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The probability of a market event under an equivalent martingale measure for a specified numéraire. It can differ from the physical probability. For a random numéraire, impose the martingale condition on asset-price ratios rather than treating the numéraire return as deterministic.

The unique equivalent martingale measure in the one-stock Cox--Ross--Rubinstein model assigns each up move probabilityand each down move probability $1-q$. Under this measure the discounted stock is a martingale.

For a fixed interior point and parameter four, the SLE angle process has zero drift and satisfies ${\theta }_t=\arg z+2{\int }_0^t{Y}_s|Z_s{|}^{-2}d{B}_s$. Its values in $(0,\pi )$ make this continuous local martingale a true bounded martingale. The identity $\log ({{\rm Y}}_t/{{\rm Y}}_0)=-[\theta {]}_t$, together with the Dambis-Dubins-Schwarz theorem, prevents its Loewner conformal radius from vanishing at a finite lifetime. Since the parameter-four trace is simple and does not meet the real boundary at positive times, finite swallowing would require a visit to the point and vanishing radius. Thus its Loewner swallowing time is infinite almost surely. This gives a global continuous bounded martingale, not just one defined before swallowing.

The space-time Probabilists' Hermite polynomial $H_n(x,t)=t^{n/2}{\mathrm{He}}_n(x/\sqrt{t})$ extends polynomially to $t=0$ and solves ${\partial }_t{H}_n+\frac{1}{2}{\partial }_x^2{H}_n=0$. Therefore $H_n(B_t,t)$ is a martingale for a standard Brownian motion, by the Itô formula and Gaussian moments.

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.

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.

For a finite stopping time $T$, use the shifted filtration ${\mathcal{G}}_s={\mathcal{F}}_{T+s}$. An martingale with a bounded $L^2$ norm shifts to the martingale $M_{T+s}-M_T$; a continuous bounded adapted integrand shifts to a predictable integrand. Their quadratic variation and integration intervals shift by subtracting the value at $T$. Both integrals have identical left-endpoint sums, and the Itô isometry passes the equality to the limit. The starting jump at $T$ is excluded.

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.

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.

The stocks $S_t=e^{B_t-t/2}$ and $S_t^{\prime }=(1+S_t)/2$ start at one, are true square-integrable zero-rate martingales, and have perfect positive correlation at each positive time. Their bounded multiplicative volatilities are $1$ and $S_t/(1+S_t)$, which differ throughout. The additive intercept is what allows the same correlation but different stochastic-exponential coefficients.

For two strictly positive zero-rate stock martingales with bounded volatilities and the same initial value, proportional terminal prices must be equal because their expectations coincide. Conditional expectation then makes the whole paths equal. Comparing their stochastic integrals by the Itô isometry gives equality of the volatility coefficients almost everywhere in time and probability. Perfect positive correlation alone does not provide the required proportionality.

For an L-diffusion with bounded coefficients and $f\in C_b^{1,2}$, $M^f$ is a continuous martingale. Freeze the time argument along a deterministic partition, apply the spatial martingale problem on each interval, and pass to the limit by the dominated convergence theorem. The same proof works if the absolute drift coefficients and the trace of the diffusivity have integrable time integrals along the path on every finite horizon. Local hypotheses alone give a local martingale conclusion.

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.

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

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 $\mathbb{E}(M_N-a{)}^{-}/(b-a)$.

The usual conditions are completeness and right continuity of a filtration: ${\mathcal{F}}_0$ contains all subsets of null events in the ambient probability space, and ${\mathcal{F}}_t={\bigcap }_{s>t}{\mathcal{F}}_s$. They allow the standard continuous-time martingale, stopping time and stochastic integration theorems to be used without repeated augmentation qualifications. An absolute continuity of measures change preserves old null sets but can introduce new ones; the usual completion under the new measure may therefore be understood when needed.