What's the relationship between these two definitions of martingales?

Question

On wikipedia, the definition of a martingale is given as follows:

A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) $X_1, X_2, X_3,\dots$ that satisfies for any time $n$,

$$\mathbb{E}(|X_n|) < \infty$$ $$\mathbb{E}(X_{n+1}\mid X_1\dots,X_n) = X_n.$$ That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation.

while in Rick Durrett's textbook Probability: Theory and Examples, the martingale is defined as follows:

In this section we will define martingales and their cousins supermartingales and submartingales, and take the first steps in developing their theory. Let $\mathcal{F}_n$ be a filtration, i.e. an increasing sequence of $\sigma$-fields. A sequence $X_n$ is said to be adapted to $\mathcal{F}_n$ if $X_n\in\mathcal{F}_n$ for all $n$. If $X_n$ is a sequence with

1. $\mathbb{E}|X_n| < \infty$
2. $X_n$ is adapted to $\mathcal{F}_n$,
3. $\mathbb{E}(X_{n+1}|\mathcal{F}_n) = X_n$ for all $n$,

then $X$ is said to be a martingale (with respect to $\mathcal{F}_n$). If in the last definition, $=$ is replaced with $\leq$ or $\geq$, then $X$ is said to be a supermartingale or submartingale, respectively.

The wikipedia definition seems quite intuitive and reasonable to me, that is, a martingale is just a process in which the expected value tomorrow given past history is always equal to today's value (with a special example being a random walk process). My questions are:

Why is it necessary to use sigma-fields instead of just past history in the definition? Do we gain anything by using the more sophisticated definition? I simply could not think of a situation in which this more sophisticated definition is needed.

I'm particularly interested in these questions as these two definitions seems non-nested, i.e., the Durrett definition does not contain the wikipedia definition as a special case, as the set of past history $\{X_1,...,X_n\}$ is not a sigma-field, it needs to include at least the empty set too.

It would be great if you could illustrate with a simple example (such as a random walk or other processes).

Answer 1

Durrett's definition is the general correct definition of a martingale, while the Wikipedia's definition is at best a "restricted definition". The qualifier "with respect to $\mathcal{F}_n$", although was placed in the parentheses, is essential to accurately define a "martingale": technically speaking, martingale is a sequence of pairs $(X_n, \mathcal{F}_n)$, not of $\{X_n\}$ alone (but the italicized sentence in the quoted paragraph preceding equation $(2)$ below may to some extent justify the latter convention). Patrick Billingsley's book Probability and Measure made this important point more explicit (Section 35):

The sequence $\{\color{red}{(X_n, \mathscr{F}_n)}: n = 1, 2, \ldots\}$ is a martingale if these four conditions hold:

1. $\mathscr{F}_n \subset \mathscr{F}_{n + 1}$;
2. $X_n$ is measurable $\mathscr{F}_n$;
3. $E[|X_n|] < \infty$;
4. with probability 1, \begin{align} E[X_{n + 1}|\mathscr{F}_n] = X_n.\tag{1} \end{align}

He continued to explain the role of $\{\mathscr{F}_n\}$ in the definition as follows:

Alternatively, the sequence $X_1, X_2, \ldots$ is said to be a martingale relative to the $\sigma$-fields $\mathscr{F}_1, \mathscr{F}_2, \ldots$. Condition 1 is expressed by saying the $\mathscr{F}_n$ form a filtration and condition 2 by saying $X_n$ are adapted to the filtration.

After that, he illustrated the relationship between Wikipedia's "definition" (which actually is just a special martingale) and the general definition above, which probably can clear out your confusion :

The sequence $X_1, X_2, \ldots$ is defined to be a martingale if it is a martingale relative to some sequence $\mathscr{F}_1, \mathscr{F}_2, \ldots$. In this case, the $\sigma$-fields $\mathscr{G}_n = \sigma(X_1, \ldots, X_n)$ always work: Obviously, $\mathscr{G}_n \subset \mathscr{G}_{n + 1}$ and $X_n$ is measurable $\mathscr{G}_n$, and if $(1)$ holds, then $E[X_{n + 1}|\mathscr{G}_n] = E[E[X_{n + 1}|\mathscr{F}_n]|\mathscr{G}_n] = E[X_n|\mathscr{G}_n] = X_n$ (by tower property of conditional expectation). For these special $\sigma$-fields $\mathscr{G}_n$, $(1)$ reduces to \begin{align} E[X_{n + 1} | X_1, \ldots, X_n] = X_n. \tag{2} \end{align} Since $\sigma(X_1, \ldots, X_n) \subset \mathscr{F}_n$ if and only if $X_n$ is measurable $\mathscr{F}_n$ for each $n$, the $\sigma(X_1, \ldots, X_n)$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale.

Some additional comments in response to specific questions you raised:

1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ is measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, it is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $\{(X_n, \mathscr{F}_n)\}$ is a martingale, then $\{(X_n, \mathscr{G}_n)\}$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ is measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". In other words, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself -- it may cover any information up to time $n$ as long as Condition 4 holds.

