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For independent Brownian motions $W,B$, the Itô formula cancels the drift after the arctangent transform. Normalizing the two noise coefficients gives a continuous local martingale $Z$ with quadratic variation $t$, hence another Brownian motion by the Lévy characterization of Brownian motion.

For a zero-start continuous local martingale $M$ with $[M{]}_{\infty }\le C$ deterministically, its stochastic exponential is a uniformly integrable martingale. For each $p>1$, ${\sup }_t\mathbb{E}\mathcal{E}(M{)}_t^p\le e^{p(p-1)C/2}$. Its terminal value is strictly positive, so it defines an equivalent infinite-horizon change of measure by the Girsanov theorem.

For orthogonal continuous local martingales $B,\vartheta$ that are standard Brownian motions, the real part $X=e^B\cos \vartheta$ and imaginary part $Y=e^B\sin \vartheta$ are continuous local martingales. The Itô formula gives $dX=XdB-Yd\vartheta$ and $dY=YdB+Xd\vartheta$, since the two diagonal second-order terms cancel. Their quadratic variations equal ${\int }_0^t{e}^{2B_s}ds$ and their quadratic covariation is zero. If $C=[B,\vartheta ]$ is nonzero, the omitted drift terms are respectively $-YdC$ and $XdC$, so the orthogonality hypothesis is essential.

A continuous local martingale $M$ with $M_0=0$ and deterministically bounded quadratic variation on every finite horizon is a square-integrable martingale, with $\mathbb{E}{M}_t^2=\mathbb{E}[M{]}_t$. Localization, the Doob L2 maximal inequality, and the resulting integrable bound on the squared supremum justify removing the localizing sequence.

A continuous semimartingale is a continuous adapted process admitting a decomposition $X=X_0+N+V$, with $N$ a zero-starting continuous local martingale and $V$ continuous adapted finite variation. The decomposition is unique because a continuous finite-variation local martingale starting at zero vanishes. Its quadratic variation is $[X]=[N]$.

Along the dyadic partitions of an interval of $[0,1]$, the sums ${\sum }_i|{\Delta }_{i}M{|}^p$ of a continuous local martingale tend to zero in probability for $p>2$. If $0<p<2$ and these sums are eventually bounded almost surely, then $M$ is constant on $[0,1]$: compare them with the sums defining quadratic variation, using the maximum increment, which tends to zero by uniform continuity.

If $M$ is a continuous local martingale and $f\in C^2(\mathbb{R})$, then finite variation of $f(M)$ forces $f(M_t)=f(M_0)$. The Itô formula and the quadratic variation of a stochastic integral give ${\int }_0^t{f}^{\prime }(M_s{)}^{2}d[M{]}_s=0$. The occupation-times formula implies that $L_t^a(M)=0$ almost everywhere where $f^{\prime }(a)\ne 0$. On the zero set of $f^{\prime }$, its derivative $f^{\prime \prime }$ vanishes almost everywhere: zeros at which $f^{\prime \prime }\ne 0$ are isolated and hence countable. Thus ${\int }_0^t|f^{\prime \prime }(M_s)|d[M{]}_s=\int |f^{\prime \prime }(a)|L_t^a(M)da=0$. Both terms in the Itô formula vanish.

Let $X$ be a continuous local martingale with $X_0=0$ and bracket $A=[X]$. On each horizon under consideration, assume $\mathbb{E}{\sup }_{s\le t}|X_s{|}^4<\infty$ and $\mathbb{E}{A}_t^2<\infty$. The Itô formula makes $X^4-6X^{2}A+3A^2$ a local martingale; these moment assumptions give an integrable running supremum and hence a true martingale. In particular,If additionally $\mathbb{E}{X}_t^2=t$ and $\mathrm{Cov}(X_t^2,A_t)=0$, the Itô isometry gives $\mathbb{E}{A}_t=t$ and thereforeIf the fourth moment equals $3t^2$ at every time under these assumptions, then $A_t=t$ on all rational times almost surely, hence at all times by continuity. The Lévy characterization of Brownian motion identifies $X$ as Brownian motion. Equality of fourth moments without the covariance hypothesis does not give this argument.

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 zero-starting continuous local martingale $X$, the stochastic exponential satisfiesThe Holder inequality and the supermartingale property of the first exponential bound its stopped $p$th moment by $(\mathbb{E}{e}^{C(p,q)X_T}{)}^{(q-1)/q}$. The subtraction in the coefficient is $\sqrt{pq}-1$, not a square root of $pq-1$.

For a continuous local martingale with infinite terminal quadratic variation, the Dambis-Dubins-Schwarz theorem represents $M_t-M_0=W_{[M{]}_t}$. Its continuous clock traverses all nonnegative times. Standard one-dimensional Brownian motion has unbounded positive and negative oscillations, by the Brownian reflection principle or recurrence of one-dimensional Brownian motion. Continuity bounds every path on compact time intervals, so these large values occur at arbitrarily late times. Pulling them back through the clock proves the displayed conclusion almost surely, and in particular ${\mathrm{limsup}}_{t\to \infty }|M_t|=\infty$.

