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Contents of PMS, Vol. 26, Fasc. 1,
pages 23 - 39
 

MONOTONICITY AND NON-MONOTONICITY OF DOMAINS OF STOCHASTIC INTEGRAL OPERATORS

Ken-Iti Sato

Abstract: A Lévy process on Rd with distribution m at time 1 is denoted by (m) (m) X = (Xt ). If the improper stochastic integral integral oo - (m) 0 f (s)dXs of f with respect to (m) X is definable, its distribution is denoted by Pf(m). The class of all infinitely divisible distributions m on d R such that Pf (m) is definable is denoted by D(Pf ). The class D(Pf ), its two extensions Dc(Pf ) and Des(Pf) (compensated and essential), and its restriction 0 D (Pf ) (absolutely definable) are studied. It is shown that Des(Pf) is monotonic with respect to f, which means that |f2|< |f1| implies Des(Pf1) < D(Pf2). Further, 0 D (Pf) is monotonic with respect to f but neither D(Pf ) nor Dc(Pf) is monotonic with respect to f. Furthermore, there exist m , f1 and f2 such that 0 < f2 < f1, m (- D(Pf1), and m / (- D(Pf2). An explicit example for this is related to some properties of a class of martingale Lévy processes.

2000 AMS Mathematics Subject Classification: 60E07, 60G51, 60H05.

Key words and phrases: Improper stochastic integral, infinitely divisible distribution, Lévy process, martingale Lévy process, monotonic.

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