Given a filtered probability space $(\Omega,\, \mathcal{F},\,(\mathcal{F}_n)_{n \in \mathbb{N}},\, P)$, a martingale is defined to be a sequence of $L^1$ random variables $X_{n}: \Omega \to \mathbb{R}$ such that $$ E[X_n | \mathcal{F}_{n-1}] = X_{n-1}. $$ In many books, it is said that the motivation of this definition is to model the wealth of a gambler playing a fair game. For example, suppose $X_1, X_2, X_3, \ldots$ represent coin tosses which are independent and give 1 dollar for the player if heads and takes him 1 dollar if tails. The gambler's wealth after $n$ coin tosses is then given by the radom variable $S_n = X_1 + \ldots + X_n$. The sequence $S_n$ is a martingale, considered the filtration given by $\mathcal{F}_n = \sigma(X_1, \ldots, X_n)$, for $$ E[S_{n}|\mathcal{F}_{n-1}] = E[X_1 + \ldots + X_n | \mathcal{F}_{n-1}] = X_1 + \ldots + X_{n-1} + E[X_n|\mathcal{F}_{n-1}] = X_1 + \ldots + X_{n-1} + E[X_n] = X_1 + \ldots + X_{n-1} = S_{n-1}, $$ where we used the fact that $X_n$ is independent from $\mathcal{F}_{n-1}$ and has expectation zero. We see that $S_n$ is a martingale, as we would guess, once one cannot hope to make more money by tossing a coin just because one knows the previous results of other coin tosses. Intuitively, this definition and example makes sense, but when we try to analize it rigorously, we have some issues:
- Why are the $S_n$ random variables? I mean, what space is the domain of the funcion?
- Well, one could naturally think that $S_n$ is either defined in $\Omega = \{H, T\}$ or in $\Omega^{n}$.
- Intuitivelly, it makes no sense for $S_n$ to be defined in $\Omega$, for then its only possible values would be $n$ ou $-n$.
- Mathematically, it makes no sense for $S_n$ to be defined in $\Omega^n$, for each $S_n$ would have a different domain, each of them different from the domain of the $X_n$, and then we would not be able to talk about the filtration, condintional expectations, etc.
The question therefore is: how to find a probability space $(\Omega, \mathcal{F}, P)$ where we can define the sequence of coin tosses $X_n$, the sequence of the gambler's wealth $S_n$, consider the filtration generated by $X_n$ and prove that, besides modeling nicely the gamble situation, $S_n$ is indeed a martingale, accordingly to our previous definition?