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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.

Let $X=(X^1,\dots ,X^d)$ be a continuous semimartingale, with $X^j=X_0^j+M^j+A^j$, where $M^j$ is a zero-starting continuous local martingale and $A^j$ is a continuous finite-variation process starting at zero. For $f\in C^{1,2}([0,\infty )\times {\mathbb{R}}^d)$, the Itô formula isHere $[M^j,M^k]$ is the quadratic covariation; the finite-variation process parts contribute no such term. The integrals against the finite-variation processes are understood pathwise, and those against the continuous local martingales are Itô integrals after localization.

For $d$-dimensional Brownian motion with independent standard coordinates, $d[B^j,B^k{]}_s={\delta }_{jk}ds$. In particular, if $f\in C^2$ is independent of time,where $\Delta$ is the Laplace operator. The same formulas hold up to exit from an open domain on which $f$ is smooth.

Let $N$ be the zero-starting continuous local martingale, and let $V_t$ be the total variation of its path on $[0,t]$. Stop when $|N|$ or $V$ reaches $n$, and also at time $n$. The stopped process is bounded, hence a true square-integrable martingale by the bounded local martingale criterion, and its total variation is bounded by $n$.

For a deterministic partition $0=t_0<\cdots <t_m=t$, martingale increments are orthogonal in $L^2$, soThe sum inside the expectation is at most the largest absolute increment times the stopped total variation. As the mesh tends to zero, path continuity makes the largest increment tend to zero. The product is bounded by $2n^2$, so the dominated convergence theorem makes the displayed expectation zero. Thus the stopped process vanishes at every fixed time. Use rational times, continuity and ${\tau }_n\uparrow \infty$ to concludeThis proves that a continuous finite-variation local martingale is constant.

Write a normalized semimartingale decomposition as $X=X_0+N+A$, with both $N_0=A_0=0$, $N$ a continuous local martingale and $A$ of finite variation. The difference of two such martingale parts is also the difference of two finite-variation parts, and hence vanishes by the result just proved. The other parts coincide too. The normalized continuous semimartingale decomposition is unique. Fixing the initial constants is necessary: without normalization a constant can be transferred between the two parts.

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.

For continuous semimartingales $U,V$, the semimartingale integration by parts formula isIndeed each partition increment satisfiesSumming telescopes the left side. The first two sums converge to the stochastic integrals by left-endpoint approximation, and the last sum converges to the quadratic covariation by polarization of the quadratic-increment formula. All limits are in uniform convergence on compacts in probability. Hence the equality holds at every time outside one null set, using continuity. In differential notation it is the Itô product rule $d(UV)=UdV+VdU+d[U,V]$.

For a vector of continuous semimartingales $X=(X^1,\dots ,X^d)$ and $F\in C^{1,2}$, the multidimensional Itô formula isOne route from the product formula is to apply it repeatedly to powers and products of the coordinates, obtaining this expression for polynomials; polarization identifies the mixed second derivatives. Time is another coordinate of finite variation, so it has no quadratic-covariation contribution. Localize to compact ranges, approximate a $C^{1,2}$ function and its indicated derivatives by smooth polynomial approximations, and pass to the limit using stochastic-integral continuity and the variation bounds on the finite-variation terms. Equivalently, second-order Taylor expansion along partitions gives the same limit, with integration by parts accounting for every quadratic term.

For every zero-starting continuous local martingale $M$, there is a unique, up to indistinguishability, continuous adapted increasing process $[M]$, with $[M{]}_0=0$, such thatThis is its quadratic variation. It also satisfies, for deterministic partitions whose mesh tends to zero,with convergence uniformly on compact time intervals in probability. The theorem includes existence and finiteness on each compact interval; monotonicity is understood pathwise outside one null set.

Uniqueness follows because the difference of two candidate increasing continuous processes is both a finite-variation process and a continuous local martingale, hence is identically zero from its zero initial value. For a continuous semimartingale $X=X_0+M+A$, its quadratic variation is $[X]=[M]$, since continuous finite-variation terms have zero quadratic variation. Polarization defines the quadratic covariation by $[M,N]=([M+N]-[M-N])/4$.

Interpret the otherwise undefined $b_k(t)$ here as the moment ${\beta }_k(t)$ defined above. For $k=1$, the process is $B_t$, and for $k=2$ it is $B_t^2-t$; both are martingales.

