On parameter-measurability of the stochastic integral with respect to the two-parameter strong martingale (Q2882770)

The aim of this article is to establish the existence of a good version of a parametrized family of two-parameters martingales given by stochastic integrals. More precisely, the main result roughly states that, given a measurable family \(\mu(0,x,\omega)\) of \(L^2\)-martingales (where \(0\) sums some measurable space, \(x\in (\mathbb{R}_+)^2\), \(\omega\in\Omega\)) and a measurable family of predictable processes \(f\) that are locally square integrable with respect to \(\mu(0,x,\omega)\), there exists a measurable version of the family of stochastic integrals NEWLINE\[NEWLINE\int_{]0,x]} f(0,x',\omega)\,\mu(0,dx',\omega),NEWLINE\]NEWLINE the quadratic variation of which is given by the integrals of \(f^2\) with respect to the quadratic variation of \(\mu\).