Dambis-Dubins-Schwarz theorem

Let $M$ be a continuous local martingale with $M_0=0$ and $[M{]}_{\infty }=\infty$. Forthe process $B_s=M_{{\tau }_s}$ is Brownian motion andWhen $[M{]}_{\infty }<\infty$, one can enlarge the probability space and continue $B$ independently beyond that time.

For independent standard Brownian motions $B,W$, Itô formula gives $dY=e^W\cos BdW-e^W\sin BdB$: the two second-order drifts cancel. Independence gives zero cross quadratic covariation, so the displayed bracket follows. The process itself is not Brownian because its bracket is random. Its inverse-bracket time change is Brownian motion starting at one, in the time-changed filtration; subtracting one gives standard Brownian motion. The Divergence of a driftless Brownian exponential clock guarantees that the inverse is finite at every new time.

Let $M$ be a zero-starting continuous local martingale and let $T<\infty$ be a stopping time with $[M{]}_{t\wedge T}\le a$ and $[M{]}_T=a$ for a deterministic $a$. The complex exponential $\exp (iu{M}_{t\wedge T}+u^2[M{]}_{t\wedge T}/2)$ is a bounded local martingale, hence a true martingale. Its terminal expectation is one, giving the characteristic function $\mathbb{E}{e}^{iu{M}_T}=e^{-u^{2}a/2}$. Conditional optional sampling between two bracket levels also gives independent Gaussian increments in inverse-bracket time; Gaussian one-time marginals alone do not establish that independence.

For a continuous local martingale $M$ with divergent bracket $K$ and a continuous drift $D$ satisfying $|D_t-D_s|\le C(K_t-K_s)$, the process $y+M+D$, $|y|<1$, exits $(-1,1)$ almost surely. In unit bracket time a Brownian increment of absolute size greater than $C+2$ forces exit within one unit, giving a geometric survival bound. The inverse bracket clock transfers this finite exit to ordinary time.

// Target: probability-and-statistics.bigb

If a continuous vector local martingale has bracket matrix $A_{t}I$ with $A_{\infty }=\infty$, inverse-clock time change gives bracket matrix $rI$. The Lévy characterization of multidimensional Brownian motion then gives Brownian motion in the common time-changed filtration. A continuous local martingale is constant on flat intervals of its bracket, so inverses using $A_t\ge r$ and $A_t>r$ give the same martingale values. The filtration at the earlier inverse is smaller; adaptation and independence from the larger past transfer the Brownian property to it. Nonzero initial values are retained by adding the initial vector back.

A continuous local martingale bounded above pathwise throughout a stochastic time interval cannot have infinite quadratic variation at that interval's endpoint. The Dambis-Dubins-Schwarz theorem represents it as Brownian motion at its quadratic-variation clock; an infinite clock would force unbounded oscillations. Its clock therefore has a finite limit, and so does the martingale. A bound below gives the same conclusion by changing sign. The bound may be random; it must hold over the entire interval.

Strict increase of a bracket does not force an infinite terminal value. Up to $L=⟨M{⟩}_{\infty }$, the inverse-clock martingale continues at a finite lifetime by its terminal limit, with bracket $u\wedge L$. On an independent product extension, add ${\int }_0^u{1}_{\{s>L\}}d{\beta }_s$. Its bracket fills $(u-L{)}^{+}$ and its cross variation with the original part vanishes. The result is Brownian for all clock times and still represents $M_t$.

For a continuous strictly increasing unbounded bracket $V$, the inverse times $T_u=\inf \{t:V_t>u\}$ satisfy $V_{T_u}=u$ and $T_{V_t}=t$. Optional sampling of the original martingale stopped at bounded bracket levels makes $M_{T_u}$ a continuous local martingale in ${\mathcal{F}}_{T_u}$. Time-changing the square-minus-bracket martingale gives its bracket $u$; the Lévy characterization of Brownian motion identifies it as Brownian.

If a continuous local martingale satisfiesfor a nonnegative predictable process $A$, then, after enlarging the probability space if necessary, there is a Brownian motion $B$ such thatAn independent Brownian motion supplies noise on the set where $A=0$.

The Brownian motion $B$ in $M_t=B_{[M{]}_t}$ is generally constructed from $M$, so it need not be independent of the quadratic variation $[M]$. For example, $M_t=(W_t^2-t)/2={\int }_0^t{W}_{s}d{W}_s$ has clock ${\int }_0^t{W}_s^{2}ds$, and their dependence can be detected by a nonzero mixed moment.