Martingales — Optional Stopping, Convergence, Inequalities¶

Job: Package Doob theory: optional sampling, $L^1$/$L^p$ convergence, maximal inequalities — the analytic core of continuous-time probability and SDE estimates.

Core¶

Setup¶

Filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})$; cadlag adapted integrable $M$. Martingale: $\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s$. Discrete time $n\in\mathbb{N}$ is the pedagogical and often the proof skeleton; continuous time needs right-continuity of filtration for optional sampling in full strength.

Optional sampling / stopping¶

Theorem (Doob OST, discrete). If $M$ is a UI martingale (or bounded stopping times, or non-negative super-martingale with caveats), and $\sigma\le\tau$ are stopping times, then $$ \mathbb{E}[M_\tau\mid\mathcal{F}\sigma]=M\sigma\quad\text{a.s.} $$ (with $M_\infty$ the a.s./$L^1$ limit when UI).

Bounded times: if $\tau\le T<\infty$ a.s. and $M$ is a martingale, $\mathbb{E}[M_\tau]=\mathbb{E}[M_0]$ under mild right-continuity. Unbounded $\tau$: need UI of $\{M_{\tau\wedge n}\}$ or domination.

Super-martingale inequality: $\mathbb{E}[M_\tau]\le\mathbb{E}[M_\sigma]$ for $\sigma\le\tau$ under conditions (non-negative super-martingale + Fatou often used for ruin).

Convergence theorems¶

Theorem	Hypotheses $\Rightarrow$ conclusion
Doob $L^1$ (submartingale)	$M_n$ $L^1$-bounded submartingale $\Rightarrow$ a.s. convergence to integrable $M_\infty$
UI martingale	UI $\Leftrightarrow$ $M_n=\mathbb{E}[M_\infty\mid\mathcal{F}_n]$ with $M_n\to M_\infty$ in $L^1$
$L^p$ martingale ($p>1$)	$\sup_n\\|M_n\\|_p<\infty$ $\Rightarrow$ $L^p$ and a.s. convergence
Reverse martingale	$M_n=\mathbb{E}[X\mid\mathcal{G}_n]$, $\mathcal{G}_n\downarrow$ $\Rightarrow$ converge a.s./$L^1$ to $\mathbb{E}[X\mid\bigcap\mathcal{G}_n]$
Continuous time	cadlag $L^1$-bounded super-martingale $\Rightarrow$ a.s. finite limits as $t\to\infty$; UI $\Rightarrow$ closable

Upcrossing inequality is the engine: number of upcrossings of $[a,b]$ controlled by $\mathbb{E}[(M_n-a)^+]$ — prevents infinite oscillation and yields a.s. convergence.

Doob maximal inequalities¶

Discrete or continuous cadlag non-negative submartingale $X$:

Weak $L^1$ (maximal): $$ \lambda\,\mathbb{P}\Bigl(\sup_{s\le t}X_s\ge\lambda\Bigr)\le\mathbb{E}[X_t 1_{{\sup_{s\le t}X_s\ge\lambda}}]\le\mathbb{E}[X_t]. $$

$L^p$ maximal ($p>1$): $$ \Bigl|\sup_{s\le t}|M_s|\Bigr|_p\le\frac{p}{p-1}|M_t|_p $$ for martingales (Doob’s $L^p$ inequality).

$L\log L$: for $p=1$, maximal function in $L^1$ needs $\mathbb{E}[|M_t|\log^+|M_t|]<\infty$ (Orlicz upgrade).

Continuous-time structure¶

Cadlag modification: right-continuous filtration + super-martingale $\Rightarrow$ cadlag version (Doob regularization).
Decomposition (Doob–Meyer): cadlag super-martingale of class (D) $= $ martingale $-$ predictable increasing process. Quadratic variation arises as compensator of $M^2$.
Local times / Tanaka: $|W_t|=\int\operatorname{sign}(W)\,dW+L_t^0$; local time is the increasing process in the semi-martingale decomposition of $|W|$.

Applications (research-facing)¶

Application	Mechanism
SDE $L^p$ estimates	Doob maximal on Itô integrals / Burkholder–Davis–Gundy
Stopping / free boundary	OST on discounted payoffs; Snell envelope = optimal super-martingale
Likelihood ratios	UI martingale $\Leftrightarrow$ absolute continuity of path measures (Kakutani, Girsanov setup)
Concentration	Azuma–Hoeffding: bounded differences martingale $\Rightarrow$ sub-Gaussian tails
Harmonic functions	$h(B_{t\wedge\tau})$ martingale $\Leftrightarrow$ $h$ harmonic (Dynkin); mean-value via OST
Branching / genetics	normalized population size martingale; extinction iff limit $0$ under $m\le 1$

Burkholder–Davis–Gundy (pointer). For cadlag local martingales, $$ c_p\,\mathbb{E}[\langle M\rangle_t^{p/2}]\le\mathbb{E}\Bigl[\sup_{s\le t}|M_s|^p\Bigr]\le C_p\,\mathbb{E}[\langle M\rangle_t^{p/2}], $$ $0<p<\infty$ (constants depend on $p$). Links maximal $L^p$ control to quadratic variation — central in SPDE energy estimates.