You asked for an example, Billingsley also provided many good ones (e.g., Example 35.1 mentioned above), in which "gambling/betting system" is quite illuminating (note that, it is said that the word "martingale" may be originated from gambling, see this interesting vignette for details):

(p. 458) If $X$ represents the fortune of a gambler after the $n$th play and $\mathscr{F}_n$ represents his information about the game at that time, $(1)$ says that his expected fortune after the next play is the same as his present fortune. Thus a martingale represents a fair game, and sums of independent random variables with mean $0$ give one example.

(p. 463) Consider again the gambler whose fortune after the $n$th play is $X_n$ and whose information about the game at that time is represented by the $\sigma$-field $\mathscr{F}_n$. If $\mathscr{F}_n = \sigma(X_1, \ldots, X_n)$, he knows the sequence of his fortunes and nothing else, but $\mathscr{F}_n$ could be larger.

The last sentence "but $\mathscr{F}_n$ could be larger." tells you that, with an example, that Durrett's definition is the correct one and the Wikipedia's definition is clearly not generalized enough. While he did not gave a specific example of a "larger $\mathscr{F}_n$", you can easily conceive some scenarios (e.g., the gambler happens to be an employee of a casino, so in addition to the basic information he should know, he also knows the secret mechanism of the roulette wheel).

Answer 2

I agree with the other two answers here, but just wanted to highlight one reason why the $\sigma$-algebra based definition in Durrett, though more sophisticated, is beneficial. This definition emphasises that the conditional expectation $E[X_{n + 1} | X_1, \ldots, X_n]$ only depends on $X_1, \ldots, X_n$ through the $\sigma$-algebra $\mathscr{F}_n=\sigma(X_1, \ldots, X_n)$ that they generate, not on the random variables themselves. So, for example, $$E[X_{n + 1} | X_1, \ldots, X_n]=E[X_{n + 1} | 2 X_1, \ldots, 2X_n]$$ because $$\sigma(X_1, \ldots, X_n)=\sigma(2X_1, \ldots, 2X_n)$$

Answer 3

What does the notation $\mathbb E[X\mid X_1,\cdots,X_n]$ mean?

Go to the basics: consider $X\in \mathcal L_1(\Omega, \boldsymbol{\mathfrak B}, \mathbb P)$ and let $\mathcal G\subset\boldsymbol{\mathfrak B}$ defined as the sigma algebra generated by a list of random variables on the same space, i.e. $\mathcal G:=\sigma(X_i, ~i\in\mathcal I). $ Then the conditional expectation of $X$ w.r.t. $\mathcal G$ is a $\mathcal G$ observable random variable $\mathbb E[X\mid\mathcal G]:=\mathbb E[X\mid X_i, ~i\in\mathcal I].$

(See the German Wikipedia on the same for the usage of the notation).

More interesting thing is how (sub)sigma algebra comes in play: sigma algebra represents information. What is$^\dagger$ information? It is the ability of deciding if an event $G$ belonging to a certain paving $\mathcal G \subset\boldsymbol{\mathfrak B}$ has been realized or not. But if we are aware of whether $G$ happens or not, then we can also comment on whether $G^\complement$ occurs or not. In the same vein, if we have the ability to know whether $G_i,~i\in\{1,2,\ldots, n\}$ occurs or not, clearly we can tell about the realization of $\cup_{i=1}^n G_i.$ This indicates the paving $\mathcal G$ that is bearing the information must be a sigma algebra.

Now we would like to add a dynamic aspect to this. That is, we would update our information at discrete units of time. Let $\mathcal G_n$ be the sigma-algebra interpreted as the "information available at time $n.$" If $T:=\{n\in\mathbb Z\mid \alpha\leq n\leq\beta\}$ is an integer interval, then a filter with time set $T$ is a list of sigma algebras $\langle \mathcal G_n\rangle_{n\in T}$ such that $$\forall m, n\in T, ~n\leq m, ~\mathcal G_n\subseteq \mathcal G_m\subseteq\boldsymbol{\mathfrak B};$$ the relation implies as time progresses, the information accumulates and increases.

Another relevant aspect needed to be highlighted is that if $\langle X_n, \mathcal G_n\rangle_{n\in T}$ is a martingale, then $\langle X_n, \sigma(X_i\mid \alpha\leq i\leq n)\rangle_{n\in T}$ is a martingale - this follows from the fact that $\sigma(X_i\mid \alpha\leq i\leq n)\subset \mathcal G_n$ and smoothing. We could have worked with $ \sigma(X_i\mid \alpha\leq i\leq n)$ instead of $\mathcal G_n$ then; however it is not worthless to have some "auxiliary information" all along.

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$^\dagger$ Billingsley explicitly pointed out information used, thus, is "informal, nonmathematical term".

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References:

$\rm [I]$ A Probability Path, Sidney Resnick, Birkhäuser, $1999, $ sec. $10.2, ~10.4.$

$\rm [II]$ Probability with a View Toward Statistics, Vol. $\rm I, $ J. Hoffman-Jørgensen, Springer Science$+$Business, $1994, $ sec. $6.1, ~7.1.$