A class of two-parameter stochastic integrators

Abstract

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 Nⁱ which behave like [M]ⁱ in parameter direction i and like M in the complementary direction. We prove that both [M]sup:i$esup and Nsup:i$esup,=l,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

Keywords:

• Two-parameter martingales
• quadratic variation
• mixed variation
• Itô's formula
• stochastic integrator
• semimartingale