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A continuous semimartingale is a continuous adapted process admitting a decomposition $X=X_0+N+V$, with $N$ a zero-starting continuous local martingale and $V$ continuous adapted finite variation. The decomposition is unique because a continuous finite-variation local martingale starting at zero vanishes. Its quadratic variation is $[X]=[N]$.

For a Radon-Nikodym derivative $L_T\ge 0$ with $\mathbb{E}{L}_T=1$, its conditional-expectation process is a nonnegative martingale. If $L_T>0$ almost surely, it defines an equivalent probability measure. A stochastic exponential satisfying the Novikov condition is a common positive density process.

For bounded measurable $a,b$, with $a$ symmetric positive semidefinite, an L-diffusion is a continuous adapted process $X$ such that $g(X_t)-g(X_0)-{\int }_0^{t}Lg(X_s)ds$ is a true martingale for every $g\in C_b^2$. The general local martingale problem may instead use compactly supported test functions and require only a local martingale. Specify the test class and the true or local convention when coefficients are unbounded.

In the one-factor Heath-Jarrow-Morton model, put $b_t^T={\int }_t^T\sigma (t,u)du$. The Itô formula gives $dP/P=r_{t}dt-b_t^{T}d{W}_t$. Thus multiplying by the discount factor yields the displayed stochastic exponential. Bounded forward volatility on a finite maturity horizon implies the Novikov condition, so the discounted bond is a true martingale.

A multiplicative factor converting a value into units of a chosen initial account. With the continuous-time bank account $B$, $D_t=1/B_t$; discounted traded asset prices are martingales under the corresponding equivalent martingale measure.

For a martingale with increments in $\{-1,0,1\}$, European call option prices satisfy $C(T+1,K)-C(T,K)=\frac{1}{2}{\sigma }^2(T,K)(C(T,K+1)-2C(T,K)+C(T,K-1))$. At positive-probability states, the variance and the upward/downward conditional transitions are determined by this identity; transitions at zero-probability states cannot be recovered.

For integer paths with increments in $\{-1,0,1\}$, the positive part obeys a telescoping identity with slope $f(x)=1_{\{x>0\}}+\frac{1}{2}{1}_{\{x=0\}}$ and a correction $\frac{1}{2}{1}_{\{S_{t-1}=K\}}(\Delta S_t{)}^2$ at the kink. For a martingale, the slope term is a martingale transform.

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.

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 $f$.

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.

A diffusion remaining in $(-\pi /2,\pi /2)$ with the displayed equation is a bounded martingale and has a terminal limit. Its expected total quadratic variation is finite by its bounded second moments. An interior limit would leave its squared diffusion coefficient bounded away from zero and force infinite quadratic variation. Thus it converges to an endpoint, and the preserved mean determines the two endpoint probabilities.

For a random walk with independent and identically distributed random variables as increments and finite moment-generating function at $\theta$, put $\kappa (\theta )=\log \mathbb{E}{e}^{\theta X_1}$. Then $e^{\theta S_n-n\kappa (\theta )}$ is a martingale by conditioning on the next independent increment. In particular, if $r>0$ and $\mathbb{E}{r}^{X_1}=1$, then $r^{S_n}$ is a martingale. Stopping at a finite interval exit can make this process bounded, allowing the dominated convergence theorem to justify passage from bounded stopping times to the exit time.

For independent and identically distributed random variables with law $N(-1,1)$, $e^{2S_n}$ is a nonnegative martingale, since the increment's moment-generating function equals one at parameter two. The Doob maximal inequality gives $\mathbb{P}(Z>t)\le e^{-2t}$ for the all-time maximum $Z\ge 0$. Integrating the tail gives the displayed bound. For $b\le 0$, $e^{bZ}\le 1$. The strong law of large numbers guarantees finiteness and attainment of the maximum because $S_n\to -\infty$.

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.

With zero interest and zero risk premium, a manager whose current wealth lies between common-tangent contacts $\ell ,h$ can improve expected utility maximization through a fair lottery paying those two values. The probability of $h$ is $(w_0-\ell )/(h-\ell )$. In a Brownian filtration, replicating a bounded terminal lottery gives a nonnegative wealth martingale throughout.

For bounded $V$ and bounded initial data $f$, a sufficiently regular classical solution bounded on every finite time slab satisfiesTo prove this, the Itô formula and Itô product rule make $u(T-t,B_t)e^{{\int }_0^{t}V(B_s)ds}$ a local martingale. Its deterministic bound on $[0,T]$ makes it a true martingale; taking expectations at the two endpoints gives the formula. This also gives uniqueness in the finite-horizon bounded class. The exponent has the same sign as the potential in the equation.

On a compact interval, squared increments of a continuous finite variation process are bounded by its modulus of continuity at the mesh size times its total variation, and therefore vanish. The Cauchy-Schwarz inequality bounds the mixed increment sum by the square root of the product of the two squared-increment sums. The local-martingale sum is bounded in probability, so the mixed term also vanishes.

For the bounded rate diffusion of linear bond pricing in a bounded short-rate diffusion, $D_t{e}^{-(T-t)}{r}_t$ is a bounded martingale with terminal value $D_T{r}_T$. Dividing its conditional expectation by $D_{t}P(t,T)$ through the Bayes formula for conditional expectation gives the displayed forward-measure expectation. It lies between zero and one.

A standardized maturity-and-quantity agreement whose quote changes generate margin payments as gains and losses are marked to market. A newly settled contract has zero value, while its quoted delivery price is generally nonzero. In the continuous-settlement idealization, futures pricing uses a pricing-measure martingale quote.