The Definition of Martingale
Definition
Let’s assume that a probability space $( \Omega , \mathcal{F} , P)$ is given.
- A sequence $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$ of sub-σ-fields of $\mathcal{F}$ is called a filtration if it satisfies the following: $$ \forall n \in \mathbb{N}, \mathcal{F}_{n} \subset \mathcal{F}_{n+1} $$
- Given a filtration $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$, a sequence $\left\{ X_{n} \right\}_{n \in \mathbb{N}}$ of ordered pairs formed by sequences of $\mathcal{F}_{n}$-measurable Lebesgue integrable random variables $X_{n}$ is called a martingale if it satisfies the following: $$ \forall n \in \mathbb{N}, E \left( X_{n+1} | \mathcal{F}_{n} \right) = X_{n} $$
- $\mathcal{F}_{n}$ being a sub-σ-field of $\mathcal{F}$ means that both are σ-fields of $\Omega$, but $\mathcal{F}_{n} \subset \mathcal{F}$ holds.
- $X_{n}$ being a $\mathcal{F}_{n}$-measurable function means that for all Borel sets $B \in \mathcal{B}(\mathbb{R})$, $X_{n}^{-1} (B) \in \mathcal{F}_{n}$ holds true.
Explanation
Submartingales and Supermartingales are referred to as follows, respectively. Remembering the inequality as sub if the right-hand side decreases, and super if it increases, may be less confusing. $$ \forall n \in \mathbb{N}, E \left( X_{n+1} | \mathcal{F}_{n} \right) \ge X_{n} \\ \forall n \in \mathbb{N}, E \left( X_{n+1} | \mathcal{F}_{n} \right) \le X_{n} $$ Of course, being both a submartingale and a supermartingale is equivalent to being a martingale. Therefore, if a theorem holds for either submartingales or supermartingales, it can also be directly applied to martingales.
Intuitively understanding martingales starts with thinking of the σ-field as a collection of events, “information”:
- Filtration: $\forall n \in \mathbb{N}, \mathcal{F}_{n} \subset \mathcal{F}_{n+1}$, meaning that having a larger σ-field implies having more information. In the definition of martingales, the process $X_{n}$ being $\mathcal{F}_{n}$-measurable means that as the actual data $x_{n}$ is observed, the σ-field $\mathcal{F}_{n}$ also expands and it is safe to assume that all information up to $n$ has been acquired.
- Martingales: $\forall n \in \mathbb{N}, E \left( X_{n+1} | \mathcal{F}_{n} \right) = X_{n}$ means assuming that knowing the information $\mathcal{F}_{n}$ up to $n$, the next scenario $X_{n+1}$ will also be similar to $X_{n}$. If the expected value of $X_{n+1}$ can be derived regardless of the previously gathered $\mathcal{F}_{n}$, such a stochastic process is no different from white noise and fails to be a subject for statistical analysis. Hence, the intuitive definition of a martingale can be seen as a “stochastic process where we can obtain mathematically and statistically better outcomes with some advantageous information.”
Origin
In a French village called ‘Martigues,’ what’s colloquially known as the ‘double or nothing strategy’ was popular. This strategy involves making a higher bet to compensate for a loss in the previous round, safeguarding against the psychological factors aside, whether this is a wise strategy still needs contemplation. Mathematically, the essence of such a strategy is summarized in the formula $$ E \left( X_{n+1} | X_{1} , \cdots , X_{n} \right) = X_{n+1} $$ pointing out the gambler’s fallacy with ‘I’ve been losing continuously, so I must win this time’, explaining why the Martingale betting is futile.
Examples
(1)
Let’s consider an autoregressive process $AR(1)$ $X_{n+1} = X_{n} + \varepsilon_{n}$. If filtration is given, then since all information regarding $X_{n}$ is known, according to the properties of conditional expectation $$ \begin{align*} E \left( X_{n+1} | \mathcal{F}_{n} \right) =& E \left( X_{n} + \varepsilon_{n} | \mathcal{F}_{n} \right) \\ =& E \left( X_{n} | \mathcal{F}_{n} \right) + E \left( \varepsilon_{n} | \mathcal{F}_{n} \right) \\ =& X_{n} + E \left( \varepsilon_{n} | \mathcal{F}_{n} \right) \\ =& X_{n} + E ( \varepsilon_{n} ) \\ =& X_{n} \end{align*} $$ , thus $\left\{ (X_{n}, \mathcal{F}_{n}) \right\}$ becomes a martingale.
(2)
Assuming $\left\{ X_{n} \right\}_{n \in \mathbb{N}}$ are independent of each other, and $E(X_{n}) = 0$ and $\displaystyle S_{n}:= \sum_{i =1}^{n} X_{i}$, then $$ \begin{align*} E(S_{n+1} | \mathcal{F}_{n} ) =& S_{n} + E( X_{n+1} | \mathcal{F}_{n} ) \\ =& S_{n} + E( X_{n+1} ) \\ =& S_{n} + 0 \end{align*} $$ , thus $\left\{ (S_{n}, \mathcal{F}_{n}) \right\}$ becomes a martingale.
Meanwhile, given a convex function $\phi$ and a martingale, one can create a submartingale as shown above.
Theorem
Given a martingale $\left\{ (X_{n}, \mathcal{F}_{n}) \right\}$ and a convex function $\phi: \mathbb{R} \to \mathbb{R}$, $( \phi (X_{n}) , \mathcal{F}_{n} )$ is a submartingale.
Proof
Conditional Jensen’s Inequality: Assuming that a probability space $( \Omega , \mathcal{F} , P)$ and a sub-σ-field $\mathcal{G} \subset \mathcal{F}$ are given, and $X$, is a random variable. For a convex function $\phi: \mathbb{R} \to \mathbb{R}$ and $\phi (X) \in \mathcal{L}^{1} ( \Omega ) $ $$ \phi \left( E \left( X | \mathcal{G} \right) \right) \le E \left( \phi (X) | \mathcal{G} \right) $$
According to Conditional Jensen’s Inequality $$ E \left( \phi (X_{n+1}) | \mathcal{F}_{n} \right) \ge \phi \left( E \left( X_{n+1} | \mathcal{F}_{n} \right) \right) = \phi ( X_{n} ) $$
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Corollary
As a corollary, setting $p \ge 1$ to $\phi (x) = | x |^{p}$, $\left\{ |X_{n}|^p , \mathcal{F}_{n} \right\}$ is always a submartingale.
See also
Various filtrations
$$ A_{1} \subset A_{2} \subset \cdots \subset A_{n} \subset \cdots $$ Universally in mathematics, structures forming nested sequences like above are referred to as filtrations.