Description of limits of ranges of iterations of stochastic integral mappings of infinitely divisible distributions

2010/12/01 by Sato, Ken-iti
#60E07 #60G51 #60H05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1012.0093

Abstract

For infinitely divisible distributions ρ on ℝ^d the stochastic integral mapping Φ_fρ is defined as the distribution of improper stochastic integral ∫₀^∞- f(s) dX_s^(ρ), where f(s) is a non-random function and \X_s^(ρ)\ is a Lévy process on ℝ^d with distribution ρ at time 1. For three families of functions f with parameters, the limits of the nested sequences of the ranges of the iterations Φ_fⁿ are shown to be some subclasses, with explicit description, of the class L_∞ of completely selfdecomposable distributions. In the critical case of parameter 1, the notion of weak mean 0 plays an important role. Examples of f with different limits of the ranges of Φ_fⁿ are also given.

Related

• Inversions of infinitely divisible distributions and conjugates of stochastic integral mappings
• Classes of infinitely divisible distributions on R^d related to the class of selfdecomposable distributions
• Monotonicity and non-monotonicity of domains of stochastic integral operators
• Transformations of infinitely divisible distributions via improper stochastic integrals
• Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes