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For an adapted process with right-continuous paths and a stopping time $T$, $X_t^T=X_{t\wedge T}$ is adapted. At fixed $t$, round $t\wedge T$ upward on finite partitions of $[0,t]$. Every approximating value is ${\mathcal{F}}_t$-measurable, and right continuity gives the limit. Completeness handles a null exceptional set if right continuity holds only almost surely. A martingale assumption and boundedness of $T$ are unnecessary.

A Stochastic process $X$ is adapted to a filtration $({\mathcal{F}}_t)$ when $X_t$ is ${\mathcal{F}}_t$-measurable at every time $t$. Its present value therefore depends only on information already available.

An announcing sequence consists of stopping times increasing to $T$ almost surely, strictly smaller than $T$ on $\{T>0\}$. Such a lifetime can be approached through stopped intervals on which a locally defined stochastic process is an ordinary adapted process. For the usual maximal local solution of a stochastic differential equation, the exit times ${\tau }_n$ from a nested compact exhaustion of the open domain, capped as $T_n={\tau }_n\wedge n$, supply this sequence. The existence of such a sequence is an extra lifetime convention; arbitrary stopping times need not admit one.

An adapted process with integrable coordinates is a martingale exactly when the displayed equality holds for every bounded stopping time. For the forward implication, expand $M_T-M_0={\sum }_{k=1}^N{1}_{\{T\ge k\}}(M_k-M_{k-1})$ and condition each term on ${\mathcal{F}}_{k-1}$. Conversely, for $A\in {\mathcal{F}}_n$ choose $T=n+1$ on $A$ and $T=n$ elsewhere. Subtract the equality for the constant stopping time $n$ to obtain $\mathbb{E}[1_A(M_{n+1}-M_n)]=0$, which is the defining conditional expectation identity.

A filtration $({\mathcal{F}}_t)$ is a Brownian filtration for $B$ when $B$ is an adapted process and each future increment $B_t-B_s$ is independent of ${\mathcal{F}}_s$. The natural filtration of a Brownian motion has this property. Arbitrary enlargement by future information need not preserve it.

Let an integrable process $(M_n)$ be adapted to $({\mathcal{F}}_n)$. It is a martingale if and only if $\mathbb{E}[M_{n\wedge \tau }]=\mathbb{E}[M_0]$ for every $n$ and every stopping time $\tau$. For the reverse implication, fix $A\in {\mathcal{F}}_n$ and take $\tau =n$ on $A$ and $\tau =n+1$ on $A^c$. Comparing its stopped expectation at $n+1$ with the expectation obtained from the deterministic stopping time $n+1$ gives $\mathbb{E}[1_A(M_{n+1}-M_n)]=0$. Since this holds for every $A\in {\mathcal{F}}_n$, it is exactly $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$.

For a continuous adapted process $X$ and $T=\inf \{s\ge 0:X_s\ge \lambda \}$,Continuity makes the supremum on $[0,t]$ attained and equals its supremum on this countable dense set. Each displayed event belongs to ${\mathcal{F}}_t$, so $T$ is a stopping time without requiring right-continuity of the filtration.

// Target: martingale.bigb

A counting process records accumulated events. It is a nonnegative integer-valued Stochastic process $N_t$, usually adapted and càdlàg, with $N_0=0$ and nondecreasing sample paths. Nonexplosion means $N_t<\infty$ at every finite time. In a simple counting process, every jump is one; broader counting conventions also allow several events at the same instant.

A rate-$\lambda$ Poisson process has independent stationary increments, mean $\lambda t$ and variance $\lambda t$. A one-event process $N_t=1_{\{T\le t\}}$, with $T>0$, instead records whether its single event has occurred. It is a counting process without the Poisson process's unrestricted repeated arrivals.

The path $N_t$ is the observed count. A counting-process intensity describes its conditional event rate, and a compensator accumulates that predictable rate or, more generally, a predictable event measure. These objects are not the same: the count has integer jumps, whereas a continuous compensator can increase between observed events. In Survival analysis, the one-event convention naturally combines a Hazard function with an at-risk 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.

