UNIVERSITY
OF WROC£AW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
43.1 42.2 42.1 41.2 41.1 40.2 40.1
39.2 39.1 38.2 38.1 37.2 37.1 36.2
36.1 35.2 35.1 34.2 34.1 33.2 33.1
32.2 32.1 31.2 31.1 30.2 30.1 29.2
29.1 28.2 28.1 27.2 27.1 26.2 26.1
25.2 25.1 24.2 24.1 23.2 23.1 22.2
22.1 21.2 21.1 20.2 20.1 19.2 19.1
18.2 18.1 17.2 17.1 16.2 16.1 15
14.2 14.1 13.2 13.1 12.2 12.1 11.2
11.1 10.2 10.1 9.2 9.1 8 7.2
7.1 6.2 6.1 5.2 5.1 4.2 4.1
3.2 3.1 2.2 2.1 1.2 1.1
 
 
WROC£AW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 34, Fasc. 2,
pages 293 - 316
 

PERSISTENCE OF SOME ITERATED PROCESSES

Christoph Baumgarten

Abstract: We study the asymptotic behaviour of the probability that a stochastic process (Z )
   tt≥0  does not exceed a constant barrier up to time T (a so-called persistence probability) when Z is the composition of two independent processes (X )
   tt∈I  and (Y )
  tt≥0  . To be precise, we consider (Z )
  tt≥0  defined by Z = X  ∘|Y |
 t        t if I = [0,∞ ) and Z = X  ∘Y
 t       t  if I = ℝ  .

For continuous self-similar processes (Y )
  tt≥0  , the rate of decay of persistence probability for Z can be inferred directly from the persistence probability of X and the index of self-similarity of Y . As a corollary, we infer that the persistence probability for iterated Brownian motion decays asymptotically like T-1∕2  .

If Y is discontinuous, the range of Y possibly contains gaps, which complicates the estimation of the persistence probability. We determine the polynomial rate of decay for X being a Lévy process (possibly two-sided if I = ℝ  ) or a fractional Brownian motion and Y being a Lévy process or random walk under suitable moment conditions.

2000 AMS Mathematics Subject Classification: Primary: 60G99; Secondary: 60G18, 60G50, 60G51, 60J65.

Keywords and phrases: Iterated process, one-sided barrier problem, one-sided exit problem, persistence, persistence probability, small deviation probability, survival probability.

Download:    Abstract    Full text   Abstract + References