I am asking this is that I am trying to show that the quadratic variation of a continuous semi martingale is just the quadratic variation of its local martingale part. That is if $X_t = X_0 + M_t+A_t$ is a continuous semi martingale, where $M$ is a local martingale and $A$ is a finite variation process, then $\langle X,X \rangle = \langle M,M \rangle$.
For clarity
I define the total variation of a function $A(t)$ (usually the path of a process) on the interval $[0,T]$ as $V(a)_{T} = \sup_{\pi} \sum_{i=0}^{n_{\pi}-1}|A_{t_{i+1}}-A_{t_i}|$ where the supremum is over partitions $\pi = \{0 = t_0 <t_1 < \cdots < t_{n_{\pi}}=T\}$. We say that a function $A$ is of finite variation of $V(A)_T < \infty$ for all $T \geq 0$.
I define the quadratic variation of a continuous local martingale as $\langle M, M \rangle_T = \lim_{n \rightarrow \infty} \sum_{i=1}^{n(\pi_n)}(M_{t_i^n}- M_{t^n_{i-1}})^2$ where the limit is in probability. This is for a sequence of partitions $\pi_n = \{0=t^n < \cdots < t^n_{n(\pi)}\}$ with $||\pi|| = \sup_{1 \leq i \leq n(\pi_n)}(t^n_i - t^n_{i-1}) \rightarrow 0$ as $n \rightarrow \infty$.
Now this is what I have for the proof so far:
$$ \sum_{i=1}^{n(\pi_n)}(X_{t_i}-X_{t_{i-1}})^2 = \sum_{i=1}^{n(\pi_n)}(M_{t_i}-M_{t_{i-1}})^2+ \sum_{i=1}^{n(\pi_n)}(A_{t_i}-A_{t_{i-1}})^2 + 2\sum_{i=1}^{n(\pi_n)}(M_{t_i}-M_{t_{i-1}})(A_{t_i} - A_{t_{i-1}}) \\= \langle M,M\rangle_t+ \sum_{i=1}^{n(\pi_n)}(A_{t_i}-A_{t_{i-1}})^2 + 2\sum_{i=1}^{n(\pi_n)}(M_{t_i}-M_{t_{i-1}})(A_{t_i} - A_{t_{i-1}}) $$
So now it remains to show that 1) $\sum_{i=1}^{n(\pi_n)}(A_{t_i}-A_{t_{i-1}})^2 \rightarrow 0$, or in other words that the quadratic variation of a finite variation process is zero. And 2) $\sum_{i=1}^{n(\pi_n)}(M_{t_i}-M_{t_{i-1}})(A_{t_i} - A_{t_{i-1}}) \rightarrow 0$.
For 1), intuitively, if something is of finite variation, then the sum of its increments converges to something finite. But then if we take the squares of the increments, this is much smaller than the increments themselves and so will go to zero. This is my attempt:
$$ \sum^{n(\pi_n)}_{i=1} (A_{t_i} - A_{t_{i-1}})^2 \leq \sup_{1 \leq i \leq n(\pi_n)}|A_{t_i} - A_{t_{i-1}}| \sum^{n(\pi_n)}_{i=1}(A_{t_i}-A_{t_{i-1}}) = ||\pi_n|| V(A)_t \rightarrow 0 $$
since the mesh size goes to zero and the total variation is finite.
For 2) this is my attempt
$$ \sum_{i=1}^{n(\pi_n)}(M_{t_i}-M_{t_{i-1}})(A_{t_i} - A_{t_{i-1}}) \leq \sup_{1 \leq i \leq n(\pi_n)} |M_{t_i} - M_{t_{i-1}}| V(A)_t. $$
But I am not sure how to finish this.
Thanks
