Martingale

An integrable adapted process $(M_n)$ is a martingale when $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$.

A nonnegative martingale takes nonnegative values almost surely at each time. In discrete time its expectations are constant, so it is bounded in $L^1$ and the Martingale convergence theorem gives a finite integrable almost sure limit. Its expectations need not converge to the expectation of that limit; uniform integrability is needed for convergence in L1.

Expose independent coordinates successively and take the conditional mean of a function after each exposure. These conditional means form a martingale from $\mathbb{E}f$ to $f$. A coordinate oscillation bound $c_i$ gives a conditional interval of length $c_i$ for the corresponding increment, by coupling the unexposed coordinates. Applying the Hoeffding lemma to these conditional ranges proves the McDiarmid inequality.

A martingale can diverge to $+\infty$ almost surely when its increments are finite but not uniformly bounded. For independent Bernoulli random variables with $\mathbb{P}(Y_n=1)=2^{-n}$,has centered integrable increments. The First Borel-Cantelli lemma makes the rare negative jumps occur only finitely often, after which every increment equals one. The expectation remains zero at every finite time despite this pathwise divergence.

An integrable adapted process is a martingale if all its values at bounded stopping times are integrable with the same expectation as $X_0$. For the converse, compare deterministic $t$ with $U=s{1}_A+t{1}_{A^c}$, where $s<t$ and $A\in {\mathcal{F}}_s$. The resulting identity $\mathbb{E}[(X_t-X_s)1_A]=0$ is precisely the defining test for conditional expectation. In the forward direction, use the optional stopping theorem for a càdlàg martingale under the usual conditions for a filtration.

If a martingale has increments with $|M_i-M_{i-1}|\le c_i$ almost surely for deterministic $c_i$, then each one-sided deviation of magnitude $t>0$ from $M_0$ has probability at most $\exp (-t^2/(2{\sum }_i{c}_i^2))$. If all $c_i$ vanish, the martingale is constant. The inequality gives concentration from bounded increments without requiring independent increments.

If a real sequence converges finitely, its successive increments tend to zero. A martingale whose increments stay bounded away from zero on every sample path cannot have a finite almost-sure limit, regardless of its zero conditional mean increments.

Once a nonnegative martingale hits zero, it stays zero almost surely. Conditional expectation on $\{M_n=0\}$ gives $\mathbb{E}[M_{n+1}{1}_{\{M_n=0\}}]=0$; nonnegativity forces the next value to vanish there. A countable intersection makes these assertions simultaneous for all times, proving absorption at the first zero.

Let a nonnegative martingale start at one, let $T$ be its first zero, and let $T_R$ be its first hit of $[R,\infty )$. Suppose $\mathbb{E}[(M_{n+1}-M_n{)}^2\mid {\mathcal{F}}_n]\ge {\sigma }^2{1}_{\{T>n\}}$ and $M_{T_R}\le \tau R$ on finite hits, for thresholds $R\ge 1$. Then $M_{T\wedge T_R\wedge n}^2-{\sigma }^2(T\wedge T_R\wedge n)$ is a submartingale. The bounded stopped values have mean one, so their second moment is at most $\tau R$. Consequently $\mathbb{E}(T\wedge T_R)\le (\tau R-1)/{\sigma }^2$. For $\tau \ge 1$ this implies the weaker bound ${\tau }^{2}R/{\sigma }^2$. The threshold restriction prevents an impossible overshoot requirement below the starting value.

Under the conditional-variance and overshoot assumptions of the absorption-time bound from conditional variance and overshoot, $\mathbb{P}(T>n)\le 2\tau /(\sigma \sqrt{n})$. Bound the probability by $\mathbb{P}(T_R<\infty )+\mathbb{E}(T\wedge T_R)/n$ and optimize $1/R+{\tau }^{2}R/({\sigma }^{2}n)$. If the resulting bound is below one, its minimizing threshold is greater than one; otherwise the probability bound is trivial.

A martingale difference sequence is an adapted sequence of integrable random variables $D_n$, $n\ge 1$, with $\mathbb{E}[D_n\mid {\mathcal{F}}_{n-1}]=0$. Its partial sums are a martingale, and martingale increments give such sequences. If the differences are square-integrable, they are uncorrelated random variables: for $j<k$, conditioning on ${\mathcal{F}}_{k-1}$ gives $\mathbb{E}{D}_j{D}_k=0$.

