an:04180444
Zbl 0716.60057
Imkeller, Peter
A class of two-parameter stochastic integrators
EN
Stochastics Stochastics Rep. 27, No. 3, 167-188 (1989).
00172268
1989
j
60H05 60G48 60G44 60E15
quadratic variation; mixed variation; It??'s formula; stochastic integrator; two-parameter martingale; multi-parameter semimartingales
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 It??'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 It?? 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.
Reviewer (Berlin)