For a positive function $g$ in the domain of a Markov jump-process generator, the generator martingale for $g(X_t)$ and semimartingale integration by parts give the displayed local martingale. Suitable finite-horizon deterministic bounds on $g$ and $Lg/g$ make it a true martingale. For a single marked exponential jump process, take $g(x)=e^{\theta x}$. Then the exponent's compensating process is $r_{\theta }(t\wedge T)$, where $r_{\theta }=(e^{\theta }-1)/\theta -1$ for $\theta \ne 0$ and $r_0=0$; it is continuous adapted and hence predictable.

For a left-continuous locally bounded adapted process $H$ and a continuous semimartingale $X$, the elementary predictable left-endpoint approximations tend pointwise to $H$. Decompose $X$ into a continuous local martingale and a continuous finite-variation process. After localization, dominated convergence with respect to the variation measure handles the finite-variation integral. The Itô isometry and Doob L2 maximal inequality handle the martingale integral. The omitted partial last interval is bounded by the local bound on $H$ times the modulus of continuity of $X$. This proves uniform convergence on compacts in probability and is the grid-sum foundation of the Itô product rule.

A locally bounded process can be made bounded by stopping it on an increasing sequence of random time intervals. For stochastic integration on positive times, one useful convention is that there exist stopping times ${\tau }_n\uparrow \infty$ almost surely and finite deterministic constants $K_n$ such thatThe interval convention matters at time zero: including zero additionally requires a deterministic bound on $H_0$, whereas the positive-time Stochastic integral does not use that value.

Every continuous adapted process with finite initial value has this positive-time localization. Set ${\tau }_n=\inf \{t\ge 0:|H_t|\ge n\}\wedge n$. If its initial magnitude already exceeds $n$, this Stopping time is zero and the indicated interval is empty. Otherwise continuity prevents overshoot at the first exit, so $K_n=n$ works. continuity on each compact time interval makes ${\tau }_n\to \infty$ pathwise.

A predictable locally bounded process is integrable against a semimartingale: the bounded stopped integrands define compatible integrals, which are joined by localization. Predictability supplies the required information condition; a magnitude bound alone does not replace it. For example, Brownian motion is not bounded on the whole infinite time axis but its continuous adapted paths give the stated localization.

// Target: probability-and-statistics.bigb

A continuous locally defined process is a pair $(X,T)$ specified on the stochastic interval $[0,T)$, with a lifetime stopping time $T$ approached by an announcing sequence for a stopping time $T_n$. Each Stopped process is an ordinary continuous adapted process. This is the convention used for continuous local differential equations; no value at $T$ is required. Restricting a globally defined process gives examples, but finite lifetimes also allow approach to a domain boundary or explosion.

A discrete-time martingale relative to a filtration $({\mathcal{F}}_n)$ is an adapted process $(M_n)$ with $\mathbb{E}|M_n|<\infty$ and $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$ almost surely. Given all information available at time $n$, the current value is the conditional prediction of the next value. A simple symmetric random walk is an example: its next independent increment has mean zero. The tower property implies $\mathbb{E}[M_m\mid {\mathcal{F}}_n]=M_n$ for $m\ge n$, hence a constant expected value. Constant unconditional means alone do not establish the conditional property. The choice of information is part of the definition.^[ref]

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.

Suppose $X_n\ge 0$ is an integrable adapted process, $Y_n\ge 0$ is finite and adapted, and $\mathbb{E}[X_{n+1}\mid {\mathcal{F}}_n]\le (1+Y_n)X_n$. The displayed correction is a nonnegative supermartingale: its next denominator is measurable at time $n$, and its reciprocal is at most one. If ${\sum }_n{Y}_n<\infty$ almost surely, the denominators converge to a finite positive limit since $\log (1+y)\le y$. The almost sure supermartingale convergence theorem then gives finite almost sure convergence of $X_n$. No integrability of the infinite product or of ${\sum }_n{Y}_n$ is needed.

