I'm working on my first ever proof that a stochastic process is a martingale, and I'm a bit confused. Is there a "standard machine" for these proofs?
To be more specific, I am trying to show that if $Y$ is a random variable with $E(Y) < \infty$, the process $X_n = E(Y|I_n)$ is a martingale (where $I_n$ is an arbitrary filtration).
The book I'm using lists the following as conditions for $X_n$ to be a martingale:
- $X_n$ is adapted to $I_n$
- $E(X_n)$ is finite
- $E(X_n|I_m) = X_m$ for $m < n$.
I can do condition 2 easily enough, by substituting $X_n = E(Y|I_n)$ and using the tower property.
I think I can do 3 similarly, by
$$ E(X_n|I_m) = E(E(Y|I_n) | I_m) \stackrel{?}{=} E(Y|I_m) = X_m. $$
Can I have some confirmation that the $\stackrel{?}{=}$ holds? And how do I prove that a process and filtration are adapted?
Here's a somewhat expanded answer, since this is a fairly straightforward question if you've covered the relevant definitions.
Let $m < n$, then we have that $\mathbb{E}(X_n|I_m) = \mathbb{E}(\mathbb{E}(Y|I_n)|I_m)$. But from the definition of a filtration, we know that $I_m \subseteq I_n$, so the law of iterated expectations applies, and it follows that $ \mathbb{E}(\mathbb{E}(Y|I_n)|I_m) = \mathbb{E}(Y|I_m) = X_m$, which proves 3.
You already know that $ \mathbb{E}(X_n) = \mathbb{E}(\mathbb{E}(Y|I_n)) = \mathbb{E}(Y) < \infty $; as I remarked, this is a special case of the law of iterated expectations.
It remains to show that the stochastic process $(X_n)_n$ is adapted to the filtration $(I_n)_n$. That means we want to show that for each $n$, $X_n = \mathbb{E}(Y|I_n)$ is an $I_n$-measurable random variable. Depending on what your definition of the conditional expectation is, the fact that $\mathbb{E}(Y|I_n)$ is measurable with respect to $I_n$ was either part of the definition itself, or should have been one of the first properties you proved once you defined it.