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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).

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3 Answers 3

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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).

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  • $\begingroup$ Thank you very much! This is very helpful! $\endgroup$ Apr 19, 2023 at 2:35
  • $\begingroup$ I couldn't read the post last night. Now that I have, it's comprehensive as usual. +1. $\endgroup$ Apr 19, 2023 at 19:52
  • $\begingroup$ If someone is interested in the historical aspect and how the name emanated (as touched here in this post), one can have a look at Jørgensen's book - there is a detailed section on the same. $\endgroup$ Apr 20, 2023 at 4:43
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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 $$\sigma(X_1, \ldots, X_n)=\sigma(2X_1, \ldots, 2X_n)$$

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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.


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


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. $6.1, ~7.1.$

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    $\begingroup$ I have always found appeals to information to be conceptually circular in this context. First one has to develop through experience a sense in which a filtration could be thought of as some kind of "information" (whatever that word really means). If there's more to it than that, then I would very much appreciate seeing how one could make sense of this explanation for a spatial stochastic process. Surely "information" has a similar meaning for variables with spatial support, but where's the filtration?? $\endgroup$
    – whuber
    Apr 18, 2023 at 18:22
  • $\begingroup$ @User1865345 I see. Thanks! This is very helpful. I did have some incomplete understanding of the conditional expectation. $\endgroup$ Apr 19, 2023 at 2:37
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    $\begingroup$ @whuber I don't really understand that objection. The typical filtration is the one generated by an observation process in time. In a spatial setting, there is an analogue, but it is indexed by a partial order, rather than a total order $(\mathcal{F}_U \subseteq \mathcal{F}_V$ if $U \subseteq V$). This appears in the formulation of Gibbsian statistical mechanical models and the (essentially equivalent) domain Markov property. Some of the nice properties of having totally ordered sets are lost but many are not. $\endgroup$ Apr 20, 2023 at 13:21
  • $\begingroup$ @User1865345 +1 for quoting Billingsley. I highly recommend anyone who is interested in understand the role of $\mathscr{F}$ in $(\Omega, \mathscr{F}, P)$ reading that section. It is very enlightening. $\endgroup$
    – Zhanxiong
    Apr 21, 2023 at 3:51

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