See it¶

Hitting a barrier for BM. Let $\tau=\inf\{t:W_t=a\}$ for $a>0$. Exponential martingale $M_t^\theta=\exp(\theta W_t-\theta^2 t/2)$ is UI on $\{t\le\tau\}$ after checking; OST $\Rightarrow$ $\mathbb{E}[e^{-\theta^2\tau/2}]=\ e^{-\theta a}$ (set $W_\tau=a$). Invert Laplace to get Lévy distribution of $\tau$. Same pattern: optional sampling of scale functions for one-dimensional diffusions.

Polya urn. Start with one red, one blue; draw and replace with two of same color. Proportion of red $M_n$ is a bounded martingale $\Rightarrow$ a.s. convergence; limit is Uniform$(0,1)$ (exchangeability / de Finetti). Prototype of “martingale convergence + identify limit law.”

Check¶

Give a martingale with $M_n\to M_\infty$ a.s. but not in $L^1$.

Simple asymmetric random walk stopped or the critical branching normalized size can fail UI. Standard: non-negative martingale with $\mathbb{E}M_n=1$ but $M_\infty=0$ a.s. (e.g. multiplicative: $M_{n+1}=M_n\cdot\xi_{n+1}$ with $\mathbb{E}\xi=1$, $\mathbb{P}(\xi=0)>0$ appropriately, or the exponential martingale of SRW at criticality). Then $\mathbb{E}|M_n-M_\infty|\not\to 0$ since $\mathbb{E}M_n=1\neq 0=\mathbb{E}M_\infty$.

Why does OST fail for $\tau=\inf\{t:W_t=1\}$ applied naively to $W$ itself with $\mathbb{E}W_\tau=\mathbb{E}W_0$?

$\mathbb{E}W_\tau=1\neq 0$, so hypotheses fail: $\tau$ is a.s. finite but $\{W_{t\wedge\tau}\}$ is not UI (mass escapes at large times). Bounded stopping times or UI needed.

Pitfalls¶

Closed vs closable: a.s. convergence alone does not give $M_t=\mathbb{E}[M_\infty\mid\mathcal{F}_t]$ without UI.
Local martingales and OST: stopping a local martingale yields a local martingale; expectations need not preserve. Strict local martingales (e.g. inverse Bessel) break “$\mathbb{E}M_\tau=\mathbb{E}M_0$.”
Predictable vs optional projection: Doob–Meyer compensators are predictable; using optional increasing processes breaks uniqueness.
Discrete inequalities $\neq$ continuous constants: same form, but cadlag proofs use first-exit from open sets; left limits matter for jumps.

Refs¶

Durrett; Karatzas–Shreve Ch. 1; Revuz–Yor Ch. II; Hall–Heyde, Martingale Limit Theory; Dellacherie–Meyer (general theory of processes).

Next¶

Foundations · Index · Probability · SPDE · Measure theory · MT / probability & stochastic — the martingale / the Itô / the Wiener

Theorem	Hypotheses \(\Rightarrow\) conclusion
Doob \(L^1\) (submartingale)	\(M_n\) \(L^1\)-bounded submartingale \(\Rightarrow\) a.s. convergence to integrable \(M_\infty\)
UI martingale	UI \(\Leftrightarrow\) \(M_n=\mathbb{E}[M_\infty\mid\mathcal{F}_n]\) with \(M_n\to M_\infty\) in \(L^1\)
\(L^p\) martingale (\(p>1\))	\(\sup_n\\|M_n\\|_p<\infty\) \(\Rightarrow\) \(L^p\) and a.s. convergence
Reverse martingale	\(M_n=\mathbb{E}[X\mid\mathcal{G}_n]\), \(\mathcal{G}_n\downarrow\) \(\Rightarrow\) converge a.s./\(L^1\) to \(\mathbb{E}[X\mid\bigcap\mathcal{G}_n]\)
Continuous time	cadlag \(L^1\)-bounded super-martingale \(\Rightarrow\) a.s. finite limits as \(t\to\infty\); UI \(\Rightarrow\) closable