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Use the standard convention that the finite-variation process $X$ has càdlàg paths and that the stochastic integral starts from zero. Write $V(X)$ for its total-variation process. The necessary admissibility condition is ${\int }_{(0,t]}|H_s|d{V}_s(X)<\infty$ almost surely for each finite $t$; it holds, for example, for locally bounded $H$. Define the pathwise Lebesgue-Stieltjes integralAn atom at $s>0$ contributes $H_s\Delta X_s$, so this convention also fixes the treatment of jumps.

Fix $t$. For a generator $H_s=1_A{1}_{(a,b]}(s)$ of the predictable sigma-algebra, with $A\in {\mathcal{F}}_a$, the Lebesgue-Stieltjes integral isIt vanishes when $t\le a$ and is ${\mathcal{F}}_t$-measurable otherwise. The total-variation process is adapted: for a càdlàg path its value at $t$ is the supremum over a countable collection of partitions of an interval using rational interior points and endpoint $t$. Thus the positive and negative Lebesgue-Stieltjes measures $d{X}^{\pm }=(dV(X)\pm dX)/2$ have ${\mathcal{F}}_t$-measurable masses on the generator intervals.

The Monotone class theorem, applied separately to these positive random measures on $[0,t]$, extends this measurability from the generators of the predictable sigma-algebra to all nonnegative predictable processes. Truncation and positive/negative decomposition then cover every admissible $H$. Consequently $H\cdot X$ is adapted.

The integrability condition is essential. Taken literally, an arbitrary previsible process cannot always be integrated against a finite-variation process: with $X_t=t$ and deterministic $H_s=s^{-1}$ for $s>0$, $H_0=0$, the proposed value at every $t>0$ diverges. The usual càdlàg convention is also what identifies $dX$ by $dX((a,b])=X_b-X_a$.

The Itô formula applied to $F(t,x)=e^{t/2}\cos x$ gives exact cancellation of the drift:On every finite interval $[0,T]$, the Itô integral is square-integrable since $\mathbb{E}{\int }_0^T{e}^s{\sin }^2{B}_{s}ds\le e^T-1$. Therefore $e^{t/2}\cos B_t$ is a true martingale, and hence a local martingale.

For the other stochastic process, $d(B_t-t^2)=d{B}_t-2tdt$. If $B_t-t^2$ were a local martingale, its difference with $B$ would make the continuous finite-variation process $-t^2$ a local martingale, contradicting the fact that a continuous finite-variation local martingale is constant. Also $\mathbb{E}(B_t-t^2)=-t^2$ is not constant. Thus $B_t-t^2$ is neither a local martingale nor a martingale.

The usual quadratic variation convention includes $⟨M{⟩}_0=0$. If $A,C$ are two continuous finite-variation processes with $A_0=C_0=0$ such that $M^2-A$ and $M^2-C$ are continuous local martingales, then $A-C$ is a continuous local martingale of finite variation. It must be constant, and its initial value is zero, so $A=C$ up to indistinguishability of stochastic processes.

Here is a proof of the fact used. Subtract the initial value of a continuous local martingale $N$ of finite variation, and stop so that both $|N|$ and its total-variation process are bounded by deterministic constants. The stopped local martingale is a bounded true martingale, so orthogonality of its increments gives, on any deterministic partition of an interval of $[0,t]$,Continuity and the dominated convergence theorem justify the last limit. Remove stopping and use continuity on a countable dense set of times to conclude $N\equiv 0$.

As literally printed, uniqueness is false without normalization. If $A$ works then $A+c$ also works for any deterministic constant $c$, since subtracting a constant preserves the local martingale property; this is a genuine omitted convention.

With normalization, existence always holds for a continuous local martingale. Its quadratic variation $[M]$, obtained as the limit in probability of squared increments, is continuous, increasing, starts at zero and satisfies $M^2-[M]$ is a continuous local martingale, so taking $⟨M⟩=[M]$ supplies the required finite-variation process.

Interpret the Stratonovich integral in its standard extension to continuous semimartingale integrands:This extension is needed because $f^{\prime }(X)$ is generally a semimartingale rather than a local martingale. Also the introductory prose in the PDF reverses integrand and integrator relative to its displayed definition; the displayed definition fixes the convention used here.

Apply the Itô formula to $f^{\prime }(X)$, using $f\in C^3$:The continuous finite-variation process in the second term has zero quadratic covariation with $X$, while the quadratic covariation rule for an Itô integral givesA second application of the Itô formula, now to $f(X)$, therefore yieldsAll unbounded coefficients are handled by stopping $X$ on compact spatial intervals, then removing the localizing sequence.

