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An adapted càdlàg process is a semimartingale exactly when it is a good integrator: on each finite horizon, for every sequence of bounded elementary predictable processes with stopping-time intervals $H^n$ whose deterministic uniform bounds tend to zero, the elementary terminal integrals $(H^n\cdot X{)}_T$ tend to zero in probability. The elementary integral is a finite sum of measurable coefficients times subsequent increments; the characterization says precisely when this operation extends continuously to stochastic integration. The theorem is used here as a standard characterization, not proved from first principles.

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 $[X]=[N]$.

Suppose $X,A$ are continuous, start at zero, and $A$ is increasing. If $e^{X-A/2}$ and $e^{-X-A/2}$ are local martingales, their logarithms first recover adaptation of $X,A$ and the semimartingale property of $X$. In the decomposition $X=N+V$, the two Itô drift measures give $dV+(d[N]-dA)/2=0$ and $-dV+(d[N]-dA)/2=0$. Thus $V=0$ and $[X]=A$. All real exponential parameters may be assumed, but these two suffice.

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.

An Itô process is a continuous semimartingale expressible as the displayed sum, with adapted coefficients locally integrable for the time integral and locally square integrable for the Itô integral. The coefficients may depend on the whole past; an Itô diffusion usually specifies them as functions of the current state and time. A continuously differentiable deterministic process is an Itô process with zero Brownian coefficient. The Itô formula and Itô product rule apply to these processes.

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.

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 an elementary previsible integrand, the integral is the increment sum ${\sum }_j{H}_j(X_{t\wedge t_{j+1}}-X_{t\wedge t_j})$. This expression uses only $H$ and $X$, so is identical under the two measures, irrespective of how the local martingale and finite-variation parts change.

We may use the stochastic dominated convergence theorem: if bounded predictable $H_n$ converge pointwise to $H$ under a common deterministic bound, their integrals against a semimartingale converge uniformly on compact intervals in probability. Also convergence in $\mathbb{P}$-probability implies convergence in $\mathbb{Q}$-probability when $\mathbb{Q}\ll \mathbb{P}$. To see the latter directly, let $D=d\mathbb{Q}/d\mathbb{P}$. For any event $E$,and let first $\mathbb{P}(E)\to 0$, then $K\to \infty$. Apply this to events involving the supremum on a finite horizon for the uniform-in-probability statement.

Let $\mathcal{C}$ be the bounded predictable integrands for which the two integral processes agree up to $\mathbb{Q}$-indistinguishability. It is a vector space containing elementary rectangle indicators generating the predictable sigma-algebra. It is closed under uniformly bounded pointwise limits: stochastic dominated convergence theorem gives convergence of the integrals under each measure, and the preceding transfer makes the $\mathbb{P}$ limit also a $\mathbb{Q}$ limit. Uniqueness of limits in probability makes the two limiting integrals equal. The functional monotone-class theorem therefore puts every bounded predictable integrand in $\mathcal{C}$.

Finally, for locally bounded $H$, stop on intervals where it has a deterministic bound. Such stopping times tend to infinity under $\mathbb{P}$ and hence also under $\mathbb{Q}$. Apply the bounded result to the stopped integrands and use the stopping property of the integral. This proves the stochastic integral under an absolutely continuous measure change identityThe decompositions $H\cdot M+H\cdot A$ and $H\cdot N+H\cdot B$ thus represent the same process under $\mathbb{Q}$, although their individual components need not agree. With only absolute continuity, equality is asserted outside a $\mathbb{Q}$-null set, not necessarily outside a $\mathbb{P}$-null set for a chosen $\mathbb{Q}$ version.

For càdlàg semimartingales, the semimartingale integration by parts formula isFor continuous processes the left limits can be replaced by the values. The bracket includes the jump products in the general formula.

For a continuous ${\mathbb{R}}^d$-valued semimartingale $X$ and $f\in C^2({\mathbb{R}}^d)$, the Itô formula isThe formula is localized when the derivatives are unbounded; continuity of $X$ makes them bounded after stopping on compact sets.

To prove the one-dimensional formula for polynomials without assuming that formula, use induction. It is immediate for constants and $X$. SupposeWrite $X=X_0+M+A$ with $M$ a continuous local martingale and $A$ finite variation. The induction hypothesis identifies the martingale part of $X^k$ as $k{X}^{k-1}\cdot M$. The stochastic-integral covariation identity, obtained by polarization from the quadratic-variation identity, givesApply semimartingale integration by parts to $X^{k}X$. Its first-order terms combine to $(k+1)X^{k}dX$, and its bracket terms combine toThis proves the induction step. Taking linear combinations proves the polynomial Itô formula from integration by parts for every polynomial $f$.

