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Monotonicity and non-monotonicity of domains of stochastic integral operators

Sato K.

Probability and Mathematical Statistics

2006

Vol. 26, Fasc. 1

23--39

A Lévy process on Rd with distribution }μ at time 1 is denoted by Xμ = {Xμt}, If the improper stochastic integral [formula] of f with respect to Xμ is definable, its distribution is denoted by Ф∫(μ). The class of all infinitely divisible distributions μ on Rd such that Ф∫(μ) is definable is denoted by D(Φ∫). The class D(Φ∫), its two extensions Dc(Φ∫) and Des(Φ∫) (compensated and essential), and its restriction D0(Φ∫)(absolutely definable) are studied. It is shown that Des(Φ∫) is monotonic with respect to ∫, which means that |f2| ≤ |f1| implies Des(Φ∫1) ⊂ Des(Φ∫2). Further D0Φf is monotonic with respect to ∫ but neither D(Φ∫) nor D0(Φ∫)is monotonic with respect to ∫. Furthermore, there exist μ, ∫1 and ∫2 such that 0 ≤∫2 ∫1, and μ∈D(Φ∫1), and μ ∉D(Φ∫2) An explicit example for this is related to some properties of a class of martingale Levy processes.