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An adapted càdlàg process is a semimartingale exactly when it is a good integrator: on each finite horizon, for every sequence of bounded elementary predictable processes with stopping-time intervals $H^n$ whose deterministic uniform bounds tend to zero, the elementary terminal integrals $(H^n\cdot X{)}_T$ tend to zero in probability. The elementary integral is a finite sum of measurable coefficients times subsequent increments; the characterization says precisely when this operation extends continuously to stochastic integration. The theorem is used here as a standard characterization, not proved from first principles.

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 Càdlàg process has sample paths that are càdlàg functions: they are right-continuous and have a finite left limit at every positive time.

Let $X$ be an integrable adapted càdlàg process under the usual conditions for a filtration. Then $X$ is a martingale if and only if $X_T$ is integrable and $\mathbb{E}{X}_T=\mathbb{E}{X}_0$ for every bounded stopping time $T$.

The forward implication is the optional stopping theorem. Conversely, fix deterministic $s<t$ and $A\in {\mathcal{F}}_s$, and use the stopping time $U=s{1}_A+t{1}_{A^c}$. Comparing $\mathbb{E}{X}_U$ with $\mathbb{E}{X}_t$ gives $\mathbb{E}[(X_t-X_s)1_A]=0$. Since this holds for every such $A$, it is the defining conditional expectation identity of a martingale. The converse uses only integrable adapted coordinates; the stated path assumptions justify the forward assertion at all bounded random times.

Let $X$ be an adapted Càdlàg process on a finite state space, and let $Q$ be a conservative transition-rate matrix. It solves the martingale problem for $Q$ when, for every real function $f$ on the state space,is a martingale for the specified filtration. On a finite state space all such test functions and their generator values are bounded.

A Continuous-time Markov chain with generator $Q$ has this property by Dynkin's formula. Conversely, fix a terminal time $t$ and put $u(s,x)=(e^{(t-s)Q}f)(x)$ for $0\le s\le t$. The backward equation gives ${\partial }_{s}u+Qu=0$. Applying the martingale identities to its finitely many time-dependent coefficients shows $u(s,X_s)$ is a martingale. ThusTesting state indicators recovers the entire conditional transition law, proving the Markov property and identifying its semigroup. The càdlàg and filtration assumptions make this a path-process characterization, rather than a statement only about separately fixed-time versions.

Relative to a filtration satisfying the usual conditions, a local martingale is an adapted càdlàg process $X$ for which there exist increasing stopping times $S_n\uparrow \infty$ almost surely such that every stopped process $X_t^{S_n}=X_{t\wedge S_n}$ is an integrable martingale. Thus, for $s\le t$,The sequence is a localizing sequence. A continuous local martingale additionally has continuous sample paths. The usual definition entails integrability of the initial value; local martingales need not be integrable at later times without stopping.

Let $\mathcal{P}$ be the predictable sigma-algebra on $\Omega \times [0,\infty )$, generated by $A\times \{0\}$ with $A\in {\mathcal{F}}_0$ and $A\times (s,u]$ with $A\in {\mathcal{F}}_s$. A stochastic process $H$ is previsible precisely when $(\omega ,t)\mapsto H_t(\omega )$ is $\mathcal{P}$-measurable.

Every left-continuous adapted process is previsible. Indeed, on intervals $(k{2}^{-n},(k+1)2^{-n}]$, put $H_t^{(n)}=H_{k{2}^{-n}}$, and put $H_0^{(n)}=H_0$. Each approximation is a predictable process, since its coefficient is ${\mathcal{F}}_{k{2}^{-n}}$-measurable; left continuity gives $H_t^{(n)}\to H_t$ for every $t$, so measurability passes to the limit.

An adapted càdlàg process need not be previsible. For a concrete example, let $\xi$ take values $0,1$ with equal probabilities, let ${\mathcal{F}}_t$ be trivial for $t<1$ and ${\mathcal{F}}_t=\sigma (\xi )$ for $t\ge 1$, and set $H_t=\xi 1_{[1,\infty )}(t)$. This is an adapted càdlàg process, but $H_1=\xi$ is not ${\mathcal{F}}_{1-}$-measurable, where ${\mathcal{F}}_{1-}=\sigma ({\bigcup }_{s<1}{\mathcal{F}}_s)$. Sections at time $1$ of the generators of $\mathcal{P}$ all belong to ${\mathcal{F}}_{1-}$, so every predictable process has an ${\mathcal{F}}_{1-}$-measurable value there; this proves the counterexample directly.

A càdlàg process $A$ is a finite-variation process when, almost surely, for every $T<\infty$,where the supremum is over every finite partition of an interval $0=t_0<\cdots <t_n=T$.

Uniform convergence on compacts in probability of $X^n$ to $X$ means that, for every $T<\infty$ and $\epsilon >0$,