A time-indexed increment $D_j$ with $\mathbb{E}(D_j\mid {\mathcal{F}}_{j-1})=0$ is a martingale difference relative to its history. Square-integrable differences at different indices are orthogonal, allowing their conditional variances to accumulate. They need not be independent.

If a martingale has increments bounded by a single deterministic constant, then $M_n/n\to 0$ almost surely. Its increments form a uniformly $L^2$-bounded martingale difference sequence; apply the strong law for uniformly L2-bounded uncorrelated random variables to their partial sums and note $M_0/n\to 0$. No square-integrability of the initial value is needed beyond the usual martingale integrability.

For a real continuous function $f$ on $[0,1]$, let $G_n$ be its secant slope on each length-$2^{-n}$ dyadic cell. Regard $([0,1],\mathcal{B},dt)$ as a probability space and use the filtration of half-open dyadic cells with the endpoint $\{1\}$ as a separate null cell. The mean of the two child slopes equals the parent slope, so $(G_n)$ is a martingale. Its integral gives the linear interpolation of $f$ on that grid. If $f$ is Lipschitz continuous, these slopes are bounded by its Lipschitz constant; the Lp martingale convergence theorem then supplies a bounded integral density for $f$.

For fair independent and identically distributed random variables ${\xi }_j$ with the Bernoulli distribution, $M_n=2^n{1}_{\{{\xi }_1=\cdots ={\xi }_n=1\}}$ is a nonnegative martingale. It has almost sure convergence to zero but $\mathbb{E}{M}_n=1$, so it lacks convergence in L1.

If $(M_n)$ is a square-integrable martingale, then its increments are orthogonal in $L^2$: for $i<j$,More generally, multiplying the later increment by any square-integrable quantity measurable before it still gives expectation zero whenever the product is integrable.

An integrable adapted process $(X_n)$ is a submartingale when

If $(X_m{)}_{m\le n}$ is a nonnegative submartingale, then for every $a>0$,Stop at the first crossing of $a$ and use the submartingale property on that event.

For a centered square-integrable martingale with $v=\mathbb{E}{M}_n^2$ and $\lambda >0$, apply the Doob maximal inequality for a nonnegative submartingale to $(M_k+c{)}^2$ for $c\ge 0$. Conditional Jensen inequality gives the submartingale property, and crossing $\lambda$ forces its square above $(\lambda +c{)}^2$. The bound $(v+c^2)/(\lambda +c{)}^2$ is minimized by $c=v/\lambda$. This extends the Cantelli inequality from one random variable to a martingale maximum, without requiring independent increments.

For a nonnegative martingale, $T_R=\inf \{n:M_n\ge R\}$ satisfies $\mathbb{P}(T_R<\infty )\le \mathbb{E}{M}_0/R$. Stop at $T_R\wedge n$ and use nonnegativity before passing to the increasing event. A limit of thresholds from below also gives the bound for $\{{\sup }_n{M}_n\ge R\}$ even when the supremum is not attained.

For independent centered random variables and $S_m={\sum }_{j\le m}{X}_j$,Apply the Doob maximal inequality for a nonnegative submartingale to $(S_m+c{)}^2$ and optimize at $c=\mathrm{Var}(S_n)/x$.

For independent centered square-integrable variables, if $S_n={\sum }_{m\le n}{X}_m$ and $V_n=\mathrm{Var}(S_n)$, thenis a martingale.

If additionally $|X_m|\le K$, thenStop the square martingale for independent centered sums at the first exit from $[-x,x]$ or at time $n$.

For an integrable random variable $Z$ and an increasing filtration $({\mathcal{F}}_n)$, the process $M_n=\mathbb{E}[Z\mid {\mathcal{F}}_n]$ is a martingale. The tower property of conditional expectation gives the martingale identity.

Expose the independent edge indicators of a binomial random graph in a fixed order, and set $M_i=\mathbb{E}[Z\mid \text{the first }i\text{ indicators}]$ for an integrable statistic $Z$. This is a martingale from $\mathbb{E}Z$ to $Z$. If changing one edge changes $Z$ by at most $c$, coupling the unexposed indicators gives $|M_i-M_{i-1}|\le c$, permitting the Azuma-Hoeffding inequality.

If $Y_1,Y_2,\dots$ are independent integrable random variables with $\mathbb{E}[Y_i]=1$, thenis a martingale for ${\mathcal{F}}_n=\sigma (Y_1,\dots ,Y_n)$, because independence gives $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n\mathbb{E}[Y_{n+1}]=M_n$.

