Monotonicity and non-monotonicity of domains of stochastic integral operators

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A Lévy process on Rᵈ with distribution μ at time 1 is denoted by X⁽μ⁾={Xₜ⁽μ⁾}. If the improper stochastic integral ∫₀∞⁻ f(s)dXₛ⁽μ⁾ of f with respect to X⁽μ⁾ is definable, its distribution is denoted by Φf(μ). The class of all infinitely divisible distributions μ on Rᵈ such that Φf(μ) is definable is denoted by D(Φf). The class D(Φf), its two extensions Dc(Φf) and Dₑ(Φf) (compensated and essential), and its restriction D⁰(Φf) (absolutely definable) are studied. It is shown that Dₑ(Φf) is monotonic with respect to f, which means that |f₂|≤ |f₁| implies Dₑ(Φf₁)⊂ Dₑ(Φf₂). Further, D⁰(Φf) is monotonic with respect to f but neither D(Φf) nor Dc(Φf) is monotonic with respect to f. Furthermore, there exist μ, f₁, and f₂ such that 0≤ f₂≤ f₁, μ∈ D(Φf₁), and μ∈ D(Φf₂). An explicit example for this is related to some properties of a class of martingale Lévy processes.