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A class of two-parameter stochastic integrators. (English) Zbl 0716.60057
Summary: Let M be a continuous square integrable two-parameter martingale. Then the quadratic i-variations \([M]^ i\) appear as integrators of terms of the second differential order in Itô’s formula, whereas terms of the third differential order are described by mixed variations \(N^ i\) which behave like \([M]^ i\) in parameter direction i and like M in the complementary direction. We prove that both \([M]^ i\) and \(N^ i\), \(i=1,2\), are stochastic integrators the integrals of which are defined on some vector space of 1- resp. 2-previsible processes. On one hand, this result shows that non-continuous previsible processes are integrable and is therefore basic for an Itô formula for non-continuous two-parameter martingales. On the other hand, the way it is derived may give a hint what multi-parameter semimartingales (martingale-like processes) are.

MSC:
60H05 Stochastic integrals
60G48 Generalizations of martingales
60G44 Martingales with continuous parameter
60E15 Inequalities; stochastic orderings
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