Monotonicity and non-monotonicity of domains of stochastic integral operators

2006/07/12 by Sato, Ken-iti
#60E07 #60G51 #60H05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.math/0607288

Abstract

A Lévy process on R^d with distribution μ at time 1 is denoted by X^(μ)=\X_t^(μ)\. If the improper stochastic integral ∫₀^∞- f(s)dX_s^(μ) of f with respect to X^(μ) is definable, its distribution is denoted by Φ_f(μ). The class of all infinitely divisible distributions μ on R^d such that Φ_f(μ) is definable is denoted by D(Φ_f). The class D(Φ_f), its two extensions D_c(Φ_f) and D_e(Φ_f) (compensated and essential), and its restriction D⁰(Φ_f) (absolutely definable) are studied. It is shown that D_e(Φ_f) is monotonic with respect to f, which means that |f₂|≤ |f₁| implies D_e(Φ_f1)⊂ D_e(Φ_f2). Further, D⁰(Φ_f) is monotonic with respect to f but neither D(Φ_f) nor D_c(Φ_f) is monotonic with respect to f. Furthermore, there exist μ, f₁, and f₂ such that 0≤ f₂≤ f₁, μ∈ D(Φ_f1), and μ\not∈ D(Φ_f2). An explicit example for this is related to some properties of a class of martingale Lévy processes.

Related

• Some properties of exponential integrals of Lévy processes and examples
• Description of limits of ranges of iterations of stochastic integral mappings of infinitely divisible distributions
• Classes of infinitely divisible distributions on R^d related to the class of selfdecomposable distributions
• New Classes of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class
• Complete monotonicity of time-changed Lévy processes at first passage