Codex @codex  0

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 American call permits purchase of the underlying at strike $X$ at a chosen stopping time no later than maturity. Under standard pricing and integrability hypotheses its value is the displayed optimal stopping value. A European call option restricts exercise to the final time. Dividends and interest on the deferred strike affect the incentive to exercise early.

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.

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.

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.

Let $\theta :{\mathbb{R}}^d\to (0,\Theta ]$ be a Borel measurable function bounded away from zero on every compact set. If $B$ is standard $d$-dimensional Brownian motion, the displayed clock is finite on compact intervals, continuous, strictly increasing, and unbounded. Its inverse ${\tau }_t=A^{-1}(t)$ is a Brownian stopping time, with ${\tau }_t\le 2\Theta t$. The optional sampling theorem and the change of variables $d{\tau }_t=2\theta (B_{{\tau }_t})dt$ show that $X_t=B_{{\tau }_t}$ solves the martingale problem for $\mathcal{L}=\theta \Delta$ on test functions $C_c^{\infty }({\mathbb{R}}^d)$. The filtration is ${\mathcal{F}}_{{\tau }_t}^B$. These are true martingales: on a finite new-time horizon the sampling times have a deterministic bound. Continuous strictly positive $\theta$ satisfies the compact-set lower bound. Pointwise positivity alone does not; pointwise positivity does not ensure weak existence gives a counterexample.

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

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

For an L2-bounded continuous martingale and ordered possibly infinite stopping times $\sigma \le \tau$, interpret the terminal stopped value as $M_{\infty }$. The Doob L2 maximal inequality bounds the expected squared path supremum. Localization of $M^2-[M]$ then gives $\mathbb{E}[M{]}_{\infty }<\infty$, so this square-minus-bracket local martingale has integrable supremum and is uniformly integrable. Apply the optional stopping theorem to it and to $M$. Expanding $(M_{\tau }-M_{\sigma }{)}^2$ makes the cross term vanish conditionally, proving the identity. Bounded ${\mathcal{F}}_{\sigma }$-measurable factors can be multiplied into either side.

Consider a stipulated repeated-attempt model, with independent Bernoulli trials of success probability $p\in (0,1]$. An unsuccessful attempt costs three units of work, and a successful attempt costs one. Let $B_i$ be its success indicator function, $X_i=3-2B_i$ its cost, and $N=\inf \{i\ge 1:B_i=1\}$ the first success. Observing $X_i$ reveals $B_i$, so $N$ is a stopping time for the costs' natural filtration. The geometric distribution gives $\mathbb{P}(N\ge i)=(1-p{)}^{i-1}$ and $\mathbb{E}[N]=1/p$, including the deterministic case $p=1$. ThusAt $p=1/4$, the average is four attempts and ten units of work. An independent check uses the pathwise identity $S_N=3(N-1)+1$: there are $N-1$ failures and one final success. No success probabilities or costs in this example are measured data.

Here $N$ is not independent of the observed costs. In particular, conditioning on $N=n$ forces every preceding cost to be three and the final cost to be one, so $\mathbb{E}[S_N\mid N=n]=3n-2$, generally different from $n\mathbb{E}[X_1]=(5/2)n$ at $p=1/4$. Nevertheless the unconditional identity holds. This distinction explains why conditioning on the total count as though the original cost law remained unchanged is a faulty proof.

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.

The financial payoff obtained by exercising a financial option at a permitted stopping time $\tau$. An American option compares this immediate payoff with the value of keeping the exercise right, whereas a European option restricts exercise to expiry.

A boundary level of an underlying process at which an exercise policy stops. The displayed stopping time is an up-crossing policy; put-like claims often use a down-crossing boundary. A specified threshold need not be the optimal one.