If $(M(x))_x$ is a collection of martingales, there is a nondecreasing processes $a$ such that $[M(x),M(y)]$ is absolutely continuous wrt $a$

Question

Let

• $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
• $T>0$
• $I:=(0,T]$
• $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
• $(X,\mathcal X)$ be a measurable space
• $M:\Omega\times\overline I\times X\to\mathbb R$ such that $M(x)=M(\;\cdot\;,\;\cdot\;,x)$ is a continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$ and $$A(x,y):=[M(x),M(y)]\;\;\;\text{for }x,y\in X,$$ where $[M(x),M(y)]$ denotes the covariation of $M(x)$ and $M(y)$

How can we show that there is a $\mathcal F$-predictable nondecreasing process $a$ such that $A(x,y)$ is absolutely continuous with respect to $a$ almost surely for all $x,y\in X$?

I guess it's an application of the martingale respresentation theorem.

Answer 1

The paper "The multiplicity of an increasing family of σ-fields" of Davis and Varaiya [Ann. Probab. 2 (1974) 958–963] has what you want, at least if $L^2(\vee_t\mathcal F_t)$ is separable.