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 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.

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.