Martingales

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We briefly recall some basic properties and inequalities concerning martingales.

Definitions

Definition 1 (Filtered space) Let $\Theta$ be a partially ordered set and $(\Omega,\mathcal{F})$ a measurable space.

A filtration on $\Omega$ indexed by $\Theta$, is an increasing collection $(\mathcal{F}_{t\in \Theta})_{t \in \Theta}$ of sub-$\sigma$-algebras of $\mathcal{F}$. Namely for any $s,t \in \Theta$ with $s\le t$, $\mathcal{F}_s \subset \mathcal{F}_t$. A filtration is separable¹ if there is a countable set $\Theta'\subset \Theta$ such that $\mathcal{F}_t = \cap_{s\in \Theta', s > t} \mathcal{F}_s$ for all $t \in \Theta$.

The triple $(\Omega,\mathcal{F},\mathcal{F}_{t\in \Theta})$ is then called a filtered space.

If $\mathbb{P}$ is a probability on $(\Omega,\mathcal{F})$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in \Theta}, \mathbb{P})$ is called a filtered probability space. It is a complete filtered probability space if $(\Omega,\mathcal{F}_t, \mathbb{P})$ is complete for every $t\in \Theta$.

Definition 2 Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in \Theta})$ be a filtered space, $S$ a measurable space and $\mathbf{X}=(X_t)_{t\in \Theta}$ a collection of measurable maps $X_t\colon \Omega \to S$. $\mathbf{X}$ is adapted to $\mathcal{F}_t$ if $X_t$ is $\mathcal{F}_t$-measurable for $t\in \Theta$. The natural filtration $\mathcal{F}_t^{\mathbf{X}}$ of $\mathbf{X}$ is the weakest filtration $\mathbf{X}$ is adapted to.

Definition 3 (Martingale) Let $(\Omega,\mathcal{F},\mathcal{F}_{t\in \Theta})$ be a filtered space, and for $t\in \Theta$ let $M_t$ be a real-valued random variable $M_t\colon \Omega \to \mathbb R$ with $M_t \in L^1(\mathbb{P})$, namely $\mathbb{E}[|M_t|]<\infty$, for $t\in \Theta$.

For $\mathbb{P}$ a probability on $(\Omega,\mathcal{F})$, $\mathbf{M}=(M_t)_{t\in \Theta}$ is

a $\mathbb{P}$-submartingale if $\mathbb{E}[M_t \mid \mathcal{F}_s] \ge M_s$, for $s,t \in \Theta$ with $s\le t$.
a $\mathbb{P}$-supermartingale if $\mathbb{E}[M_t \mid \mathcal{F}_s] \le M_s$, for $s,t \in \Theta$ with $s\le t$.
a $\mathbb{P}$-martingale if it is both a supermartingale and a submartingale, namely if $\mathbb{E}[M_t \mid \mathcal{F}_s] = M_s$, for $s,t \in \Theta$.

If $\mathbb{P}$ is understood from the context, then it is usually omitted in the notation, for instance a $\mathbb{P}$-martingale is just called martingale in this case.

Example 1 Let $(X_i)_{i\ge 1}$ be a sequence of independent real-valued integrable random variables, and $\mathcal{F}_n:=\sigma(X_1,\ldots,X_n)$ be the smallest $\sigma$-algebra such that $(X_1,\ldots,X_n)$ is measurable.

Then $M_n:=\sum_{i=1}^n X_i$ is a submartingale iff $\mathbb{E}[X_i]\ge 0$ for all $i\ge 1$, a supermartingale iff $\mathbb{E}[X_i]\le 0$ for all $i$. And thus a martingale iff $\mathbb{E}[X_i]= 0$ for all $i$. Indeed, for $k < n$, using independence \[ \mathbb{E}[M_n \mid \mathcal{F}_{k}] = \mathbb{E}[ M_k + (X_{k+1} + \ldots X_n) \mid (X_1,\ldots,X_k)]= = M_k+ \sum_{i=k+1}^n \mathbb{E}[X_i] \] So for instance if the expectations are non-negative, $M_n$ is a submartingale. On the other hand, if $M_n$ is a submartingale, then one needs $\mathbb{E}[X_n]\ge 0$ for all $n\ge 1$, taking $k=n-1$ above.

