2 questions on cross-variation

Question

Let $M$ and $N$ be continuous local martingales and $X=\exp(N_t - \frac{\langle N_t\rangle}{2})$.

1. Why does it hold that: $$d\langle M,X \rangle=X d\langle M,N\rangle$$
2. What's the easiest way to see that is $M$ is a local martingale and $V$ a bounded variation process $$\langle M,V \rangle=0$$ I tried the latter using the identity $$\langle M,V \rangle=\frac{1}{4}(\langle M+V \rangle-\langle M-V \rangle)$$ but something doesn't quite add up

Answer 1

1. Hint: By Ito's lemma $dX_t=X_t\,dN_t\,.$

2. $\langle M,V\rangle_t=\lim_{n\to\infty}\sum_{i=1}^n(M_{t_{i+1}}-M_{t_i})(V_{t_{i+1}}-V_{t_i})\,.$ Therefore, by the continuity of $M$ and the bounded variation of $V$,

\begin{align} |\langle M,V\rangle_t|&\le\lim_{n\to\infty}\sum_{i=1}^n|M_{t_{i+1}}-M_{t_i}|\,|V_{t_{i+1}}-V_{t_i}|\\ &\le\underbrace{\lim_{n\to\infty}\max_n|M_{t_{i+1}}-M_{t_i}|}_{0}\cdot\lim_{n\to\infty}\sum_{i=1}^n|V_{t_{i+1}}-V_{t_i}|=0\,. \end{align}