For $k\ge 3$, the Itô formula and the derivative of the Brownian moment recursion yieldThe stochastic integral is a true martingale on finite horizons, since the required $2k-2$ moment is integrable in time. The continuous finite-variation term is not identically zero. If it were, its continuous derivative would vanish at every time, whereas at a fixed $t>0$, the nondegenerate Gaussian $B_t$ does not almost surely satisfy the polynomial equation $B_t^{k-2}={\beta }_{k-2}(t)$. Uniqueness of continuous semimartingale decomposition therefore rules out even the local martingale property. HenceFor example $\mathbb{E}(B_t^3\mid {\mathcal{F}}_s)=B_s^3+3(t-s)B_s$, so merely subtracting its zero expectation cannot make the cubic power a martingale. The centered Brownian powers need not be martingales distinction is resolved by subtracting the random drift compensator $k(k-1){\int }_0^t{B}_s^{k-2}ds/2$ instead of the deterministic mean.

For $x\le 0$, $f_n(x)=|x|$; for $x>0$, only the interval $(0,1/n)$ contributes an error to the integral of the derivative. HenceBy the Itô formula, the continuous processesare nondecreasing, since $f_n^{\prime \prime }\ge 0$. Parts (a) and (b), together with the uniform approximation of absolute value, show that $C^{(n)}$ converges uniformly on compacts in probability toThe limit is continuous: the martingale integral is continuous, and the bounded-integrand finite-variation integral is continuous because $A$ is continuous. For any rational $s<t$, convergence in probability and $C_t^{(n)}-C_s^{(n)}\ge 0$ imply $Z_t-Z_s\ge 0$ almost surely. Intersect these countably many events and use continuity to extend to all real $s<t$. ThusSince $|X|={\mathrm{sgn}}_{-}(X)\cdot M+({\mathrm{sgn}}_{-}(X)\cdot A+Z)$ is a continuous local martingale plus continuous finite variation, $|X|$ is a continuous semimartingale. This is the smooth convex approximation proof of the Tanaka formula. With the specified sign value $-1$ at zero, $Z$ is the right local time at zero, rather than an unspecified symmetric normalization.

For an arbitrary continuous semimartingale starting at zero, use its continuous decomposition $X=M+A$, with both parts starting at zero, and letThe continuity and local finiteness of variation imply ${\rho }_k\uparrow \infty$ almost surely. Up to ${\rho }_k$ the required deterministic bound holds, and the stopped local martingale is bounded, hence a true $L^2$-bounded martingale. Apply the previous argument to $X^{{\rho }_k}$ for each integer $k$. Stopping the defining integrals gives precisely $Z_{t\wedge {\rho }_k}$. On the intersection of their full-probability events, each stopped $Z$ is nondecreasing; every finite interval is eventually contained before ${\rho }_k$, so $Z$ is nondecreasing globally. Therefore

Define the quadratic covariation of two continuous semimartingales by polarization:The available quadratic variation convergence theorem and the polarization identity imply that the sums of products of simultaneous grid increments converge in uniform convergence on compacts in probability to $[X,Y]$. The quadratic covariation is a continuous finite-variation process, vanishing at time zero.

For a grid ending at $t_n=h_n\lfloor t/h_n\rfloor$, the elementary algebraic identitytelescopes. The first two sums converge by part (a), with the continuous processes $X,Y$ as integrands; continuous paths are left-continuous and locally bounded. The last sum converges to the quadratic covariation by the preceding polarization argument. Finally, $X_{t_n}{Y}_{t_n}\to X_t{Y}_t$ locally uniformly by continuity. HenceThis is the Itô product rule, or stochastic integration by parts. Uniqueness of the limit in uniform convergence on compacts in probability, followed by continuity, makes the identity hold simultaneously for all $t$ outside one null set. The proof used finite-grid algebra, part (a), and the permitted quadratic variation theorem; it did not use the Itô formula.

Use the local convention for the martingale problem, which is appropriate for arbitrary measurable coefficients. A continuous adapted process $X$ solves $\mathbf{M}(a,b)$ if the coefficient integrals are locally finite and, for every $f\in C_c^{\infty }(\mathbb{R})$,is a local martingale. Here locally finite means ${\int }_0^T(|a(X_s)|+|b(X_s)|)ds<\infty$ almost surely for each $T<\infty$. Suitable localization upgrades the identities to true martingale identities. With bounded coefficients and bounded test derivatives, this distinction disappears on finite horizons.

A Markov diffusion generated by $L$ is a time-homogeneous Markov process with continuous paths whose infinitesimal generator on the stated test domain is $L$. Its transition operators satisfy the generator limit, and Dynkin formula gives precisely the preceding martingale problem identities. Some terminology uses L-diffusion for the test-function identities alone; the Markov property is an additional assertion unless uniqueness supplies it.

