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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.

A Compensator of a counting process $N$ is a predictable increasing process $\Lambda$, normalized by ${\Lambda }_0=0$, such that $N-\Lambda$ is a local martingale. Under the standard local-integrability and usual-filtration conditions, the predictable compensator is unique up to indistinguishability. It describes accumulated conditional event rate, not an event probability.

For a rate-$\lambda$ Poisson process, ${\Lambda }_t=\lambda t$ and $N_t-\lambda t$ is a martingale. If $T$ is exponential of rate $\lambda$ and $N_t=1_{\{T\le t\}}$ counts a single event, its predictable intensity is $\lambda 1_{\{N_{t-}=0\}}$ andThe rate shuts off after the event. This pathwise accumulated intensity can exceed one even before an event occurs; it differs from the deterministic probability $\mathbb{P}(T\le t)=1-e^{-\lambda t}$.

A compensator need not be absolutely continuous in time. The deterministic count $N_t=1_{\{t\ge 1\}}$ has compensator $\Lambda =N$, because its jump time is known in advance and $N-\Lambda =0$. No ordinary Rate function integrated against $dt$ produces that step. When ${\Lambda }_t={\int }_0^t{\lambda }_{s}ds$, the predictable derivative ${\lambda }_s$ is the counting-process intensity. For an observed survival event under independent censoring it takes the familiar form ${\lambda }_s=Y_s{h}_s$.

In discrete time, let $V$ be a supermartingale and let $\sigma \le \tau$ be bounded stopping times. Then the stopped values are integrable andFor a martingale, equality holds. The same bounded-time statement holds for a càdlàg supermartingale under the usual conditions for a filtration. In discrete time, expand stopped increments and use that $1_{\{\tau \ge k\}}$ is known at time $k-1$; continuous time follows by approximation and the path regularity.

Almost-sure finiteness of $\tau$ alone is insufficient. A simple symmetric random walk started at zero hits one almost surely, but its value there is one rather than its initial expectation zero. Conditions such as uniform integrability of a stopped uniformly integrable martingale, or bounded-increment optional stopping with integrable time, justify appropriate unbounded-time versions.

A precise continuous Optional time-change theorem is the following. Let $M$ be a continuous local martingale with $M_0=0$ for a filtration satisfying the usual conditions. Let $({\tau }_s{)}_{s\ge 0}$ be a family of random times: every ${\tau }_s$ is a finite Stopping time, and its sample paths in $s$ are continuous and nondecreasing, with ${\tau }_0=0$. Thenis a continuous local martingale for the time-changed filtration ${\mathcal{G}}_s={\mathcal{F}}_{{\tau }_s}$. The stopping-time sigma-algebra, rather than the deterministic-time sigma-algebra at $s$, describes the information available after the clock has advanced to ${\tau }_s$.

For a proof, localize $M$ by increasing stopping times $R_n$ so that $M^{R_n}$ is bounded. Optional sampling makes $M_{{\tau }_s}^{R_n}$ a bounded martingale for $\mathcal{G}$. The new-clock stopping times $S_n=\inf \{s:{\tau }_s\ge R_n\}$ tend to infinity, because each ${\tau }_s$ is finite and $R_n\to \infty$. continuity of the clock identifies the process stopped at $S_n$ with the corresponding old-clock Stopped process. These are localizers for $N$.

The quadratic variation transforms by $⟨N{⟩}_s=⟨M{⟩}_{{\tau }_s}$. Applying the same argument to $M^2-⟨M⟩$ proves the identity. A Finite-variation process remains of finite variation on each compact new-time interval, since its variation is bounded by the old variation up to the finite time ${\tau }_s$.

For example, the deterministic clock ${\tau }_s=cs$, $c\ge 0$, turns standard Brownian motion into a Gaussian martingale of variance $cs$. General random time substitutions require the stopping-time and filtration conditions; path continuity alone establishes continuity, not the martingale claim.

The optional stopping theorem for a right-continuous martingale $M$ on a filtered probability space satisfying the usual conditions for a filtration states that, for bounded stopping times $\sigma \le \tau$,For a submartingale the conditional equality becomes $\mathbb{E}[M_{\tau }\mid {\mathcal{F}}_{\sigma }]\ge M_{\sigma }$, and for a supermartingale it becomes the reverse inequality.

For a uniformly integrable martingale, boundedness of the stopping times can be removed, with the terminal limit used if a stopping time is infinite. A useful application is to a bounded stopped martingale $M_{t\wedge \tau }$ with $\tau <\infty$ almost surely: apply the bounded-time theorem first and then the bounded convergence theorem to obtain $\mathbb{E}{M}_{\tau }=\mathbb{E}{M}_0$. Finiteness of $\tau$ alone is not a sufficient hypothesis for an arbitrary martingale.

The existence and pathwise uniqueness theorem for a stochastic differential equation states that locally Lipschitz drift $b$ and diffusion $\sigma$ on an open domain give a pathwise unique strong solution of a stochastic differential equation up to a maximal lifetime $\xi$. The initial value is measurable with respect to the initial sigma-algebra, and the filtration satisfies the usual conditions for a filtration. On ${\mathbb{R}}^d$, if $T_n$ are exits from increasing balls, then $T_n\uparrow \xi$; a finite lifetime entails escape from every compact subset. The coefficients need not satisfy a global growth bound for this local result.