A semimartingale is an adapted process of the form $X=X_0+M+A$, where $M$ is a local martingale and $A$ is an adapted finite-variation process. A sequence of processes $X^n$ converges to $X$ in uniform convergence on compacts in probability, abbreviated ucp, when for every $T<\infty$ and $\delta >0$,

Each $A^{ϵ}$ is continuous, adapted, and increasing. From ucp convergence one can choose a subsequence that converges uniformly almost surely on every compact interval. Its limit $A$ is therefore also continuous and increasing, hence a finite-variation process. The stochastic integral $M_t={\int }_0^t\mathrm{sgn}(B_s)d{B}_s$ is a continuous local martingale, and part e gives the semimartingale decompositionConsequently $|B|$ is a semimartingale. In fact, comparison with the Tanaka formula identifies $A_t$ as the local time of a semimartingale $L_t^0(B)$.

A càdlàg process $A$ is a finite-variation process when, almost surely, for every $T<\infty$,where the supremum is over every finite partition of an interval $0=t_0<\cdots <t_n=T$.

Uniform convergence on compacts in probability of $X^n$ to $X$ means that, for every $T<\infty$ and $\epsilon >0$,

Define the deterministic function $f(t)=\mathbb{E}[M_t^2]$. The independent increments from part (i) show that $f$ is increasing and that $M_t^2-f(t)$ is a martingale. Mean-square continuity follows from path continuity and the Gaussian laws, so $f$ is continuous.

The Itô formula also says that $M_t^2-[M{]}_t$ is a local martingale. Their difference $[M{]}_t-f(t)$ is therefore a continuous finite-variation process that is also a local martingale. By the theorem that a continuous finite-variation local martingale is constant, and because the difference starts at zero,for all $t\ge 0$ almost surely.

Write the $d$-dimensional continuous semimartingale as $X=X_0+M+A$, where $M$ is a continuous local martingale and $A$ is a continuous finite-variation process. For $f\in C^2({\mathbb{R}}^d)$, Itô formula states

Apply Itô formula to $Z_t^{(A)}=\exp (A{X}_t-A^{2}t/2)$. Its semimartingale decomposition isThe second term is a continuous finite-variation process. Since $Z^{(A)}$ is assumed to be a local martingale, uniqueness of the semimartingale decomposition makes this term identically zero. Both $A$ and $Z^{(A)}$ are nonzero, so the quadratic variation of $X$ is $[X{]}_t=t$.

The Lévy characterization of Brownian motion now says that $X_t-X_0$ is a Brownian motion. Consequentlyis a constant multiple of an exponential Brownian martingale. It is therefore a true martingale for every $a\in \mathbb{R}$.

Use the continuous semimartingale decomposition $X=X_0+M+A$, where $M$ is a continuous local martingale and $A$ is a continuous adapted finite-variation process. Pointwise limits preserve predictability, so $H$ is predictable; it is bounded by the common bound for the $H_n$.

Localize so that $[M{]}_{\infty }$ and the total variation $V(A{)}_{\infty }$ are bounded. The Doob L2 maximal inequality and the Itô isometry giveby the dominated convergence theorem. For the finite-variation part,almost surely, again by dominated convergence, now for each sample path. Hence the two integrals converge uniformly in probability after every localization. Part (b) removes the localization and provesu.c.p.

A random measure assigns a measure ${\mu }_{\omega }$ to each outcome so that $\omega \mapsto {\mu }_{\omega }(A)$ is a random variable for every measurable $A$. A finite-variation process gives pathwise signed random measures through Lebesgue–Stieltjes integration.

A semimartingale has a decomposition $X=X_0+M+A$ into a local martingale $M$ and an adapted finite-variation process $A$. Under standard normalizations the decomposition is unique.

In singular stochastic control, a control is a finite-variation process rather than only a rate integrated against time. Its increments can have atoms or a part singular with respect to time. The Hamilton-Jacobi-Bellman equation becomes a variational inequality with a constraint on directional derivatives of the value function.

For a continuous local martingale $M$, at most one continuous adapted increasing process $A$ starting at zero makes $M^2-A$ a local martingale. The difference of two candidates is a continuous finite-variation process and a local martingale, hence constant; its initial value is zero. Without the initial-value normalization only uniqueness up to an initial constant is possible.