An adapted integrable process $(M_t{)}_{t\ge 0}$ is a continuous-time martingale when $\mathbb{E}[M_t\mid {\mathcal{F}}_s]=M_s$ for every $s\le t$.

A martingale whose values over its whole time interval form a uniformly integrable family has an integrable terminal limit and is closed by that limit. Conversely, conditional expectations of an integrable terminal variable form a uniformly integrable martingale, by uniform integrability of conditional expectations. For continuous martingales, one may equivalently use uniform integrability of the family stopped at finite stopping times.

A càdlàg martingale is a continuous-time martingale represented by a càdlàg process: its paths are right-continuous and have finite left limits at positive times, on one common probability-one event. This path regularity permits approximation of stopping times from above.

A continuous martingale is a continuous-time martingale whose paths are continuous outside one null event. Continuity does not remove the integrability or conditional-expectation requirements in the martingale definition.

A continuous martingale $M$ is L2-bounded when ${\sup }_{t\ge 0}\mathbb{E}|M_t{|}^2<\infty$. The L2 martingale convergence theorem gives a terminal value $M_{\infty }$ with convergence in $L^2$ and almost sure convergence. The resulting conditional expectation identity is $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$. Boundedness on each separate finite horizon is a weaker condition. The subspace with $M_0=0$ is commonly denoted ${\mathcal{M}}_0^2$.

For a L2-bounded continuous martingale, the terminal and maximal norms satisfyThe first inequality follows from almost sure convergence. Apply the Doob L2 maximal inequality on $[0,T]$ and then the monotone convergence theorem as $T\to \infty$ for the second. These are equivalent norms on the space modulo indistinguishability of stochastic processes; neither inequality needs a zero initial value.

An adapted process $M$ is a local martingale when there are stopping times $T_n\uparrow \infty$ almost surely such that each stopped process $M_{t\wedge T_n}$ is a martingale. Every martingale is local, while a positive local martingale is a supermartingale.

A uniformly bounded local martingale is a true uniformly integrable martingale: a localizing sequence gives uniformly dominated stopped values, so their conditional identities pass to the limit. A deterministic bound on each compact time interval gives a true martingale on each such interval, even when the bound grows with time.

An integrable discrete-time local martingale is a true martingale. Its stopped increments have the form $1_{\{\tau \ge t\}}(X_t-X_{t-1})$, and the dominated convergence theorem removes a localizing sequence from the conditional increment identity.

The Fatou lemma makes every time value of a nonnegative discrete-time local martingale integrable, assuming the usual integrable initial value. The integrable discrete-time result then makes it a true martingale.

A strict local martingale is a local martingale that is not a true martingale. The reciprocal of a three-dimensional Bessel process is an example: it is a nonnegative continuous local martingale whose expectation decreases strictly.

A localizing sequence for a local martingale $M$ is an increasing sequence of stopping times $T_n\uparrow \infty$ almost surely such that every stopped process $M^{T_n}$ is a true martingale. One can often choose the times so that the stopped processes are bounded.

A continuous local martingale is a local martingale whose sample paths are continuous almost surely. Its quadratic variation determines much of its pathwise behaviour.

Suppose $X,A$ are continuous, start at zero, and $A$ is increasing. If $e^{X-A/2}$ and $e^{-X-A/2}$ are local martingales, their logarithms first recover adaptation of $X,A$ and the semimartingale property of $X$. In the decomposition $X=N+V$, the two Itô drift measures give $dV+(d[N]-dA)/2=0$ and $-dV+(d[N]-dA)/2=0$. Thus $V=0$ and $[X]=A$. All real exponential parameters may be assumed, but these two suffice.

A process has conditionally symmetric increments if, for all deterministic $s\le t$ and bounded measurable $g$,Equivalently, the conditional characteristic function of every increment is invariant under changing the sign of its argument. This property is stronger than unconditional symmetry and is different from independent increments. It supplies identities between positive and negative exponential terminal martingales.

Suppose $X$ is a continuous local martingale starting at zero, with conditionally symmetric increments, and terminal conditional expectations have continuous martingale versions. ThenTo prove this, set $M_t=\mathbb{E}[e^{i\theta X_T}\mid {\mathcal{F}}_t]$. Symmetry makes $e^{-2i\theta X_t}{M}_t$ a martingale, and the Itô product rule gives $d⟨M,X⟩=i\theta Md⟨X⟩$. Consequently $e^{-i\theta X_t-{\theta }^2⟨X{⟩}_t/2}{M}_t$ is a bounded local martingale, hence a martingale. Evaluating at the endpoints proves the formula. If also $X_T\sim N(0,T)$ for all $T$, the values of the bracket Laplace transform at $1$ and $2$ force $⟨X{⟩}_T=T$ almost surely. Continuity and the Lévy characterization of Brownian motion then identify $X$ as Brownian motion.

