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A diffusion remaining in $(-\pi /2,\pi /2)$ with the displayed equation is a bounded martingale and has a terminal limit. Its expected total quadratic variation is finite by its bounded second moments. An interior limit would leave its squared diffusion coefficient bounded away from zero and force infinite quadratic variation. Thus it converges to an endpoint, and the preserved mean determines the two endpoint probabilities.

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 $M_0\in L^2$.

For a semimartingale with càdlàg paths $X$ and $f\in C^2$, the Itô formula uses the left-limit gradient in $\int f^{\prime }(X_{s-})d{X}_s$, the continuous quadratic variation in its second-order integral, and the jump correction ${\sum }_{s\le t}[f(X_s)-f(X_{s-})-f^{\prime }(X_{s-})\Delta X_s]$. In several dimensions use the corresponding gradient, Hessian and continuous bracket matrix. The correction is the part of each jump not captured by first-order linearization.

For continuous local martingales, the total-variation process of their quadratic covariation satisfiesIt is the Cauchy-Schwarz inequality for the matrix-valued measure formed by their quadratic variations and quadratic covariation.

A continuous local martingale $M$ with $M_0=0$ is standard Brownian motion exactly when its quadratic variation satisfies $[M{]}_t=t$.

The characterization asserts that a continuous local martingale starting at zero with quadratic variation $t$ is Brownian motion.

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.

Local time records how much a stochastic process visits a level. Its normalization depends on the process: when the state space is a discrete space, it is occupation time at a state, while local time of a semimartingale is an occupation density weighted by quadratic variation. Brownian local time is a special case.

For Schramm–Loewner evolution with $\kappa =4$ and $Z_t=g_t(z)-2W_t$, the Itô formula for the complex logarithm gives $d\log Z_t=-2Z_t^{-1}d{W}_t$ before swallowing. The drift cancels exactly at this parameter. Its real part is $\log |Z_t|$ and its imaginary part is the SLE4 angle martingale. The real-part quadratic variation is $4\int X_t^2/|Z_t{|}^{4}dt$, which also gives a direct non-swallowing argument.

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.

By the usual definition of a Lévy process, $X_0=0$ almost surely. The two martingale assumptions giveFor a continuous Lévy process, the jump measure in the Lévy–Khintchine formula is zero, and the process has the form of a Brownian Gaussian component plus a deterministic drift. Thus its characteristic function isThe first moment makes $m=0$, and the second moment makes ${\sigma }^2=1$. ConsequentlyEquivalently, continuity and the martingale identity $X_t^2-t$ identify the quadratic variation as $t$, so the Lévy characterization of Brownian motion gives the same conclusion. A continuous nonzero drift is excluded by the mean-zero martingale assumption.

Here is the Brownian change-of-drift form of the Girsanov theorem. Let $W$ be a standard Brownian motion on $[0,T]$ and $\theta$ predictable with ${\int }_0^T{\theta }_s^{2}ds<\infty$ almost surely. Suppose the stochastic exponentialis a true martingale with $\mathbb{E}{Z}_T=1$. Define $dQ=Z_{T}dP$. Then $Q$ is equivalent to $P$ andThe same statement holds in $d$ dimensions with scalar products, squared norms and a vector drift. Novikov's condition $\mathbb{E}\exp (\frac{1}{2}{\int }_0^T{\theta }_s^{2}ds)<\infty$ is a standard sufficient condition for the true-martingale hypothesis. Bounded $\theta$ is enough for all applications below.

For the proof, Itô formula gives $d{Z}_t=-Z_t{\theta }_{t}d{W}_t$. The Itô product rule gives the cancellationThe two drift terms cancel because $d[Z,W{]}_t=-Z_t{\theta }_{t}dt$. Stop $W^Q$ when its absolute value reaches $n$, at a time ${\tau }_n$ capped by $T$. The same product calculation makes $Z_t{W}_{t\wedge {\tau }_n}^Q$ a local martingale under $P$. Its absolute value is at most $n{Z}_t$; the density martingale is uniformly integrable over this finite horizon, so the stopped product is a true martingale. Bayes' conditional identity then givesAs $n$ increases, these stopping times exhaust the horizon by continuity. Thus $W^Q$ is a continuous local martingale under $Q$. Adding a finite-variation drift does not change quadratic variation, so $[W^Q{]}_t=t$.

