Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process such that for every $(\omega,r) \in \Omega \times \mathbb{R}_+,y \to X_r^y(\omega)$ is continuous and for every $y \in \mathbb{R}^d,(X^y_r)_{r \in \mathbb{R}_+}$ is $L^p$-martingale relative to $\left(\sigma(X_u^y),u \in [0,r]\right)_{r \in \mathbb{R}_+}.$ Suppose that for every $r \in \mathbb{R}_+,\int_{\mathbb{R}^d}|f(y)|E[|X_r^y|]dy<\infty,\mathcal{G}_r:=\sigma\left(\bigcup_{k \in \mathbb{N}}\bigcup_{\Vert y \Vert \leq k}\sigma(X^y_u,u \in [0,r])\right),$ where $\Vert \cdot \Vert$ is the Euclidean norm.
Prove that for every $r \in \mathbb{R}_+,(\omega,y) \to X_r^y(\omega)$ is $\left(\mathcal{G}_r\otimes \mathcal{B}(\mathbb{R}^d),\mathcal{B}(\mathbb{R})\right)$-measurable.
Prove that $Y:=\left(\int_{\mathbb{R}^d}f(y)X_r^ydy\right)_{r \in \mathbb{R}_+}$ is a martingale relative to $(\mathcal{G}_r)_{r \in \mathbb{R}_+}.$ Compute the quadratic variation $[Y].$
Attempt:
Let $Y_r:=\int_{\mathbb{R}^d}f(y)X_r^ydy$
This will follow by considering dyadic approximation of each component of $y.$
For every $(r,\omega) \in \mathbb{R}_+ \times \Omega,Y_r(\omega)$ is well defined because $f \in C^{\infty}_c(\mathbb{R}^d,\mathbb{R})$ and $y \to X^y_r(\omega)$ is continuous.
for every $r \geq 0,Y_r \in L^1(\Omega)$ since $\int_{\mathbb{R}^d}|f(y)|E[|X_r^y|]dy<\infty.$
Since for every $r \geq 0,Y_r=\int_{\mathbb{R}^d}\max(f(y)X_r^y,0)dy-\int_{\mathbb{R}^d}\max(-f(y)X_r^y,0)dy,$ The measurability of $Y_r$ follows by part 1. and Fubini-Tonelli theorem for positive functions.
Let $0\leq r \leq u.\mathcal{G}_r$ is generated by a $\pi$-system. For $k \in \mathbb{N},\Vert y \Vert \leq k,O \in \sigma(X^y_u,u \in [0,r]),$ we have $E\left[1_{O}\int_{\mathbb{R}^d}f(y)X_u^ydy\right]=\int_{\mathbb{R}^d}f(y)E[1_{O}X_r^y]dy=E\left[1_O\int_{\mathbb{R}^d}f(y)X_r^ydy\right].$
Is the above attempt correct? How to compute the quadratic variation? Is it $\int_{\mathbb{R}^d}|f(y)|^2[X^y]_rdy?$ Why?