In probability, a martingale is given by a sequence of integrable
random variables $(S_n)$ and an increasing sequence of
$\sigma$-algebras ${\cal F}_n$ such that
$S_n$ is ${\cal F}_n$-measurable and
$E(S_{n+1} \mid {\cal F}_n) = S_{n}$.
This is an important notion because there are many results concerning
convergence of martingales sequences, e.g. if it is bounded in $L^2$
then it converges in $L^2$ norm and $a.e.$
If $X_i$ is a sequence of i.i.d. random variables and
${\cal F}_n = \sigma(X_i, i\leq n)$, then the following sequences
are martingales:
-
$S_n – E(S_n)$,
-
$ \exp(S_n)/E(\exp(S_n))$,
-
$(S_n)^2-E(S_n^2)$,
These are used in the theory of random walks to compute e.g. the mean time before reaching a given state.
Are there any other interesting examples of discrete time martingales?
Best Answer
$\newcommand{\bN}{\mathbb{N}}$ $\newcommand{\eF}{\mathscr{F}}$ $\newcommand{\si}{\sigma}$
Branching processes Set $\bN_0=\{0,1,2,\dotsc\}$. Fix a probability measure $\mu$ on $\bN_0$ such that $$ m:=\sum_{k\in\bN_0}k\mu\bigl(\,\{k\}\,\bigr)<\infty $$ and $\mu(\{k_0\})>0$ for some $k_0>1$. Consider next a sequence $(X_{n,j})_{j,n\in\bN_0}$ of i.i.d. $\bN_0$-valued random variables with common probability distribution $\mu$. Fix $\ell\in \bN_0$, $\ell>0$, set $Z_0=\ell$. For $n\in\bN_0$ define $$ Z_{n+1}=\sum_{j=1}^{Z_n} X_{n,j},\;\;\eF_n=\si\bigl(\, X_{k,j};\, k\in\bN_0, k<n\,\bigr). $$ The random variable $Z_n$ can be interpreted as the population of the $n$-th generation of a species that had $\ell$ individuals at $n=0$ and such that the number of offsprings of a given individuals is a random variable with distribution $\mu$.
Then $Y_n=m^{-n}Z_n$ is a martingale.
Polya's urn scheme An urn contains $r>0$ red balls and $g>0$ green balls. Fix an integer $c\geq 0$. Every unit of time, we draw a ball, and we replace it by $c+1$ balls of the same color as the one drawn. Denote by $R_n$ and $G_n$ the number of red and respectively green balls in the urn after the $n$-th draw, and set $$ X_n:=\frac{R_n}{R_n+G_n},\;\;\eF_n=\si(R_0,G_0,\dotsc , R_n, G_n). $$ Then $(X_\bullet)$ is an $\eF_\bullet$-martingale.
Markov chains Suppose that $(X_n)_{n\in\bN_0}$ is a Markov chain with countable state space $E$ and transition matrix $P=\big(P(i,j)\big)_{i,j\in E}$ $\newcommand{\bP}{\mathbb{P}}$ $\newcommand{\bR}{\mathbb{R}}$ $$ P(i,j)=\bP(X_{n+1}=j|X_n=i). $$ For any function $f: E\to \bR$ we define $Pf:E\to\bR$ $$ Pf(i)=\sum_jP(i,j)f(j). $$
Then the sequence $$Y_n= f(X_n)-\sum_{k=0}^{n-1}\Big( Pf(X_k)-f(X_k)\;\Big) $$
is a martingale.
Doob martingale Suppose that $f:[0,1]\to\bR$ is an integrable function. Denote by $\eF_n$ the sigma algebra generated by the intervals $I_{k,n}:=\big(\;(k-1)/2^n, k/2^n\;\big)$, $k=1,\dotsc ,2^n$.
Define $f_n:[0,1]\to\bR$
$$ f_n(x)= 2^n\int_{I_{k,n}} f(t) dt ,\;\;x\in I_{k,n}. $$
Then the sequence $(f_n)$ is an $\eF_n$-martingale.