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With continuous cash settlement and a money-market risk-neutral measure, discounted futures gains have zero drift; under the necessary true-martingale integrability, the futures quote is a martingale ending at the terminal spot price. The forward contract discount-weighted expectation is a different valuation expression.

If a continuous mean martingale $M$ and a nonnegative finite-variation process $V$ satisfy $d⟨M⟩=-dV$, then the Itô formula shows $\exp (iM-V/2)$ is a bounded complex martingale. Its terminal expectation establishes an entire Gaussian distribution with the initial mean and variance. This is stronger than matching only the first two moments in a random-domain field construction.

A model evolving the entire instantaneous forward rate curve. In the displayed one-factor risk-neutral measure dynamics the drift restriction makes every suitably integrable discounted zero-coupon bond price a martingale. The multi-factor form replaces the product of volatilities by their dot product.

Let $D=\mathbb{H}\setminus K$ and stop planar Brownian motion from $z\in D$ on reaching height $R$ or leaving $D$. Then $\mathrm{Im}{g}_K(z)={\lim }_{R\to \infty }R{\mathbb{P}}_z(\mathrm{Im}{B}_{{\tau }_R}=R)$. The stopped harmonic height is a bounded martingale. The bounded difference $g_K(z)-z$ makes its top-boundary value differ from $R$ by a uniform constant, while optional stopping of the imaginary coordinate bounds the top exit probability by $\mathrm{Im}z/R$. Killing on the hull is essential.

A centered continuous Gaussian process that is also a martingale has independent increments. Every future increment has zero covariance with every finite vector of past values by the martingale property; uncorrelated jointly normal variables are independent.

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.

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 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 ${\mathcal{M}}_0^2$.

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 standard Brownian motion started at zero and first exit ${\tau }_x$ from $(-x,x)$, the Laplace transform is $1/\cosh (x\sqrt{2\lambda })$ for $\lambda >0$. The stochastic process $e^{-\lambda t}\cosh (\sqrt{2\lambda }{B}_t)$ is the average of two copies of the Exponential martingale for Brownian motion. Its values stopped at $t\wedge {\tau }_x$ are bounded, so bounded-time application of the optional stopping theorem followed by dominated convergence evaluates the transform without assuming uniform integrability of an unstopped exponential martingale on the infinite horizon.

A continuous vector-valued local martingale $M$ starting at zero is standard multidimensional Brownian motion if its quadratic covariations satisfy $[M^i,M^j{]}_t={\delta }_{ij}t$. For each deterministic vector $\xi$, the Itô formula makes $\exp (i\xi \cdot M_t+|\xi {|}^{2}t/2)$ a complex local martingale. Its absolute value is bounded on each finite horizon, so it is a true martingale. Its conditional expectation identity givesThis conditional characteristic function identifies a centered multivariate normal distribution with covariance $(t-s)I$, independent of the past. Thus the coordinates are independent Brownian motions. The converse follows immediately from the standard coordinate quadratic covariations.

Stop a continuous local martingale at increasing level-and-time stopping times to obtain bounded martingales. Their limits of discrete quadratic variation sums agree before the earlier stopping time, because the sums commute exactly with stopping. These continuous limits patch into a continuous adapted nondecreasing process. On each compact interval, the chance that the stopping time occurs early tends to zero, giving uniform convergence on compacts in probability for the original sums.

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 compound Poisson process with rate $\lambda$ and mark $Y$ is a martingale exactly when $\mathbb{E}|Y|<\infty$ and $\mathbb{E}Y=0$.

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.

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 $\mathbb{E}{D}_j{D}_k=0$.

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.

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.