On a probability space $(\Omega ,\mathcal{F},\mathbb{P})$, a standard one-dimensional Brownian motion is a real-valued stochastic process $B=(B_t{)}_{t\ge 0}$ with $B_0=0$ almost surely, continuous sample paths almost surely, and independent increments, such thatHere the second parameter is the variance. Thus its Brownian increments have the normal distribution with mean zero and variance equal to elapsed time. Equivalently, relative to a specified filtration $({\mathcal{F}}_t)$, require $B$ to be an adapted process and $B_t-B_s$ to be independent of ${\mathcal{F}}_s$ with this normal distribution. Taking the natural Brownian filtration recovers the first definition.

Put $M_t=B_t^3-3t{B}_t$. It is an adapted process and is integrable at every finite time, because normal random variables have finite absolute moments of every order. Fix $0\le s<t$ and let $D=B_t-B_s$. Then $D$ is independent of ${\mathcal{F}}_s$, has mean zero, variance $t-s$, and third moment zero. Expanding the cube inside the conditional expectation givesAlso $\mathbb{E}[B_t\mid {\mathcal{F}}_s]=B_s$. ThereforeThese are exactly the defining conditions for a martingale.

A process is a supermartingale relative to the given filtration if $X_n$ is ${\mathcal{F}}_n$-measurable and integrable for each $n$, andIteration gives the equivalent inequality $\mathbb{E}(X_m\mid {\mathcal{F}}_n)\le X_n$ for every $m\ge n$. adapted process and integrability are part of the definition, not consequences of the displayed inequality alone.

For the continuous adapted process $X$, the supremum on each compact time interval is attained. Let $D_t=(\mathbb{Q}\cap [0,t])\cup \{t\}$. Continuity gives ${\sup }_{s\le t}{X}_s={\sup }_{r\in D_t}{X}_r$, and henceEach event inside the union is in ${\mathcal{F}}_r\subseteq {\mathcal{F}}_t$, so this proves carefully that $T$ is a stopping time. Merely checking rational times for $X_r\ge \lambda$ would miss a path that touches the level at an isolated irrational time; the approximation from below avoids this issue. This is the closed-level hitting times of continuous adapted processes criterion. For $\lambda =0$, nonnegativity makes $T=0$ and the later inequality is immediate.

For $\lambda >0$, put $A=\{T\le t\}$ and $\tau =T\wedge t$. On $A$, continuity implies $X_{\tau }\ge \lambda$. To justify the optional sampling step without extra filtration assumptions, round $\tau$ upwards to a finite dyadic grid in $[0,t]$, obtaining stopping times ${\tau }_n\downarrow \tau$. The event $A$ belongs to ${\mathcal{F}}_{\tau }$, and hence to ${\mathcal{F}}_{{\tau }_n}$: directly, $A\cap \{\tau \le u\}=\{T\le u\}$ for $u<t$, while for $u\ge t$ it is $A$. Discrete optional sampling for the sampled submartingale givesAs $n\to \infty$, continuity gives $X_{{\tau }_n}\to X_{\tau }$. Nonnegativity and the Fatou lemma therefore yieldConsequently the Doob maximal inequality for a nonnegative submartingale takes the precise form

For the second-moment bound, suppose first that $\mathbb{E}{X}_t^2<\infty$; otherwise the inequality with an infinite right side is automatic. Set $M_R=X_t^{\ast }\wedge R$. The layer cake representation and the preceding tail estimate giveThe integrands are nonnegative, so the interchange is justified by the Tonelli theorem. Since $M_R$ is bounded, division is legitimate when its norm is nonzero, giving $\Vert M_R{\Vert }_2\le 2\Vert X_t{\Vert }_2$; the zero-norm case is immediate. Finally, monotone convergence as $R\to \infty$ givesThis truncated layer-cake proof of the L2 maximal inequality proves square integrability of the maximum rather than assuming it during the Cauchy-Schwarz inequality step.