. Such covariance exactly coincides with the covariance of the fractional Ornstein-Uhlenbeck process. Applying the Lamperti transform we obtain a self-similar random process with continuous time, stationary in the wide sense increments, and one dimensional distributions scaling the distribution of a term of the the initial subordinated sequence of the random loads. The Central Limit Theorem for vectors allows us to obtain in a limit, in the sense of convergence of finite dimensional distributions, the fractional Gaussian Brownian motion and the fractional Ornstein- Uhlenbeck process.