To prove the Brownian conclusion rather than just assert it, for every real $u$ apply Itô formula to $F_t=\exp (iu{W}_t^Q+u^{2}t/2)$. Its differential is $iu{F}_{t}d{W}_t^Q$, and its modulus is bounded on the fixed horizon, so it is a true martingale. ConsequentlyThe conditional characteristic function identifies a centered normal increment with variance $t-s$, independent of the past. Together with continuity and $W_0^Q=0$, this proves the Brownian assertion. For vector $u$, the same argument uses $\Vert u{\Vert }^2$ and gives the multivariate version. If $|\theta |\le K$, stopping $Z$ and applying Itô formula to $Z^2$ gives a uniform second-moment bound $\mathbb{E}{Z}_{t\wedge \tau }^2\le e^{K^{2}t}$ by Gronwall inequality; this proves uniform integrability of the stopped densities and the needed true-martingale property in the bounded case.

In the Black-Scholes model with its usual augmented Brownian filtration, write $d{S}_t=S_t(\alpha dt+\sigma d{W}_t)$, $B_t=e^{\rho t}$, with $\sigma >0$. Take $\theta =(\alpha -\rho )/\sigma$, the market price of risk. Under $Q$, the preceding change of drift givesso $S_t/B_t$ is a martingale. The Martingale representation theorem then makes integrable Brownian-market claims attainable and gives their risk-neutral pricing as discounted conditional expectations. The physical drift $\alpha$ has disappeared from the price dynamics.

For the joint terminal-value and maximum law, first use the Brownian reflection principle at level $a>0$. Reflecting a path after its first hit of $a$ preserves its law as a Brownian motion and sends a terminal value $y<a$ to $2a-y>a$; the symmetry and strong Markov property of the post-hit increments justify this map. Thus, with ${\phi }_t(y)=(2\pi t{)}^{-1/2}{e}^{-y^2/(2t)}$,Take the constant-drift exponential density $e^{\mu W_t-{\mu }^{2}t/2}$. By Girsanov theorem, the coordinate process under this tilted measure has the law of $W_s+\mu s$ under $P$. Consequently, for $t>0$ and $x\le a$,Completing the squares givesIntegrating provesHere $\Phi$ is the standard normal distribution function. The first variable is the terminal Brownian value, distinct from its running maximum.

Set $T=t_0$, $a=\log (b/S_0)/\sigma >0$ and $\mu =(\rho -{\sigma }^2/2)/\sigma$. Under the risk-neutral measure, $\log (S_s/S_0)/\sigma =W_s^Q+\mu s$. Taking $x=a$ in the joint law gives the probability of not hitting the upper level. Complementing it and discounting the unit payment at maturity gives the one-touch option priceThe exponential weight is also $(b/S_0{)}^{2\rho /{\sigma }^2-1}$. The payment occurs at $T$, even if the barrier was hit earlier, so its discount factor is $e^{-\rho T}$, not a hitting-time discount factor.

For an Itô process $d{X}_t=a_{t}dt+b_{t}d{W}_t$ and $F\in C^{1,2}$, the differential Itô formula iswhere the derivatives are evaluated at $(t,X_t)$. A multidimensional version replaces $b_t^2{F}_{xx}$ by the contraction of the Itô diffusion covariance matrix with the spatial Hessian.

For a heuristic derivation, partition time and use the Euler increment $\Delta X=a\Delta t+b\sqrt{\Delta t}Z$, where $Z$ is standard normal. A second-order Taylor expansion givesThe term $b^2(\Delta W{)}^2$ has order $\Delta t$, so it survives summation; ordinary first-order calculus would incorrectly discard it. For independent Brownian motion increments,This gives the quadratic variation $[W{]}_t=t$. The summed mixed $dtdW$ terms and $d{t}^2$ terms vanish, while the linear random increments converge to an Itô integral. After localizing bounded coefficients and derivatives, Taylor remainders are negligible and the surviving terms give the formula. Thus $(dW{)}^2=dt$ is shorthand for a statement about summed quadratic variation, not an equality of individual random increments.