A continuous local martingale converges to a finite limit on the event that its terminal bracket is finite. Stopping at each integer bracket level gives an L2-bounded martingale and its almost sure limit. On the finite-bracket event, some such stopped process coincides with the original process forever. This supplies terminal values for a finite DDS lifetime.

The polynomial $X^4-6X^2⟨X⟩+3⟨X{⟩}^2$ has zero Itô drift. Finite fourth maximal and second bracket moments make it a true martingale. If $\mathbb{E}{X}_t^2=t$ and $\mathrm{Cov}(X_t^2,⟨X{⟩}_t)=0$, then $\mathbb{E}{X}_t^4=3t^2-3\mathrm{Var}(⟨X{⟩}_t)$. Equality at every time forces a deterministic clock and hence Brownian motion by the Lévy characterization of Brownian motion.

If a local martingale has an integrable running absolute supremum on each fixed finite horizon, localization and conditional dominated convergence make it a true martingale. This upgrades polynomial Itô local martingales when maximal moments and bracket moments control all terms.

A continuous local martingale $M$ with $M_0=0$ and deterministically bounded quadratic variation on every finite horizon is a square-integrable martingale, with $\mathbb{E}{M}_t^2=\mathbb{E}[M{]}_t$. Localization, the Doob L2 maximal inequality, and the resulting integrable bound on the squared supremum justify removing the localizing sequence.

A continuous local martingale with finite variation is almost surely constant. After localization, its quadratic variation is both zero, because it has finite variation, and the quantity controlling its $L^2$ increments, because it is a martingale.

Let $M$ be a continuous local martingale with $M_0=0$ and $[M{]}_{\infty }=\infty$. Forthe process $B_s=M_{{\tau }_s}$ is Brownian motion andWhen $[M{]}_{\infty }<\infty$, one can enlarge the probability space and continue $B$ independently beyond that time.

A continuous local martingale bounded above pathwise throughout a stochastic time interval cannot have infinite quadratic variation at that interval's endpoint. The Dambis-Dubins-Schwarz theorem represents it as Brownian motion at its quadratic-variation clock; an infinite clock would force unbounded oscillations. Its clock therefore has a finite limit, and so does the martingale. A bound below gives the same conclusion by changing sign. The bound may be random; it must hold over the entire interval.

Strict increase of a bracket does not force an infinite terminal value. Up to $L=⟨M{⟩}_{\infty }$, the inverse-clock martingale continues at a finite lifetime by its terminal limit, with bracket $u\wedge L$. On an independent product extension, add ${\int }_0^u{1}_{\{s>L\}}d{\beta }_s$. Its bracket fills $(u-L{)}^{+}$ and its cross variation with the original part vanishes. The result is Brownian for all clock times and still represents $M_t$.

For a continuous strictly increasing unbounded bracket $V$, the inverse times $T_u=\inf \{t:V_t>u\}$ satisfy $V_{T_u}=u$ and $T_{V_t}=t$. Optional sampling of the original martingale stopped at bounded bracket levels makes $M_{T_u}$ a continuous local martingale in ${\mathcal{F}}_{T_u}$. Time-changing the square-minus-bracket martingale gives its bracket $u$; the Lévy characterization of Brownian motion identifies it as Brownian.

If a continuous local martingale satisfiesfor a nonnegative predictable process $A$, then, after enlarging the probability space if necessary, there is a Brownian motion $B$ such thatAn independent Brownian motion supplies noise on the set where $A=0$.

The Brownian motion $B$ in $M_t=B_{[M{]}_t}$ is generally constructed from $M$, so it need not be independent of the quadratic variation $[M]$. For example, $M_t=(W_t^2-t)/2={\int }_0^t{W}_{s}d{W}_s$ has clock ${\int }_0^t{W}_s^{2}ds$, and their dependence can be detected by a nonzero mixed moment.

For every $p>0$ there are constants $c_p,C_p>0$ such that a continuous local martingale $M$ with $M_0=0$ satisfiesfor every stopping time $T$ for which the quantities are finite.

