I know that for a stochastic process $\{X_n\}_{n\geq 0}$ to be a martingale, we need it to be:
- $E[X_n]\leq +\infty$
- Adapted to a filtration $\{\mathcal{F}_n\}_{n\geq 0}$
- If $n\leq m$ then $E[X_m\mid F_n]=X_n$
I have been given a stochastic process $\{X_n\}_{n\geq 0}$ with independent increments such that $E[X_n]=0$ for all $n$. The two first points are evident if we consider $\{\mathcal{F}_n\}_{n\geq 0}$ the natural filtration. However, I'm having trouble with the third point. The interpretation of the last point should be that the expected value of the next observations, given the last observations, should be the value of the observation that is immediately before. Given this interpretation, it's obvious that if the expected value is $0$, meaning we don't expect the outcome to change from our initial condition, it should be obvious that $\{X_n\}_{n\geq 0}$ is a martingale. However, I am failing at how to translate this to more mathematical terms. Any hints would be greatly appreciated. I need to show it's a martingale in respect to the natural filtration