Suppose now $a={\sigma }^2>0$. Use smooth compactly supported functions agreeing with $x$ and $x^2$ on $[-R,R]$, and stop before exiting this interval. Removing these stops shows thatis a continuous local martingale. Consequently $X$ is a continuous semimartingale. The martingale problem applied to the second cutoff function says thatis a local martingale. The Itô formula for $X^2$ also says thatis a local martingale. Subtracting, the continuous finite-variation process $[N{]}_t-{\int }_0^{t}a(X_s)ds$ is a local martingale and starts at zero. The theorem that a continuous finite-variation local martingale is constant givesDefineThe integrand is predictable and is stochastically integrable, since its integral against the bracket is exactly $t$. Thus $B$ is a continuous local martingale with $B_0=0$ and $[B{]}_t=t$. The Lévy characterization of Brownian motion states that these conditions make $B$ a Brownian motion relative to the given filtration. Associativity of stochastic integrals recovers $N=\int \sigma (X)dB$, and thereforeThe original process is a weak solution of a stochastic differential equation, with its driving Brownian motion constructed on the original space.

Conversely, the Itô formula applied to any weak solution with locally integrable coefficients gives the martingale problem identities. A Markov diffusion with generator $L$ gives them by Dynkin formula. A well-posed martingale problem, with the usual measurable path-space setup, identifies a unique family of diffusion laws and yields the Strong Markov property. Without uniqueness, the test-function identities alone do not assert that every chosen solution is Markov. This specifies both the equivalence with weak equations and the role of uniqueness in identifying a diffusion.

For a continuous semimartingale $X$, its quadratic variation is the continuous increasing zero-starting process obtained as the limit, uniformly on compact time intervals in probability, of sums of squared increments along deterministic partitions whose mesh tends to zero. If $X=X_0+N+V$ is its continuous local-martingale/finite-variation decomposition, then $[X]=[N]$; finite-variation terms and their cross sums vanish.

Write $D=d\mathbb{Q}/d\mathbb{P}$. If $\mathbb{P}(E_n)\to 0$, then for every $R>0$,First send $n$ to infinity and then $R$ to infinity. Thus convergence in probability under $\mathbb{P}$ implies convergence in probability under $\mathbb{Q}$, also for a supremum on a compact time interval. The same squared-increment sums converge under $\mathbb{Q}$ to the $\mathbb{P}$ version of $[X]$. Since $X$ is also a semimartingale under $\mathbb{Q}$, these sums converge to its $\mathbb{Q}$ quadratic variation. Uniqueness of limits in probability gives equality at every fixed time; continuity and countably many rational times giveAbsolute continuity suffices; equivalence is not required. This is quadratic variation under an absolutely continuous measure change.

Use just the parameters $1$ and $-1$. Positivity and adaptation of their local martingales giveThus both processes are adapted. A positive continuous local martingale has a semimartingale logarithm, by Itô formula after localization away from zero. Since $X=\log Z^{(1)}+A/2$ and $A$ has finite variation, $X$ is a continuous semimartingale. Write $X=N+V$, where $N$ is a zero-starting continuous local martingale and $V$ is continuous finite variation, also starting from zero.

Apply Itô formula to each exponential:Because $Z^{(\theta )}$ is itself a local martingale, the finite-variation term is zero by part (a). Divide its finite signed measure by the strictly positive $Z^{(\theta )}$. For $\theta =1,-1$, respectively,Adding and subtracting show $V=0$ and $[N]=A$. Therefore the exponential criterion for a continuous local martingale and its bracket givesThe semimartingale property was established before applying Itô's formula to $X$; it was not assumed from the outset.

The relation between $B$ and $W$ is the linear equation $d{B}_s=d{W}_s-\beta B_{s}ds$ with $B_0=0$. Multiplication by the deterministic integrating factor $e^{\beta s}$ and the Itô product rule giveHenceThe second equality follows by semimartingale integration by parts. It expresses $B$ as an explicit function of the continuous path of $W$, using only an ordinary time integral. This inversion holds before changing measure; $W$ is already a continuous semimartingale under the original measure.

If $Q\ll P$ and a continuous semimartingale is a semimartingale under both measures, its quadratic variations agree $Q$-indistinguishably. The same squared-increment sums converge uniformly on compacts in probability under both measures: absolute continuity transfers the original convergence, and uniqueness of the limit identifies the two continuous versions. Equivalence of measures is unnecessary.