= Optional time-change theorem
A precise continuous <Optional time-change theorem> is the following. Let $M$ be a <continuous local martingale> with $M_0=0$ for a <filtration (probability theory)>[filtration] satisfying the <usual conditions for a filtration>[usual conditions]. Let $(\tau_s)_{s\geq0}$ 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 $\tau_0=0$. Then
$$
N_s=M_{\tau_s}
$$
is 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 time>[stopping times] $R_n$ so that $M^{R_n}$ is bounded. <optional stopping theorem>[Optional sampling] makes $M^{R_n}_{\tau_s}$ a bounded <martingale> for $\mathcal G$. The new-clock <stopping time>[stopping times] $S_n=\inf\{s:\tau_s\geq R_n\}$ 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 $\langle N\rangle_s=\langle M\rangle_{\tau_s}$. Applying the same argument to $M^2-\langle M\rangle$ 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 $\tau_s=cs$, $c\geq0$, turns standard <Brownian motion> into a Gaussian <martingale> of <variance> $cs$. General random time substitutions require the stopping-time and <filtration (probability theory)>[filtration] conditions; path <continuity> alone establishes <continuity>, not the <martingale> claim.
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