Show independence of stochastic integral and stochastic process

Question

Let $ M_t $ and $ N_t$ be two continuous local martingales with respect to a filtration $ \mathcal{F}_t $. Suppose that $ M_t $ and $ N_t$ are independent and set $X_t = \int_0^t M_s^4 \mathrm{d} M_s $ for all $t \geq0$.

Now I have to show that $X_t$ is independent of $N_t$. Can I get some help?

Answer 1

Hints

1. Recall that the independence of two stochastic processes $(M_t)_{t \geq 0}$ and $(N_t)_{t \geq 0}$ is equivalent to the independence of the corresponding canonical $\sigma$-algebras $\mathcal{F}_{\infty}^M$ and $\mathcal{F}_{\infty}^N$, $$\mathcal{F}_{\infty}^M := \sigma(M_s; s \geq 0).$$
2. Let $f$ be an $\mathcal{F}_t^M$-adapted process such that $$X_t := \int_0^t f(s) \, dM_s$$ is well-defined. Then $X_t$ is $\mathcal{F}_t^M$-adapted. In particular, $\mathcal{F}_t^X \subseteq \mathcal{F}_t^M$. Conclude from the first step that $(X_t)_{t \geq 0}$ and $(N_t)_{t \geq 0}$ are independent.