For $p\ge 2$ and a continuous local martingale starting at zero, the Itô formula for $|X{|}^p$, the Doob Lp maximal inequality and Hölder's inequality yield $\mathbb{E}{\sup }_{s\le t}|X_s{|}^p\le C_p\mathbb{E}⟨X{⟩}_t^{p/2}$. One possible constant is $C_p=[p(p-1)(p/(p-1){)}^p/2{]}^{p/2}$. Stop at increasing absolute-value levels to remove an initial boundedness assumption. This proves the upper half of the Burkholder-Davis-Gundy inequalities in this range.

Every nonnegative local martingale is a supermartingale. In particular, one that starts from zero is identically zero: its nonnegative value at every time has expectation at most zero.

For $M_0=1$ and $M_t\to 0$, stop at the infimum of strict crossings above $a>1$. Continuity bounds the stopped process by $a$ and makes its value at a finite crossing time exactly $a$. Its expectation remains one, and bounded convergence at infinity yields $\mathbb{P}(\sup M>a)=1/a$. The maximum has density $m^{-2}$ on $m>1$, with no endpoint atom.

A nonnegative local martingale is a supermartingale. If its integrable terminal limit has the same expectation as its initial value, conditional Fatou lemma gives $\mathbb{E}[Z_{\infty }\mid {\mathcal{F}}_t]\le Z_t$, and equality of expectations makes this an equality. It is therefore a uniformly integrable martingale.

For a square-integrable martingale starting at zero,For a continuous local martingale stopped so that its quadratic variation is integrable, the Itô isometry makes the right-hand side $4\mathbb{E}[M{]}_t$.

For $p>1$ and a martingale $(M_k{)}_{k\le n}$,

A martingale bounded in $L^2$ converges almost surely and in $L^2$ to a square-integrable limit. Orthogonality of its increments makes their squared $L^2$ norms summable.

If every increment ${\xi }_n=M_n-M_{n-1}$ of a discrete martingale takes values in $\{-1,1\}$, thenIterated conditioning shows that the increments are independent symmetric signs, so $M_n-M_0$ is a simple symmetric random walk.

If $(M_n)$ is a martingale and $U_N[a,b]$ counts its completed upcrossings of $[a,b]$ by time $N$, thenEquivalent conventions may add an initial endpoint term.

An upcrossing of an interval $[a,b]$ consists of an observation at or below $a$ followed by a later observation at or above $b$. Successive completed upcrossings use disjoint buy-sell pairs.

The upcrossing count $U_N[a,b]$ counts completed upcrossings by time $N$, allowing the first purchase at time zero. For a martingale, the Doob upcrossing inequality bounds its expected value by $\mathbb{E}(M_N-a{)}^{-}/(b-a)$.

A discrete-time martingale with ${\sup }_n\mathbb{E}|M_n|<\infty$ converges almost surely to an integrable random variable. If the martingale is uniformly integrable, convergence also holds in $L^1$.

An $L^2$-bounded real martingale has a square-integrable terminal value, converges to it almost surely and in $L^2$, and satisfies $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$. The $L^2$ Cauchy property follows from orthogonality of increments and monotonicity of $\mathbb{E}{M}_t^2$. This terminal representation permits optional sampling at finite unbounded stopping times and uniform-integrability control of stopped squares by conditional Jensen.

For independent fair Bernoulli variables, this nonnegative martingale starts at one, doubles on every success, and becomes zero permanently on the first failure. Its expectation is always one, but it tends to zero almost surely because the probability of success forever is zero. Thus it satisfies the Martingale convergence theorem while failing convergence in L1. This illustrates how rare large values obstruct uniform integrability.

A submartingale with ${\sup }_n\mathbb{E}{Y}_n^{+}<\infty$ converges almost surely to a finite integrable random variable. The positive-part bound also bounds negative parts because $\mathbb{E}{Y}_n\ge \mathbb{E}{Y}_0$. The result specializes to an $L^1$-bounded martingale, and hence to every nonnegative martingale. It does not by itself give convergence in L1; uniform integrability supplies that stronger conclusion.

A martingale satisfying ${\sup }_n\mathbb{E}|M_n{|}^p<\infty$, for $1<p<\infty$, has almost sure convergence and convergence in Lp to a limit $M_{\infty }\in L^p$. Also $M_n=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_n]$. The Doob Lp maximal inequality makes ${\sup }_n|M_n|$ integrable to the power $p$; the Martingale convergence theorem gives the almost-sure limit, and the dominated convergence theorem gives convergence in the Lp norm. Boundedness in $L^1$ alone does not imply convergence in L1.

