Proving this process is a martingale

Question

Let $X_j, j \geq 1$ be $\mathcal{L}_{1}$ random variables and $\mathscr{F}_n = \sigma \left(X_j, 1 \leq j \leq n\right), n \geq 0$ be the natural filtration. Define the process $Z = \lbrace Z_n, n \geq 0\rbrace$ by $$Z_0:=0, \qquad Z_n := \sum_{j=0}^{n-1} \left(X_{j+1} - \mathbb{E}\left[X_{j+1}|\mathscr{F}_j\right]\right)$$

I want to show that $Z$ is a martingale.

I have the following: \begin{align} \mathbb{E} \left[Z_n | \mathscr{F}_{n-1}\right] &= \mathbb{E} \left[\sum_{j=0}^{n-1} \left(X_{j+1} - \mathbb{E}\left[X_{j+1}|\mathscr{F}_j\right]\right)\right]\\ &= \sum_{j=0}^{n-1} \mathbb{E} \left[X_{j+1} - \mathbb{E}\left(X_{j+1}|\mathscr{F}_j\right)\right|\mathscr{F}_{n-1}]\\ &= \sum_{j=0}^{n-1} \left(\mathbb{E} \left[X_{j+1}|\mathscr{F}_{n-1}\right] - \mathbb{E}\left[\mathbb{E}\left[X_{j+1}|\mathscr{F}_j\right]\right|\mathscr{F}_{n-1}]\right)\\ &= \sum_{j=0}^{n-2} X_{j+1} + \mathbb{E} \left[X_n\right] - \sum_{j=0}^{n-2} \mathbb{E}\left[X_{j+1}|\mathscr{F}_j\right] - \mathbb{E}\left[X_n\right]\\ &= Z_{n-1} \end{align} so that $Z$ is a martingale. Is this a sufficient proof? Clearly through the process I made use of properties of conditional expectations, and the fact that $\mathbb{E}\left[X_{j+1}|\mathscr{F}_j\right]$ is $\mathscr{F}_{n-1}$-measurable for $1 \leq j \leq n-1$ so that $$\mathbb{E}\left[\mathbb{E}\left[X_{j+1}|\mathscr{F}_j\right]\right|\mathscr{F}_{n-1}] = \mathbb{E}\left[X_{j+1}|\mathscr{F}_j\right]$$

Answer 1

It is almost sufficient. Here are some remarks.

• The conditional expectation with respect to $\mathcal F_{n-1}$ is missing in the right hand side of the first line.
• We have to say that each $Z_n$ is integrable.
• And also justify the fact that $Z_n$ is $\mathcal F_n$-measurable.