Chapter 5 — Discrete-Time Martingales

This page develops discrete-time filtrations, martingales, stopping times, and stopped processes. The mathematical discussion remains separate from the Python API and worked examples.

1. Filtrations and adapted processes

A filtration is an increasing sequence of sigma-fields

\[\mathcal F_0\subseteq\mathcal F_1\subseteq\cdots\subseteq\mathcal A.\]

A process \((X_n)\) is adapted if \(X_n\) is \(\mathcal F_n\)-measurable for every \(n\). Its natural filtration is

\[\mathcal F_n^X=\sigma(X_0,\ldots,X_n).\]

2. Martingale

An integrable and adapted process \((X_n)\) is a martingale if

\[\boxed{E(X_{n+1}\mid\mathcal F_n)=X_n\quad\text{a.s.}}\]

Equivalently,

\[E(X_{n+1}-X_n\mid\mathcal F_n)=0.\]

For every \(A\in\mathcal F_n\),

\[E(\mathbf1_A X_{n+1})=E(\mathbf1_A X_n).\]

3. Doob martingale

For \(X\in L^1\), the process

\[M_n=E(X\mid\mathcal F_n)\]

is a martingale. This is the Doob martingale.

4. Jensen's inequality and constructions

For a convex function \(\varphi\),

\[\varphi(E(X))\le E(\varphi(X)).\]

A sum

\[X_n=\varepsilon_1+\cdots+\varepsilon_n\]

is a martingale when

\[E(\varepsilon_{n+1}\mid\mathcal F_n)=0.\]

A central example is the symmetric random walk \(X_n=\sum_{i=1}^n\xi_i\), with independent increments taking values in \(\{-1,1\}\), together with exponential transformations.

5. Properties

For a martingale,

\[E(X_n)=E(X_0),\]

and, for \(m<n\),

\[E(X_n\mid\mathcal F_m)=X_m.\]

6. Submartingales and supermartingales

A supermartingale satisfies

\[E(X_{n+1}\mid\mathcal F_n)\le X_n,\]

while a submartingale satisfies

\[E(X_{n+1}\mid\mathcal F_n)\ge X_n.\]

For a biased random walk with step \(+1\) and probability \(p\), the process is a martingale when \(p=1/2\), a supermartingale when \(p<1/2\), and a submartingale when \(p>1/2\).

If \(\varphi\) is convex and the required integrability conditions hold, \((\varphi(X_n))\) is a submartingale whenever \((X_n)\) is a martingale. In particular, \((|X_n|)\) and, when integrable, \((X_n^2)\), are submartingales.

7. Stopping times

A variable

\[T:\Omega\to\mathbb N\cup\{+\infty\}\]

is a stopping time if

\[\{T=n\}\in\mathcal F_n,\]

which is equivalent to the conditions \(\{T\le n\}\in\mathcal F_n\) and \(\{T>n\}\in\mathcal F_n\).

If \(S\) and \(T\) are stopping times for the same filtration, the operations \(S+T\), \(S\wedge T\), and \(S\vee T\) also give stopping times. The first hitting time of a set is a fundamental example.

8. Stopped process

For an adapted process \((X_n)\) and a stopping time \(\tau\), the stopped process follows the original path until the stopping time and then remains at its stopped value. One representation is

\[X_n^\tau=X_n\mathbf1_{\{n<\tau\}}+X_\tau\mathbf1_{\{\tau\le n\}}.\]

Another form is

\[X_n^\tau=X_0+\sum_{k=0}^{n-1}(X_{k+1}-X_k)\mathbf1_{\{\tau>k\}}.\]

9. Stopped martingale

If \((X_n)\) is a martingale and \(\tau\) is a stopping time, then the stopped process \((X_n^\tau)\) is again a martingale.

10. Terminal variable

When \(P(\tau<\infty)=1\), the terminal variable of the stopped process satisfies

\[X_n^\tau\xrightarrow[n\to\infty]{\mathrm{a.s.}}X^\tau.\]

11. Summary

\[\text{Filtration}\rightarrow\text{adapted process}\rightarrow\text{martingale}\rightarrow\text{Jensen}\]

\[\rightarrow\text{sub/supermartingale}\rightarrow\text{stopping time}\rightarrow\text{stopped process}\rightarrow\text{stopped martingale}.\]