A uniformly integrable martingale $M_n$ has almost sure convergence and convergence in L1 to an integrable random variable $M_{\infty }$, and $M_n=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_n]$.

Stopping a uniformly integrable discrete-time martingale at any stopping time preserves uniform integrability, including when the stopping time can be infinite. For $C={\sup }_n\mathbb{E}|X_n|$, the optional stopping theorem and the Markov inequality giveChoose $R$ first, then $K$, to make this uniformly small.

For a decreasing sequence of sigma-algebras ${\mathcal{F}}_n$ and an integrable random variable $X$,almost surely and in $L^1$.

A nonnegative supermartingale, or more generally a supermartingale whose negative parts have uniformly bounded expectations, converges almost surely to a finite integrable random variable.

If ${\mathcal{F}}_{\infty }=\sigma ({\bigcup }_n{\mathcal{F}}_n)$ and $Z$ is bounded, thenalmost surely and in $L^1$. Testing the almost-sure martingale limit on the algebra ${\bigcup }_n{\mathcal{F}}_n$ identifies it with the terminal conditional expectation.

If bounded random variables $X_n$ converge almost surely to $X$, thenalmost surely and in $L^1$. For $Z_n={\sup }_{m\ge n}|X_m-X|$, the variables $\mathbb{E}[Z_n\mid {\mathcal{F}}_n]$ form a nonnegative supermartingale with expectations tending to zero.

A right-continuous continuous-time martingale with ${\sup }_{t\ge 0}\mathbb{E}|M_t|<\infty$ has an almost-sure finite limit as $t\to \infty$.

An integrable adapted process $(V_n)$ is a supermartingale when $\mathbb{E}[V_{n+1}\mid {\mathcal{F}}_n]\le V_n$.

A local supermartingale becomes a supermartingale after stopping along an increasing sequence of stopping times tending to infinity. Localization allows an Itô formula comparison before global integrability is established. Additional lower bounds or uniform integrability are needed to pass from the local comparison to an unrestricted expectation inequality.

For a positive stock satisfying $dS=S\sigma dW$, the Itô formula gives $d\sqrt{S}=\frac{1}{2}\sqrt{S}\sigma dW-\frac{1}{8}\sqrt{S}{\sigma }^{2}dt$. Localization and the conditional Fatou lemma make this nonnegative local supermartingale a true supermartingale.

For a right-continuous nonnegative supermartingale $M$ and $R>0$, the first level-hit time $\sigma$ and the optional stopping theorem give $R\mathbb{P}(\sigma \le T)\le \mathbb{E}[M_{\sigma \wedge T}]\le \mathbb{E}[M_0]$. Let $T\to \infty$ and first use levels $R-\epsilon$, then $\epsilon \downarrow 0$, to include a supremum not attained. For a continuous nonnegative local martingale on a stochastic interval, apply the argument on increasing localized compact subintervals, then pass to their limit. Letting $R\to \infty$ shows that its running supremum is finite almost surely.

If a nonnegative supermartingale reaches zero at a discrete stopping time, its later nonnegative values have conditional expectation at most zero and must also be zero almost surely.

A discrete-time supermartingale $X$ is a martingale exactly when $\mathbb{E}[X_n]=\mathbb{E}[X_0]$ for every $n$. If expectations are constant, the nonnegative variablehas expectation zero and therefore vanishes almost surely.

If a supermartingale $V$ dominates an adapted reward process $Z$, optional sampling gives $\mathbb{E}{Z}_{\tau }\le V_0$ for every bounded stopping time $\tau$.

A random time $\tau$ is a stopping time when $\{\tau \le n\}\in {\mathcal{F}}_n$ for every $n$.

For independent trials of success probability $p\in (0,1]$, let $T$ be the first index completing two consecutive successes. If $e_0,e_1$ are the mean remaining times with no trailing success and with one trailing success, conditioning gives $e_0=1+(1-p)e_0+p{e}_1$ and $e_1=1+(1-p)e_0$. Solving yields $e_0=(1+p)/p^2$, so fair trials have mean waiting time six. Disjoint pairs give a geometric tail bound and justify finiteness before solving the equations. Overlapping pairs are not independent trials.

The stopped stochastic process follows $X$ until the stopping time $\tau$ and thereafter keeps the value $X_{\tau }$ when $\tau$ is finite. For a pathwise right-continuous adapted process, the stopped process is adapted, by progressive measurability and measurable evaluation at $t\wedge \tau$. When $X$ is a martingale, further stopping and integrability conditions determine whether its stopped process is a martingale.