The Itô formula decomposes the smooth transform into a continuous local martingale and a finite-variation process:If $f(M)$ has finite variation, then $N$ also has finite variation. A continuous finite-variation local martingale is constant, so $N=0$ andIt remains to show that the second-order term vanishes; the preceding observation alone does not establish this.

Let $Z=\{a:f^{\prime }(a)=0\}$. By the occupation-times formula for the local time of a semimartingale,Therefore $L_t^a(M)=0$ for almost every $a\notin Z$. Also $f^{\prime \prime }(a)=0$ for almost every $a\in Z$: if $f^{\prime }(a)=0$ and $f^{\prime \prime }(a)\ne 0$, continuity of $f^{\prime \prime }$ makes $f^{\prime }$ strictly monotone in a neighborhood of $a$, so that zero is isolated. Such isolated zeros form a countable set. Localizing to bounded paths and finite quadratic variation, then using the occupation-times formula once more, givesBoth terms in the Itô formula now vanish. Apply this on all rational times and use continuity to obtain the simultaneous conclusionThis is the finite-variation smooth transform of a continuous local martingale property.

The multidimensional Itô formula for continuous semimartingales states that a $C^2$ transformation has first-order stochastic and finite-variation integrals plus one-half its Hessian contracted with the quadratic covariations. A continuous finite-variation process has zero quadratic variation and zero quadratic covariation with a continuous local martingale. Thus in the present caseThe first integrand is continuous adapted, hence predictable, and locally bounded by stopping the two coordinate paths on exiting compact sets. The standard construction of the Itô integral then makes this integral a zero-starting continuous local martingale. The remaining integrals are pathwise Lebesgue-Stieltjes integrals. On each compact time interval their integrands are bounded along the continuous paths, so their total variations are bounded by a constant times $V_t(A)+[M{]}_t$, which is finite. Therefore the normalized terms areThe process $A^f$ is continuous adapted of finite variation and $A_0^f=0$. When $f(M_0,A_0)$ is integrable, $M^f$ is a local martingale in the definition of part (a). Uniqueness follows from part (a): the difference of two normalized decompositions is both a continuous local martingale and a continuous finite-variation process starting at zero, hence vanishes.

An initial-value qualification is necessary under that definition. The printed assumptions do not themselves guarantee $\mathbb{E}|f(M_0,A_0)|<\infty$. Take $M=0$, $A_t=G$ with $G$ a standard normal random variable known at time zero, and $f(x,a)=e^{a^2}$. All path assumptions hold, but $\mathbb{E}{e}^{G^2}=\infty$. Any decomposition with $A_0^f=0$ would force the nonintegrable value $M_0^f=e^{G^2}$, contradicting the local-martingale definition. Without an integrability hypothesis, the always-valid continuous semimartingale decomposition is $X=X_0+N+A^f$, with $N$ the zero-starting integral above. Alternatively one may use the convention that a local martingale permits an arbitrary finite initial value and localizes its centered increments. With deterministic initial data, as in part (c), there is no issue. This is initial-value integrability in normalized semimartingale decompositions.

Subtract $M_0$, which does not affect either the integral or its quadratic variation. Let $N=M-M_0$. Choose a localizing sequence ${\tau }_n$ for $N$. By local boundedness of the previsible process $H$, there are increasing stopping times ${\rho }_n\uparrow \infty$ and finite deterministic constants $K_n$ with $|H_s|\le K_n$ on $0<s\le {\rho }_n$. PutContinuity and finiteness on compact intervals imply ${\sigma }_n\uparrow \infty$ almost surely. The stopped process $N^{{\sigma }_n}$ is a bounded square-integrable martingale, its bracket is bounded by $n$, and its stopped integrand is bounded by $K_n$. The indicator $1_{\{s\le {\sigma }_n\}}$ is predictable, so the stopped stochastic integral identity is valid.