The intended setting is $X=M+A$ with $A$ continuous and adapted. These conditions are needed for the requested continuous-semimartingale conclusion; finite variation alone does not supply them. For example, $M=0$, $A_t=1_{\{t\ge 1\}}$ satisfies the printed deterministic bound and starts at zero, but $f_n(X)$ has a jump for sufficiently large $n$. We use the continuous adapted interpretation throughout this question.

Define $f_n(x)={\int }_0^x\varphi (ny)dy$. Then $f_n\in C^2$, $|f_n^{\prime }|\le 1$ and $f_n^{\prime \prime }(x)=n{\varphi }^{\prime }(nx)\ge 0$. The Itô formula gives the canonical continuous semimartingale decompositionThe initial value is zero. The first term is a continuous martingale, since $M$ is bounded and the integrand is bounded. The second is continuous adapted finite variation, hence predictable. This decomposition is unique because a continuous finite-variation local martingale is constant. The phrase Doob-Meyer here must mean this semimartingale decomposition: $V^{(n)}$ need not be nondecreasing, and the hypotheses do not make $f_n(X)$ a submartingale as required by the Doob-Meyer decomposition theorem. Indeed $f_n^{\prime }(0)=-1$, so the bounded deterministic drift $A_t=t\wedge \delta$ makes $f_n(A_t)$ initially decrease, excluding the submartingale property.

The sign process is predictable because it is a Borel function of the continuous adapted, and therefore predictable, process $X$. Its paths need not themselves be left-continuous. Pointwise, including at $x=0$,For a fixed horizon $T$, $\mathbb{E}[M{]}_T=\mathbb{E}{M}_T^2\le K^2$. The dominated convergence theorem for the finite measure $\mathbb{E}d[M]$ on $[0,T]$ givesThe Itô isometry, extended from simple predictable integrands by their density, and the Doob L2 maximal inequality now implyHence

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.

Use the Chordal Loewner equation with Loewner driving function ${\xi }_t=\sqrt{\kappa }{W}_t$, and set $Z_t=g_t(z)-{\xi }_t=X_t+i{Y}_t$. Before the Loewner swallowing time,The upper-half-plane branch of the logarithm has imaginary part $h_t$. The Itô formula givesand therefore the SLE angle process satisfiesAt $\kappa =4$ the drift vanishes. Since $0<h_t<\pi$, the stopped local martingale is a true bounded martingale, and the SLE4 angle martingale is global for each fixed point almost surely, using fixed-interior-point avoidance of SLE4 for the simple parameter-four Loewner trace.

Conversely, take $\kappa >0$. If this semimartingale were a local martingale, uniqueness of its finite-variation decomposition would force $(\kappa -4)X_t{Y}_t=0$ throughout every compact interval before swallowing. Because $Y_t>0$, for $\kappa \ne 4$ this would force $X_t$ to be identically zero on such an interval. Its Brownian component has quadratic variation $\kappa t$, which makes that impossible. Thus for positive $\kappa$, the martingale parameter is exactly four.

If the degenerate value $\kappa =0$ is admitted, there is one exception to a fixed-point reading of the assertion: for $z$ on the positive imaginary axis the deterministic flow stays on that axis until swallowing, and $h_t=\pi /2$ is constant. For all starting points simultaneously, the unique parameter giving the martingale property is still four.

The mark-to-market value of holdings $H_t$ in the asset-price vector is the dot product $X_t=H_t\cdot P_t$. A self-financing strategy pays for every rebalancing from within the portfolio. Over a short interval, the gains on the currently held assets are $H_t\cdot d{P}_t$, while consumption removes $c_{t}dt$ units of wealth. ThereforeThe holdings are predictable and integrable against the price semimartingale; the consumption rate is nonnegative and suitably measurable. This is the continuous-time self-financing-with-consumption convention.

Write ${\delta }_n=2^{-n}$ and let $Q_n(U)$ denote the printed squared-increment sum for a process $U$, on grid times $k{\delta }_n$. We construct its limit first for bounded martingales, then use localizing sequences and the finite variation part of a semimartingale.

