any guidance on how to prove the linearity of quadratic variation would be greatly appreciated.
Denote quadratic variation of a continuous stochastic process $X_t$ as $$\left[X,X\right]_t = p-\lim_{max_{i}|t_{i+1}-t_i| \to 0}\sum_{i=0}^{n-1}{|X_{t_{i+1}}-X_{t_i}|}^2$$ and given that the quadratic covariation of two continuous square integrable martingales $X_t$ and $Y_t$ is defined as \begin{align} \left[X,Y\right]_t &= \frac{1}{4}\bigl(\left[X+Y,X+Y\right]_t - \left[X-Y,X-Y\right]_t\bigr) \\ &=\frac{1}{2}\bigl(\left[X+Y,X+Y\right]_t-\left[X,X\right]_t - \left[Y,Y\right]_t\bigr) \end{align}
show that $$\left[aX+bY,Z\right]_t = a\left[X,Z\right]_t + b\left[Y,Z\right]_t$$
EDIT: I am mainly hesitant because of the modulus and square exponent. Perhaps, let's try to show a simplier example: $[X+Z,Z] = [X,Z]+[Z,Z]$ (I drop off the index)
LHS: $$[X+Z,Z]=\frac{1}{2}[X+2Z,X+2Z]-\frac{1}{2}[X+Z,X+Z]-\frac{1}{2}[Z,Z]$$
RHS: \begin{align} [X,Z]+[Z,Z] &= \frac{1}{2}([X+Z,X+Z]-[X,X]-[Z,Z]) + [Z,Z] \\ &= \frac{1}{2}[X+Z,X+Z] - \frac{1}{2}[X,X] + \frac{1}{2}[Z,Z] \end{align}
Should I be approaching this problem as such?
Thanks!
One approach would be to first show $$[f,g]_t = \lim_{\delta_n \to \,0}\sum_{i=0}^{n-1}(f_{t_{i+1}} - f_{t_{i}})(g_{t_{i+1}} - g_{t_{i}})$$ then we get$$[\alpha f+ \beta g,h]_t = \lim_{\delta_n \to \,0}\sum_{i=0}^{n-1}\left[(\alpha f + \beta g)_{t_{i+1}} - (\alpha f + \beta g)_{t_{i}})(h_{t_{i+1}} - h_{t_{i}})\right]$$ and $$\alpha[f,h]_t = \alpha\left[\lim_{\delta_n \to \,0}\sum_{i=0}^{n-1}(f_{t_{i+1}} - f_{t_{i}})(h_{t_{i+1}} - h_{t_{i}})\right]$$ $$\beta[g,h]_t = \beta\left[\lim_{\delta_n \to \,0}\sum_{i=0}^{n-1}(g_{t_{i+1}} - g_{t_{i}})(h_{t_{i+1}} - h_{t_{i}})\right]$$
Finally, we obtain the desired linearity of quadratic variation after a couple more lines of simplification.