From Williams' Probability with Martingales
- Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed to be constant.
- How do we know $|M_n^T| \le c + K$?
This is what I tried:
$$M_n^T = M_{T \wedge n} = M_0 + \sum_{k=1}^{T \wedge n} (M_k - M_{k-1})$$
$$\to |M_n^T| \le |M_0| + \sum_{k=1}^{T \wedge n} |M_k - M_{k-1}|$$
$$\le |M_0| + \sum_{k=1}^{T \wedge n} |X_k|$$
$$\le c + \sum_{k=1}^{T \wedge n} |X_k|$$
$$\le c + \sum_{k=1}^{T \wedge n} K$$
$$\le c + (T \wedge n - 1 + 1)(K)$$
$$ = c + (T \wedge n)(K)$$
I'm stuck. How can I approach this?