Continuity and monotonicity of the bounded-martingale limit. Let $U$ be a uniformly bounded continuous martingale. The given result supplies a limit $R^U$ in uniform convergence on compacts in probability. Each $Q_n(U)$ is continuous and adapted. A subsequence converges almost surely uniformly on each compact time interval, by choosing summable error probabilities and diagonalizing. Its limit therefore has a continuous version; in the usual completed filtration this version is adapted.

The sums $Q_n(U)$ themselves can decrease between grid points, because the last partial increment is being squared. To establish monotonicity, use instead the quadratic variation from completed grid incrementsThese step processes are nondecreasing, and pathwise uniform continuity giveswhere ${\omega }_U(\delta ;T)=\sup \{|U_t-U_s|:s,t\le T,|t-s|\le \delta \}$. Thus the same almost surely uniform subsequence of $S_n$ converges to $R^U$, proving that $R^U$ is nondecreasing. Also $R_0^U=0$.

Localization of a continuous local martingale. Subtract the initial value, which does not affect increments, so that $M_0=0$. SetContinuity gives ${\tau }_m\uparrow \infty$ almost surely and makes $M^{{\tau }_m}$ bounded. The bounded local martingale criterion makes it a true martingale. Let $R^m$ be its continuous, adapted, nondecreasing limit.

The discrete sums commute exactly with stopping:For $\ell \le m$, uniqueness of the probability limit and stability of uniform convergence on compacts in probability under stopping giveup to indistinguishability. Taking a common null set for the countably many pairs, patch these processes into $R^M$ by setting $R_t^M=R_t^m$ when $t\le {\tau }_m$. Compatibility makes this definition independent of $m$. It is continuous, adapted, and nondecreasing. For each $T,\epsilon >0$,Let $n\to \infty$ and then $m\to \infty$. This proves the localization and patching of quadratic variation.

Adding finite variation. Decompose the continuous semimartingale as $X=X_0+M+A$, where $M$ is a continuous local martingale starting at zero and $A$ is continuous, adapted, and locally of finite variation. For every compact interval, the sum of the absolute increments of $A$ is at most its total variation of a function. Hence, pathwise,The Cauchy-Schwarz inequality bounds the mixed increment sum byThe first factor is bounded in probability, since $Q_n(M)\to R^M$ uniformly on compacts in probability; the second tends to zero almost surely. Thus the mixed term tends to zero in probability. Expanding the square provesThe constructed $R=R^M$ is continuous, nondecreasing, and adapted. This is the quadratic variation of $X$, and expresses the fact that finite-variation terms do not change quadratic variation.

This part's conditional-expectation representation is its own hypothesis; it does not need the incorrect unrestricted claim in part (b). Fix $T$, and writeThe assumed representation makes $M(T)$ a martingale. The stochastic Fubini theorem givesApply the Itô formula to $e^{-I}$ and use the equation for $U$ from part (a), including the cross-variation. The result isUniqueness of the continuous semimartingale decomposition makes the drift vanish. Initially this is a $dtd\mathbb{P}$ statement; the assumed continuity in time and maturity extends it to the continuous versions simultaneously. Let $T\downarrow t$ to obtainSubstitute back and differentiate the maturity integrals using their continuous integrands:This is the forward drift restriction for square-root stock claims. The $-{\sigma }_t/2$ term comes from the product cross-variation and must be retained. For the uninformative zero-stock case, the representation does not identify $f$; as usual a positive initial stock price is understood.

For a continuous local martingale $M$, its stochastic exponential isThe Itô formula gives $d{Z}_t=Z_{t}d{M}_t$ and $Z_0=1$, so $Z$ is a strictly positive local martingale. The finite-horizon Girsanov theorem says: if $Z$ is a true martingale on $[0,T]$ and $d\tilde{\mathbb{P}}=Z_{T}d\mathbb{P}$ on ${\mathcal{F}}_T$, then for every continuous local martingale $N$ under $\mathbb{P}$,For an infinite-horizon change of measure, assume $Z$ is a uniformly integrable martingale and use its terminal density $Z_{\infty }$. In particular, if $M=\int \theta dB$, the process $\tilde{B}=B-\int \theta ds$ is Brownian motion under the new probability measure, by the Lévy characterization of Brownian motion.

The quadratic variation cannot change. Adding the continuous finite-variation process $-[N,M]$ leaves $[\tilde{N}]=[N]$. Also the same pathwise partition sums defining $[N]$ have the same limit under an absolutely continuous change of measure; convergence in probability transfers by integrability of the density. Thus the bracket of the unchanged process $N$, now a semimartingale, is also unchanged. The local martingale property itself need not survive. This is quadratic covariation under an absolutely continuous measure change.