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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.

Interpret $f$ as a conformal bijection onto $D^{\prime }$. If it is merely an injective map into a larger target, the correct exit domain is $f(D)$ instead. Let $\zeta$ be the exit time from $D$, write $f=u+iv$, and identify planar Brownian motion with $B^1+i{B}^2$. The Cauchy-Riemann equations give $u_x=v_y$, $u_y=-v_x$, and both $u,v$ are harmonic functions. The Itô formula, localized on compact subsets of $D$, yieldsThus both coordinates are continuous local martingales, withSince $f^{\prime }$ never vanishes, $A$ is strictly increasing before $\zeta$. Let ${\eta }_s=\inf \{t:A_t>s\}$, defined for $s<A_{\zeta -}$. The optional time-change theorem gives continuous local martingale coordinates for ${\beta }_s=f(B_{{\eta }_s})$, with bracket matrix $sI$. The Lévy characterization of multidimensional Brownian motion makes $\beta$ a standard planar Brownian motion started at $f(z)$, up to its lifetime. Therefore conformal invariance of planar Brownian motion takes the formTo identify the lifetime, exhaust $D$ by relatively compact open sets $D_n$ with nested closures. The images $f(D_n)$ exhaust $D^{\prime }$ because $f$ is a homeomorphism onto $D^{\prime }$. Each stopped transformed path exits $f(D_n)$ at precisely the clock value of its exit from $D_n$. Increasing these exit times gives the exit lifetime of Brownian motion from $D^{\prime }$, including a possibly infinite lifetime. This identifies $A_{\zeta -}$ with that lifetime and needs no global extension of $f$ to the boundary.

The Itô formula for $f(t,x)=e^{x+at}$ gives the stochastic differential equationWrite $N_t={\int }_0^t{Z}_{u}d{B}_u$ for its continuous local martingale part. Its clock is the quadratic variationBecause $Z_t>0$, $A$ is strictly increasing. For $s<A_{\infty }$, its inverse ${\tau }_s$ satisfies $A_{{\tau }_s}=s$ and $d{\tau }_s/ds=Z_{{\tau }_s}^{-2}$. The Dambis-Dubins-Schwarz theorem, or the Lévy characterization of Brownian motion after the optional time-change theorem, makes ${\beta }_s=N_{{\tau }_s}$ a Brownian motion for the time-changed filtration. With $Y_s=Z_{{\tau }_s}$,This is the Bessel process equation with dimension $\boxed{2+2a}$, on the positive stochastic interval. This establishes the Exponential Brownian-to-Bessel time change.

The lifetime convention is important: the inverse is defined only for $s<A_{\infty }$. If $a<0$, the strong law for Brownian motion gives $B_t/t\to 0$, so $Z_t\to 0$ exponentially and $A_{\infty }<\infty$. Then $Y_s\to 0$ as $s\uparrow A_{\infty }$, and this is its first hit of zero after continuous extension. If $a>0$ the clock diverges by the same strong law. If $a=0$ it diverges by infinite occupation time of one-dimensional Brownian motion on a bounded interval, since $e^{2B_t}$ is bounded below there. Thus the lifetime is infinite when $a\ge 0$. Negative dimensions are interpreted by the same positive-state equation killed at zero; no post-zero Bessel process behavior is asserted.

The strictly increasing continuous clock $[M]$ has a finite continuous inverse ${\tau }_s$. The time change of a continuous process therefore preserves continuity and adaptedness. Moreover,The optional time-change theorem makes $M_{{\tau }_s}$ a continuous local martingale for ${\tilde{\mathcal{F}}}_s={\mathcal{F}}_{{\tau }_s}$, while composition with the increasing map $s\mapsto {\tau }_s$ preserves the finite variation of $A$. Hence $\tilde{X}$ is a continuous semimartingale in the time-changed filtration.

A time-changed filtration records the information available after a new clock has reached an old time. Let $({\mathcal{F}}_t)$ be a filtration and let $({\tau }_s)$ be a nondecreasing family of finite stopping times. DefineThe second expression is the stopping-time sigma-algebra. If $s\le u$ and $A\in {\mathcal{F}}_{{\tau }_s}$, thenThus ${\mathcal{G}}_s\subseteq {\mathcal{G}}_u$, as required for a filtration.

For the deterministic clock ${\tau }_s=2s$, the new information is ${\mathcal{F}}_{2s}$, not ${\mathcal{F}}_s$. A continuous adapted process sampled at these stopping times is adapted to the new filtration. Preservation of the local martingale property requires further hypotheses such as those in the optional time-change theorem; merely substituting arbitrary random times does not establish it.