One construction truncates the coefficients outside each compact region to globally Lipschitz coefficients, solves the resulting global equations, and patches their solutions until the exits. Pathwise uniqueness makes solutions from larger truncations agree on the smaller stopped interval. Their union is the maximal local solution of a stochastic differential equation, and any extension would contradict the exhaustion of the domain by the exit times.

Let $D=d\mathbb{Q}/d\mathbb{P}$ on $\mathcal{F}$. The Radon-Nikodym derivative is nonnegative and ${\mathbb{E}}_{\mathbb{P}}D=1$. For $A\in {\mathcal{F}}_t$,Uniqueness of the restricted Radon-Nikodym derivative therefore identifiesFor $s\le t$, the tower property of conditional expectation gives ${\mathbb{E}}_{\mathbb{P}}(Z_t\mid {\mathcal{F}}_s)=Z_s$, and ${\mathbb{E}}_{\mathbb{P}}{Z}_t=1$. Thus $Z$ is a nonnegative true martingale, indeed a uniformly integrable martingale, since it consists of conditional expectations of one integrable random variable. This is the Radon-Nikodym density martingale. Under the usual conditions for a filtration, it has the standard càdlàg version; continuity is the additional assumption in part (c), not a consequence of absolute continuity alone.

Use the standard càdlàg martingale versions and usual conditions for a filtration where optional sampling and semimartingale decompositions are invoked.

Fix deterministic $0\le s<t$ and $A\in {\mathcal{F}}_s$. The pasting ordered stopping times argument shows that $U=s{1}_A+t{1}_{A^c}$ is a bounded stopping time. The given stopped-expectation property, applied to $U$ and to the deterministic stopping time $t$, yieldsThis holds for every $A\in {\mathcal{F}}_s$. Since $X_s$ is ${\mathcal{F}}_s$-measurable and both time values are integrable, it is exactly the defining test for conditional expectation:Hence $X$ is a martingale. This proof of the characterization of a martingale by bounded continuous-time stopped expectations uses only deterministic two-time pastings; the càdlàg assumption and the usual conditions for a filtration are stronger than needed for this implication.

A continuous martingale is an adapted process with almost surely continuous paths, $\mathbb{E}|M_t|<\infty$ for every $t$, andA continuous local martingale is continuous and adapted and has a localizing sequence of stopping times $T_n\uparrow \infty$ almost surely such that every $M_t^{T_n}=M_{t\wedge T_n}$ is a true martingale. The initial value is normally integrable; the frequently used class ${\mathcal{M}}_{c,\mathrm{loc}}$ is normalized to start at zero.

The usual conditions for a filtration are completeness and right continuity. Completeness means that ${\mathcal{F}}_0$ contains every subset of each $\mathbb{P}$-null set in the ambient $\mathcal{F}$; right continuity means ${\mathcal{F}}_t={\bigcap }_{u>t}{\mathcal{F}}_u$. These conditions concern the filtered probability space, whereas continuity of a martingale concerns its sample paths.

Work on a filtered probability space with the usual conditions for a filtration. A continuous martingale belongs to the class of L2-bounded continuous martingales whenThis means a uniform bound over the entire time interval, not merely being square-integrable at each individual time. The L2 martingale convergence theorem gives a terminal variable $M_{\infty }$ and almost sure convergence and convergence in the Lebesgue space $L^2$; moreover $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$ by conditional expectation.

The predictable sigma-algebra $\mathcal{P}$ on $\Omega \times [0,\infty )$ is the smallest sigma-algebra making every left-continuous adapted process measurable. Equivalently, it is generated bywhere the sample coordinate comes first. A previsible process is precisely a process measurable for this predictable sigma-algebra. Values at time zero matter for its definition, although they make no contribution to integration against a continuous quadratic variation starting at zero.

The raw natural filtration isEquivalently it is the smallest sigma-algebra making every observation $B_u$, $u\le t$, measurable. As $s\le t$ enlarges the set of observations, ${\mathcal{F}}_s^B\subseteq {\mathcal{F}}_t^B$. It is the smallest filtration to which the process is adapted. If the usual conditions for a filtration are imposed, take the completion by the probability-zero sets and its right-continuous augmentation. Brownian independent increments remain valid for this usual augmentation. Adding arbitrary future information is not part of the generated filtration.

Let an integrable adapted reward process $Z$ be dominated by a supermartingale $V$ over a finite discrete horizon. For every bounded stopping time $\tau$ within that horizon, the optional stopping theorem givesWhen $V_0$ is a deterministic constant, the latter bound is $\mathbb{E}{Z}_{\tau }\le V_0$. An expectation is required on the right when $V_0$ is random. The Snell envelope is the least such supermartingale majorant and turns this upper bound into the optimal expected reward. For continuous time, use a càdlàg supermartingale under the usual conditions for a filtration and ensure integrability of the stopped reward.