# Martingale Definition notation

I am reading Stochastic Calculus by Shreve and am a bit confused by the notation when he first introduces a Martingale with the definition: $$E_n(X_{n+1})=X_n$$ What I don't understand is why the $$X_n$$ is capitalized. I thought that when we refer to a specific value a random variable takes we would write $$x$$ as opposed to $$X$$. Doesn't the $$X_n$$ here refer to a known value at time $$n$$?

• Usually capital letters are used to refer to the random variables, and non-capitalized the possible specific values it can take. So, for example for a positive random variable $E[X] = \int_0^\infty x p(x)$ where $p(x)$ is the probability density of $X$. And $X_n$ indeed refers to a known unique value (today's value) of the random variable. $X_{n+1}$ refers to the future (unknown) values of the random variable.
– user34971
Oct 12, 2020 at 12:24

You've got to make clear for yourself what the notation here means. The operator $$\mathbb{E}_{n}$$ is an abbreviation for a conditional expectation, given some sigma algebra say $$\mathcal{F}_{n}$$ of a filtration $$\lbrace \mathcal{F}_{n}\rbrace_{n\ge 1}$$, i.e. $$$$\mathbb{E}_{n}[X]:=\mathbb{E}[X|\mathcal{F}_{n}].$$$$ And this guy is not deterministic but random! Namely it's defined to be the random variable which integrates against all $$\mathcal{F}_{n}$$-measurable random variables in the same way as $$X$$ does. Now $$\mathbb{E}_{n}[X_{n+1}]$$ being random, you should be less surprised about $$X_{n}$$ (which is random) being capitalized on the right hand side of your equation. It is an identity between two random variables (which btw therefore is only asked to be true $$\mathbb{P}$$-almost surely).