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The Definition of Martingale 📂Probability Theory

The Definition of Martingale

Definition

Let’s assume that a probability space (Ω,F,P)( \Omega , \mathcal{F} , P) is given.

  1. A sequence {Fn}n∈N\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}} of sub-σ-fields of F\mathcal{F} is called a filtration if it satisfies the following: ∀n∈N,Fn⊂Fn+1 \forall n \in \mathbb{N}, \mathcal{F}_{n} \subset \mathcal{F}_{n+1}
  2. Given a filtration {Fn}n∈N\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}, a sequence {Xn}n∈N\left\{ X_{n} \right\}_{n \in \mathbb{N}} of ordered pairs formed by sequences of Fn\mathcal{F}_{n}-measurable Lebesgue integrable random variables XnX_{n} is called a martingale if it satisfies the following: ∀n∈N,E(Xn+1∣Fn)=Xn \forall n \in \mathbb{N}, E \left( X_{n+1} | \mathcal{F}_{n} \right) = X_{n}
  • Fn\mathcal{F}_{n} being a sub-σ-field of F\mathcal{F} means that both are σ-fields of Ω\Omega, but Fn⊂F\mathcal{F}_{n} \subset \mathcal{F} holds.
  • XnX_{n} being a Fn\mathcal{F}_{n}-measurable function means that for all Borel sets B∈B(R)B \in \mathcal{B}(\mathbb{R}), Xn−1(B)∈FnX_{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. ∀n∈N,E(Xn+1∣Fn)≥Xn∀n∈N,E(Xn+1∣Fn)≤Xn \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”:

  1. Filtration: ∀n∈N,Fn⊂Fn+1\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 XnX_{n} being Fn\mathcal{F}_{n}-measurable means that as the actual data xnx_{n} is observed, the σ-field Fn\mathcal{F}_{n} also expands and it is safe to assume that all information up to nn has been acquired.
  2. Martingales: ∀n∈N,E(Xn+1∣Fn)=Xn\forall n \in \mathbb{N}, E \left( X_{n+1} | \mathcal{F}_{n} \right) = X_{n} means assuming that knowing the information Fn\mathcal{F}_{n} up to nn, the next scenario Xn+1X_{n+1} will also be similar to XnX_{n}. If the expected value of Xn+1X_{n+1} can be derived regardless of the previously gathered Fn\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(Xn+1∣X1,⋯ ,Xn)=Xn+1 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)AR(1) Xn+1=Xn+εnX_{n+1} = X_{n} + \varepsilon_{n}. If filtration is given, then since all information regarding XnX_{n} is known, according to the properties of conditional expectation E(Xn+1∣Fn)=E(Xn+εn∣Fn)=E(Xn∣Fn)+E(εn∣Fn)=Xn+E(εn∣Fn)=Xn+E(εn)=Xn \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 {(Xn,Fn)}\left\{ (X_{n}, \mathcal{F}_{n}) \right\} becomes a martingale.

(2)

Assuming {Xn}n∈N\left\{ X_{n} \right\}_{n \in \mathbb{N}} are independent of each other, and E(Xn)=0E(X_{n}) = 0 and Sn:=∑i=1nXi\displaystyle S_{n}:= \sum_{i =1}^{n} X_{i}, then E(Sn+1∣Fn)=Sn+E(Xn+1∣Fn)=Sn+E(Xn+1)=Sn+0 \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 {(Sn,Fn)}\left\{ (S_{n}, \mathcal{F}_{n}) \right\} becomes a martingale.

Meanwhile, given a convex function ϕ\phi and a martingale, one can create a submartingale as shown above.

Theorem

Given a martingale {(Xn,Fn)}\left\{ (X_{n}, \mathcal{F}_{n}) \right\} and a convex function ϕ:R→R\phi: \mathbb{R} \to \mathbb{R}, (ϕ(Xn),Fn)( \phi (X_{n}) , \mathcal{F}_{n} ) is a submartingale.

Proof

Conditional Jensen’s Inequality: Assuming that a probability space (Ω,F,P)( \Omega , \mathcal{F} , P) and a sub-σ-field G⊂F\mathcal{G} \subset \mathcal{F} are given, and XX, is a random variable. For a convex function ϕ:R→R\phi: \mathbb{R} \to \mathbb{R} and ϕ(X)∈L1(Ω)\phi (X) \in \mathcal{L}^{1} ( \Omega ) ϕ(E(X∣G))≤E(ϕ(X)∣G) \phi \left( E \left( X | \mathcal{G} \right) \right) \le E \left( \phi (X) | \mathcal{G} \right)

According to Conditional Jensen’s Inequality E(ϕ(Xn+1)∣Fn)≥ϕ(E(Xn+1∣Fn))=ϕ(Xn) 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≥1p \ge 1 to ϕ(x)=∣x∣p\phi (x) = | x |^{p}, {∣Xn∣p,Fn}\left\{ |X_{n}|^p , \mathcal{F}_{n} \right\} is always a submartingale.

See also

Various filtrations

A1⊂A2⊂⋯⊂An⊂⋯ 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.