Remark 1. $M_t \in L^1(\mathbb{P})$ is a martingale w.r.t. a given filtration iff it is a martingale w.r.t. its natural filtration, namely iff $\mathbb{E}[M_t \mid (M_r)_{r\le s}]= M_s$ for $s\le t$.

Remark 2. If $f \colon \mathbb R \to \mathbb R$ is convex (concave) and $M_t$ is a submartingale (supermartingale), then $f(M_t)$ is a submartingale (supermartingale), provided $f(M_t)\in L^1(\mathbb{P})$. This is a consequence of Jensen inequality. In particular linear combinations of martingales are martingales.

Remark 3. If $\Theta=\mathbb{N}$ (or any other countable set), the filtration is automatically separable, and a martingale (sub/supermartingale) $(M_n)$ is called a discrete martingale (sub/supermartingale) in this case.

Remark 4. If $\Theta=[0,\infty)$ (or some other real interval), the filtration is separable iff \[ \mathcal{F}_s= \mathcal{F}^+_s:= \cap_{t>s} \mathcal{F}_t \] since we can always take the intersection on the rational numbers. In this case, the filtration is usually called right-continuous. A martingale (sub/supermartingale) $(M_t)_{t\ge 0}$ is called a continuous-time martingale (sub/supermartingale) in this case.

If $(M_n)$ is a discrete martingale (sub/supermartingale) on a discrete filtered space $(\Omega,\mathcal{F})$, $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n\in \mathbb{N}}, \mathbb{P})$, we can define for $t\in [n,n+1)$: \[ \mathcal{F}_t:=\mathcal{F}_n \qquad M_t:=M_n \] to get a continuous-time martingale (sub/supermartingale). So the theory for continuous-time martingales covers indeed the theory for discrete martingales.

Continuous-time martingales and Stopping times

Doob convergence Theorem

For a real number $a\in \mathbb{R}$, we use the notation $a^+:=\max(x,0)$, $a^-:=\max(-a,0)$, so that $a=a^+-a^-$ and $|a|=a^+ + a^-$. For $\mathbf{X}:=(X_t)_{t\in \mathbb{R}}$ we also write \[ X_{t^+}:= \lim_{s\downarrow t} X_s, \qquad X_{t^-}:= \lim_{s\uparrow t} X_s \] and we define the random sets² \[ I^+\equiv I_{+}(\mathbf{X}):= \{t\in \mathbb{R} \st X_{t}=X_{t^+} \}, \qquad I_{-}\equiv I^-(\mathbf{X}):= \{t\in \mathbb{R} \st X_{t}=X_{t^-} \} \]

Theorem 1 Let $\mathbf{X}:=(X_t)_{t\in \mathbb{R}}$ be a continuous time supermartingale such that \[ \varlimsup_{t} \mathbb{E}[X_t^-]<\infty \tag{1}\] Then $X_t$ admits left and right limits, with probability $1$: \[ \mathbb{P}(X_{t^+}, X_{t^-} \text{ exist for all $t$})=1 \]

Moreover there exists a random variable $X\in L^1(\Omega)$ such that \[ \lim_{t\to \infty, t\in I_{+}\cup I_{-}} X_t=X \quad \text{a.s.} \] In particular, under Equation 1, a right-continuous (or left-continuous) supermartingale admits an a.s. limit.

Footnotes

When $\Theta$ is a totally ordered, separable space, in particular when $\Theta \subset \mathbb{R}$, a separable filtration is also called right-continuous. Indeed in this case one usually introduces the right-augmented filtration \[ \mathcal{F}_t^+:= \cap_{s>t} \mathcal{F}_s \] and separability is equivalent to $\mathcal{F}_t=\mathcal{F}^+_t$.↩︎
to be precise, in the notation $X_t=X_{t^+}$ we mean that the limit must exist and equal $X_t$, or equivalently $\varlimsup_{s\downarrow t} |X_{s}-X_t|=0$.↩︎