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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.

For càdlàg processes on a finite horizon, compact containment together with uniformly small increments over pairs of bounded stopping times whose separation tends to zero is a sufficient criterion for tightness in the Skorokhod J1 topology. The increment condition is in probability, not merely at deterministic times. If in addition the largest jump tends to zero in probability, every limiting law is supported on continuous paths.

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.

A bounded stopping time is a stopping time bounded almost surely by a deterministic finite constant. In discrete time, a bound $T\le N$ makes $M_T$ integrable whenever $M_0,\dots ,M_N$ are integrable random variables, and the optional stopping theorem applies to a martingale without additional limiting hypotheses. Almost-sure finiteness alone is weaker than boundedness.

For a domain $D$, the Brownian exit time is $T_D=\inf \{t\ge 0:B_t\notin D\}$. It is a stopping time, and it has finite expectation when $D$ is bounded.

Let $\tau$ be the first attainment of the maximum of standard Brownian motion on $[0,T]$, with $T>0$. Time reversal shows $\tau <T$ with probability one. Then $Y_t\le 0$ for $0\le t\le T-\tau$, an initial interval of positive length. Standard Brownian motion has probability zero of this property by the Brownian reflection principle, so $Y$ is not Brownian. Moreover $\tau$ is not a stopping time for the natural Brownian filtration: for $0<t<T$, $\mathbb{P}(\tau \le t\mid {\mathcal{F}}_t)=2\Phi ((M_t-B_t)/\sqrt{T-t})-1$, which is strictly between zero and one on $\{B_t<0\}$. The Strong Markov property is therefore inapplicable at this random time.

Reflecting a Brownian motion $B$ after a stopping time $T$ by ${\hat{B}}_t=2B_T-B_t$ for $t>T$, and leaving it unchanged for $t\le T$, preserves its path probability distribution. For finite $T$, the Strong Markov property and symmetry of the future Brownian motion prove this. When $T=\infty$, the path is left unchanged.

For every $p>0$ there are constants $c_p,C_p>0$ such that a continuous local martingale $M$ with $M_0=0$ satisfiesfor every stopping time $T$ for which the quantities are finite.

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.

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 $\mathbb{E}[M_{n+1}\mid {\mathcal{F}}_n]=M_n$.

The random time at which credit default occurs. In an observed default model it is a stopping time for the market filtration.

An elementary predictable process may use finitely many ordered stopping times ${\tau }_j$, with bounded ${\mathcal{F}}_{{\tau }_j}$-measurable coefficients $h_j$ on $({\tau }_j,{\tau }_{j+1}]$. Its elementary stochastic integral is ${\sum }_j{h}_j(X_{t\wedge {\tau }_{j+1}}-X_{t\wedge {\tau }_j})$. The stopped indicators are left-continuous and adapted, so the process is predictable; common refinement shows the sum is independent of its representation. Deterministic endpoints recover the usual simple predictable process. These integrands give the precise good-integrator test in the Bichteler-Dellacherie theorem.

Suppose a nonnegative stopping time $T$ and constants $L>0$, $0<q\le 1$ satisfy $\mathbb{P}(T\le nL+L\mid {\mathcal{F}}_{nL})\ge q$ on $\{T>nL\}$. Take $L$ to be a positive integer in discrete time. ThenOn the surviving event, conditional survival is at most $1-q$. The tower property of conditional expectation therefore givesInduction proves the geometric bound. Partitioning the tail integral into intervals of length $L$ then gives $\mathbb{E}T\le L{\sum }_{n\ge 0}\mathbb{P}(T>nL)\le L/q$; in discrete time, group the tail sum into $L$ successive integer indices instead. This proves finite expected hitting times when every surviving state has a uniform positive chance of escaping within a fixed time. It works in discrete or continuous time without assuming independence of successive survival events.

A one-dimensional Brownian motion spends an infinite total time in every nonempty open interval almost surely. To prove this for $(-1,1)$ after first hitting zero, alternate exits from that interval with returns to zero. By recurrence of one-dimensional Brownian motion all these stopping times are finite, and the Strong Markov property makes the successive exit durations independent and identically distributed random variables that are strictly positive. Some fixed positive duration is exceeded with positive probability, so the Borel-Cantelli lemma makes the sum of the durations infinite. Translate and scale for other intervals. In particular, if a nonnegative function is bounded below by a positive constant on such an interval, its infinite-horizon Brownian occupation time integral diverges. Recurrent visits alone, without this duration argument, would not prove divergence of an integral.

Let $X_n$ be integrable independent and identically distributed random variables, and let $T\ge 1$ be a stopping time for their natural filtration, with $\mathbb{E}T<\infty$. Since $\{T\ge n\}$ is measurable before $X_n$ is observed, independence and the Tonelli theorem give $\mathbb{E}|X_T|\le {\sum }_n\mathbb{E}[|X_n|1_{\{T\ge n\}}]=\mathbb{E}|X_1|\mathbb{E}T$. The event $T=n$ need not be independent of $X_n$, so the selected increment can be biased.

An $L^2$-bounded real martingale has a square-integrable terminal value, converges to it almost surely and in $L^2$, and satisfies $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$. The $L^2$ Cauchy property follows from orthogonality of increments and monotonicity of $\mathbb{E}{M}_t^2$. This terminal representation permits optional sampling at finite unbounded stopping times and uniform-integrability control of stopped squares by conditional Jensen.

A localizing sequence for a local martingale $M$ is an increasing sequence of stopping times $T_n\uparrow \infty$ almost surely such that every stopped process $M^{T_n}$ is a true martingale. One can often choose the times so that the stopped processes are bounded.

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 local supermartingale becomes a supermartingale after stopping along an increasing sequence of stopping times tending to infinity. Localization allows an Itô formula comparison before global integrability is established. Additional lower bounds or uniform integrability are needed to pass from the local comparison to an unrestricted expectation inequality.