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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.

The limit is in probability, or more strongly in uniform convergence on compacts in probability; a pathwise limit for every continuous local martingale is not implicit. The original PDF uses $\lceil t{2}^n\rceil$, whereas the supplied TeX changes this to $\lfloor t{2}^n\rfloor$; both versions have the same limit by continuity.

Put $\delta =2^{-n}$ and $m=\lfloor t/\delta \rfloor$. For the completed intervals, the algebraic identityseparates a left-point sum from a quadratic covariation sum.

The results from stochastic calculus used are these: predictable left-step approximations to a continuous adapted integrand converge in their Itô integrals in uniform convergence on compacts in probability; and products of increments of continuous semimartingales on deterministic partitions of an interval with vanishing mesh converge to their quadratic covariation in the same sense. The first follows after localization from the Itô isometry and the Doob L2 maximal inequality, since the integrand approximations converge uniformly on compact intervals; the second is the deterministic-partition characterization of quadratic covariation.

Thus the two sums converge to ${\int }_0^t{Y}_{s}d{X}_s$ and $\frac{1}{2}⟨Y,X{⟩}_t$. For the ceiling convention, at most one additional full interval extends past $t$; on $[0,T+1]$ its absolute contribution is at most ${\sup }_{s\le T+1}|Y_s|$ times the modulus of continuity of $X$ at scale $\delta$, which tends to zero almost surely. Therefore

Start the Bessel process at $x>0$ and let ${\tau }_0$ be its first hit of zero. Away from zero its stochastic differential equation isApply the Itô formula to $f(x)=x^{2-\delta }$, localizing first to a compact interval in $(0,\infty )$. Its drift cancels:Thus the Bessel power local martingale satisfiesFor $\delta =2$, this power is simply the constant one. For $\delta >2$, it is a continuous local martingale for all time, as zero is inaccessible.

Here is a proof of that inaccessibility and of the stronger minimum assertion. For $0<r<x<R$, stop the power at ${\tau }_r\wedge {\tau }_R$. The stopped process is bounded, so the optional stopping theorem givesExit from the bounded interval occurs almost surely. Taking $r\downarrow 0$ with $R$ fixed shows that zero cannot be hit before $R$, hence cannot be hit in finite time. Taking $R\to \infty$ instead givesIf the all-time infimum were zero, every level $r>0$ below $x$ would be hit, by continuity. Letting $r\downarrow 0$ proves the all-time minimum of a transient Bessel process assertion:The first claim in the printed question needs a qualification when $\delta <2$: it holds before the first hit of zero, or for the process stopped there, rather than for the usual reflecting continuation. For example, at $\delta =1$ the proposed power is $X$ itself, a Reflected Brownian motion; the Tanaka formula contains a nonzero local time of a semimartingale term, so it is not a local martingale after reflection. A positive starting point is also needed for the negative power when $\delta >2$; an entrance process started at zero would give an infinite value at time zero.

With the notation supplied in the question, ${\tilde{U}}_t={\psi }_t(U_t)$ and $d{U}_t=\sqrt{\kappa }d{B}_t$. The Itô formula and ${\partial }_t{\psi }_t(U_t)=-3{\psi }_t^{\prime \prime }(U_t)$ giveFor $\kappa =6$, the drift vanishes. The resulting continuous local martingale has quadratic variationIf $s=\tilde{a}(t)/2$ is the usual half-plane-capacity time, then $⟨\tilde{U}⟩=6s$. The Dambis-Dubins-Schwarz theorem therefore givesfor a standard Brownian motion $\tilde{B}$. The mapped hulls are consequently an ${\mathrm{SLE}}_6$, which proves locality.

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 $L_t^0(B)$.

For a continuous local martingale $M$ with $M_0=0$, its stochastic exponential isThe Itô formula gives $d\mathcal{E}(M{)}_t=\mathcal{E}(M{)}_{t}d{M}_t$, so $\mathcal{E}(M)$ is a positive continuous local martingale. Every nonnegative local martingale is a supermartingale, because localization and the Conditional Fatou lemma turn the localized martingale equality into the supermartingale inequality.

The integrand $\mathrm{sign}({\beta }_s)$ is a bounded previsible process, so $B$ is a continuous local martingale. The quadratic variation of a stochastic integral isbecause the Brownian zero set has zero Lebesgue measure. Since $B_0=0$, the Lévy characterization of Brownian motion shows that $B$ is a standard Brownian motion.

The Dambis-Dubins-Schwarz theorem states that if $M$ is a continuous local martingale with $M_0=0$ and $[M{]}_{\infty }=\infty$, then, forthe process $W_s=M_{{\tau }_s}$ is a standard Brownian motion and $M_t=W_{[M{]}_t}$. If $[M{]}_{\infty }<\infty$, one obtains the same representation after enlarging the probability space and continuing $W$ independently beyond $[M{]}_{\infty }$.

The Kunita-Watanabe inequality applied to the continuous local martingales $M,N$ gives directlyEquivalently, with the clock and densities from part (i), positivity of the covariance matrix gives $|c|\le \sqrt{ab}$, and the Cauchy-Schwarz inequality gives

The cash-discounted stock is a positive continuous local martingale, because its dynamics contain no drift. Applying Itô formula to ${\pi }_t=U(t,v_t,S_t)$, the displayed partial differential equation cancels its drift exactly, leaving another local martingale. Since $U$ is bounded, $\pi$ is in fact a true martingale.

Thus the physical measure itself is an equivalent local martingale measure relative to cash for all three traded assets. The continuous-time fundamental theorem of asset pricing rules out arbitrage, more precisely no free lunch with vanishing risk, in the usual admissible class.

The bounded continuous local martingale $X$ is a square-integrable martingale. Since $M^{(n)}$ is a discrete predictable transform of $X$, it has mean zero. The identity $A_t^{(n)}=X_t^2-M_t^{(n)}$ therefore givesuniformly in $t$ and $n$.

For Brownian motion started at radius $x>0$, the displayed process is a continuous local martingale. It is a bounded true martingale when $0<a\le x$, but a strict local martingale when $a=0$ or $a>x$. In the latter case the boundary is outside the starting point, so stopping does not remove the singularity at the origin. Brownian scaling gives $\mathbb{E}\Vert B_t{\Vert }^{2-d}=O(t^{-(d-2)/2})$; when $a>x$, the stopped expectation tends to $a^{2-d}<x^{2-d}$.

A strict local martingale is a local martingale that is not a true martingale. The reciprocal of a three-dimensional Bessel process is an example: it is a nonnegative continuous local martingale whose expectation decreases strictly.

For a convergent continuous local martingale and $r>1$, factor $\mathcal{E}(X)$ into $\mathcal{E}(rX{)}^{1/r^2}$ and an ordinary exponential. Apply the Holder inequality and then the concave Jensen inequality to the fractional power $2/(r+1)$. The displayed lower bound is informative when the terminal exponential moment is finite.

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.