With the local martingale definition requiring each stopped process itself to be a martingale, the initial value is integrable. A normalized continuous semimartingale decomposition $X=M+A$ with $A_0=0$ can therefore have a local-martingale part only if $X_0$ is integrable. A smooth transformation need not preserve this condition: take $M=0$, $A_t=G$ for an initially known standard normal random variable, and $f(x,a)=e^{a^2}$. Then $f(M_0,A_0)$ is finite almost surely but has infinite expectation. The always-valid decomposition is instead $X=X_0+N+V$ with a zero-starting continuous local martingale $N$ and a continuous finite-variation process $V$ starting at zero.

For a continuous local martingale starting at zero, finite expected terminal quadratic variation is equivalent to being an L2-bounded continuous martingale. Stop at bounded path and bracket levels and use $\mathbb{E}(M_t^{\tau }{)}^2=\mathbb{E}[M{]}_{t\wedge \tau }$. Uniform integrability removes the localization; Doob L2 maximal inequality justifies convergence of squares. The terminal identity is $\mathbb{E}{M}_{\infty }^2=\mathbb{E}[M{]}_{\infty }$. For nonzero initial values one must additionally assume $M_0\in L^2$.

This is an exponential-moment criterion for the stochastic exponential to be a true martingale. In the infinite-horizon stopped-moment form, a zero-starting convergent continuous local martingale satisfying ${\sup }_T\mathbb{E}{e}^{M_T/2}<\infty$ has a uniformly integrable martingale $\mathcal{E}(M)$. The half-threshold for the exponential-martingale Hölder bound first controls strict scalings. The terminal scaling inequality for stochastic exponentials and terminal expectation criterion for a nonnegative local martingale then include the endpoint scaling.

For a left-continuous locally bounded adapted process $H$ and a continuous semimartingale $X$, the elementary predictable left-endpoint approximations tend pointwise to $H$. Decompose $X$ into a continuous local martingale and a continuous finite-variation process. After localization, dominated convergence with respect to the variation measure handles the finite-variation integral. The Itô isometry and Doob L2 maximal inequality handle the martingale integral. The omitted partial last interval is bounded by the local bound on $H$ times the modulus of continuity of $X$. This proves uniform convergence on compacts in probability and is the grid-sum foundation of the Itô product rule.

A continuous local martingale $M$ with $M_0=0$ is standard Brownian motion in its given filtration if and only if its quadratic variation satisfies $[M{]}_t=t$ for every $t\ge 0$, simultaneously almost surely.

For the nontrivial direction, the Itô formula makes $\exp (i\theta M_t+{\theta }^{2}t/2)$ a complex local martingale. Its absolute value is deterministically bounded on each finite horizon, so it is a true martingale. Conditioning at $s<t$ givesThe conditional characteristic-function criterion for Brownian increments identifies a centered normal distribution independent of ${\mathcal{F}}_s$. This turns a bracket computation into a proof of the complete Brownian motion law.

A scale-invariant law of unparameterized chords from zero to infinity in the complex upper half-plane is local when a boundary-preserving conformal map between neighborhoods of zero, fixing zero, transports its curve stopped on leaving the first neighborhood to the same-law curve stopped on leaving the second. Equality concerns the stopped curves up to increasing reparameterization. For SLE with parameter six, the transformed SLE driving function is a continuous local martingale with bracket six times the image capacity clock. The Lévy characterization of Brownian motion makes that driver Brownian in the new clock, proving the local stopped-curve invariance.

Stop a continuous local martingale at increasing level-and-time stopping times to obtain bounded martingales. Their limits of discrete quadratic variation sums agree before the earlier stopping time, because the sums commute exactly with stopping. These continuous limits patch into a continuous adapted nondecreasing process. On each compact interval, the chance that the stopping time occurs early tends to zero, giving uniform convergence on compacts in probability for the original sums.

For a continuous local martingale and a locally bounded predictable process, stop at a localizing time, an integrand bound, a time bound, and level bounds for the centered martingale and its bracket. The stopped martingale is bounded and its bracket and integrand are deterministically bounded. The Itô isometry, applied to $1_F{1}_{(s,t]}H$ for $F\in {\mathcal{F}}_s$, gives the conditional second-moment identity. Expanding the square proves that $(H\cdot M{)}^2-H^2\cdot [M]$ is a martingale after stopping. Uniqueness of an increasing square compensator identifies the bracket, and increasing stopping times patch the identities into the displayed local result.

Started at radius one, planar Brownian motion avoids the origin, and its logarithmic radius is a continuous local martingale because the logarithmic potential is harmonic away from zero. It is a strict local martingale: at deterministic $t>0$, the angular average of a logarithmic potential gives $\mathbb{E}\log |B_t|={\int }_1^{\infty }{e}^{-r^2/(2t)}dr/r>0$, whereas the initial value is zero. Absolute integrability follows from integrability of $r|\log r|$ near zero and the Gaussian tail. Its positive expectation does not contradict nonnegative-local-martingale bounds, since this process takes both signs.