Fix $n$ and abbreviateThe Itô isometry gives $\mathbb{E}{J}_t^2=\mathbb{E}{Q}_t\le K_n^{2}n$, so $J$ is a square-integrable martingale. For $F\in {\mathcal{F}}_s$, apply the same isometry to the predictable integrand $1_F{1}_{(s,t]}H$, stopped at ${\sigma }_n$. This yieldsAlso $\mathbb{E}[J_t-J_s\mid {\mathcal{F}}_s]=0$. Expanding $J_t^2-J_s^2$ therefore proves that $J^2-Q$ is a true martingale. The process $Q$ is continuous adapted increasing and starts at zero. By the characterization of continuous martingale quadratic variation as its increasing square compensator, and uniqueness of an increasing square compensator, we have $[J]=Q$.

Stochastic integration and quadratic variation commute with stopping; for the latter this follows either from squared-increment sums or from the same square-compensator characterization. Hence the desired identity holds up to every ${\sigma }_n$. Since these times increase to infinity, they exhaust every finite time interval almost surely, and the continuous identities patch to giveThis is the localized isometry proof of stochastic-integral quadratic variation. No unstopped second-moment or bracket-integrability assumption has been made.

The one-dimensional Lévy characterization of Brownian motion states: a continuous adapted process $L$, with $L_0=0$, is standard Brownian motion relative to its filtration if and only if it is a local martingale and $[L{]}_t=t$ for all $t$ almost surely.

For necessity, the centered independent Gaussian increments of Brownian motion give both the martingale property and the martingale $L_t^2-t$. The uniqueness characterization of quadratic variation then gives $[L{]}_t=t$.

For sufficiency fix $\theta \in \mathbb{R}$. The Itô formula givesThe second-order term cancels the time derivative because $d[L{]}_t=dt$. Thus $E$ is a complex local martingale. Its modulus is the deterministic quantity $e^{{\theta }^{2}t/2}$, bounded on every finite horizon, so its real and imaginary parts are true martingales by the bounded local martingale criterion. For $s<t$,Uniqueness of characteristic functions says that the increment has normal distribution $N(0,t-s)$ independently of ${\mathcal{F}}_s$. Indeed multiplying the identity by any bounded ${\mathcal{F}}_s$-measurable variable gives the product characteristic-function identity expressing independence. Iterating over deterministic times gives independent increments. Together with continuity and $L_0=0$, these are precisely the defining properties of standard Brownian motion. If the starting value is a fixed $x$, the same argument identifies $L-x$ as standard Brownian motion.

Fix a sequence of deterministic partitions with mesh tending to zero and define the completed-increment sumsFor a continuous semimartingale, these sums converge to $[X{]}_t$ in uniform convergence on compacts in probability, the standard path-increment formula for quadratic variation. The sums use only the path of $X$, and do not mention a filtration. If the same process is a semimartingale for two filtrations, the same sums have both proposed quadratic variations as limits. Uniqueness of limits in probability, followed by continuity, makes these limits indistinguishable.

For the change of measure, let $R=d\mathbb{Q}/d\mathbb{P}$. If $\mathbb{P}(A_n)\to 0$, thentruncate $R$ at a large constant, use $\mathbb{E}[R{1}_{\{R>K\}}]\to 0$, and then let $\mathbb{P}(A_n)\to 0$. Apply this argument to the event ${\sup }_{s\le t}|S_s^{(n)}-[X{]}_s^{\mathbb{P}}|>\epsilon$. The same sums converge to the P-version of the bracket in Q-probability, and the Q-semimartingale formula makes them converge to the Q-version. ThereforeThis is quadratic variation under an absolutely continuous measure change. With only $\mathbb{Q}\ll \mathbb{P}$, the equality is asserted under Q: a Q-version is unconstrained on a Q-null set that might have positive P-probability.