Girsanov's theorem corollary

Question

Trying to understand the proof of the corollary on the page

http://en.wikipedia.org/wiki/Girsanov_theorem

It remains for me the show the equality of the quadratic variations

$[W, X]_t = 2[[W, X], W]_t$

where $W_t$ is a Wiener process and $X_t$ is a continuous process.

I do not see how to prove this. I have tried the polarization identity $2[X, Y]_t = [X + Y]_t - [X]_t - [Y]_t$ to no avail.

Answer 1

Recall that

$$[X,Y]_t = \lim_{|\Pi| \to 0} \sum_{t_j \in \Pi} (X_{t_{j+1}}-X_{t_j})(Y_{t_{j+1}}-Y_{t_j}); \tag{1}$$

here $\Pi = \{0=t_0 < \ldots < t_n=t\}$ denotes a partition of $[0,t]$ with mesh size $$|\Pi| := \max_j |t_j-t_{j-1}|.$$

Lemma Let $(X_t)_{t \geq 0}$ be a stochastic process with continuous sample paths and $(Y_t)_{t \geq 0}$ be a stochastic process with sample paths of bounded variation. Then $[X,Y]_t = 0$ for all $t \geq 0$.

Proof: Since $(Y_t)_{t \geq 0}$ has sample paths of bounded variation, we have $$\begin{align*} \left|\sum_{t_j \in \Pi} (X_{t_{j+1}}-X_{t_j})(Y_{t_{j+1}}-Y_{t_j})\right|&\leq \sup_{|s-r| \leq |\Pi|; s,r \in [0,t]} |X_s-X_r| \cdot \sum_{t_j \in \Pi} |Y_{t_{j+1}}-Y_{t_j}| \\ &\leq \sup_{|s-r| \leq |\Pi|; s,r \in [0,t]} |X_s-X_r| \cdot \text{BV}(Y,[0,t]). \end{align*}$$ As $s \mapsto X_s(\omega)$ is uniformly continuous on $[0,t]$ for each fixed $\omega$, we get $[X,Y]_t = 0$ by letting $|\Pi| \to 0$.

Corollary Let $(X_t)_{t \geq 0}$ be a stochastic process with continuous sample paths. Then $[W,[X,W]]_t = [[X,W],[X,W]]_t = 0$ for all $t \geq 0$.

Proof: It follows from the polarization identity that $[X,W]_t$ is a stochastic process with sample paths of bounded variation. Since both $(W_t)_{t \geq 0}$ and $([X,W]_t)_{t \geq 0}$ have continuous sample paths, it follows from the previous lemma that $$[[X,W],[X,W]]_t = [W,[X,W]]_t=0.$$