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For a Bessel process of dimension $\delta$, started at $x>0$, the Itô formula giveson its stochastic interval before hitting zero. Thus the power is a local martingale after localization away from zero. For $\delta \ge 2$ this interval is the whole lifetime; at $\delta =2$ the power is constant. The unqualified assertion fails for a reflecting Bessel process of dimension one, which acquires local time of a semimartingale at zero.

The occupation time of a measurable set $A$ through time $t$ is ${\int }_0^t{1}_{\{B_s\in A\}}ds$. The occupation-times formula expresses such integrals through local time of a semimartingale. Recurrence and the Strong Markov property imply that one-dimensional Brownian motion spends an unbounded total time in every nonempty open interval.

For a standard Brownian motion $B$, its Brownian running maximum $M$, and its local time of a semimartingale $L^0$ at zero, the process pairs satisfyIn particular, $M-B$ and $|B|$ are both Reflected Brownian motion and have the same law as processes.

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 Brownian motion started at zero, let $f_n$ equal $|x|$ outside $(-1/n,1/n)$ and $n{x}^2/2+1/(2n)$ inside. Its derivative is the clipped sign function $g_n(x)=\max \{-1,\min \{1,nx\}\}$. The Itô integral $M_t^{(n)}={\int }_0^t{g}_n(B_s)d{B}_s$ converges uniformly on bounded time intervals in $L^2$ to a continuous martingale $M$, by the Itô isometry and Doob L2 maximal inequality: the squared maximal error is at most $8\sqrt{2T/\pi }/n$. The Itô formula gives $\frac{n}{2}{\int }_0^t{1}_{\{|B_s|<1/n\}}ds=f_n(B_t)-f_n(0)-M_t^{(n)}$. Along $n=k^2$, the maximal errors are summable, so First Borel-Cantelli lemma gives uniform almost-sure convergence. The ratio-one interpolation for shrinking-window occupation integrals extends it to every integer $n$. The limit is $|B_t|-M_t=L_t^0$, the local time of a semimartingale in the Tanaka formula convention. The unnormalized factor $n$ gives twice this limit.

For a continuous semimartingale $X$ with local time of a semimartingale $L_t^a(X)$,for every nonnegative measurable function $f$.

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.

For a discrete-time martingale relative to the filtration $({\mathcal{F}}_n)$, each $M_n$ is ${\mathcal{F}}_n$-measurable, $\mathbb{E}|M_n|<\infty$, and $\mathbb{E}[M_n\mid {\mathcal{F}}_{n-1}]=M_{n-1}$ for $n\ge 1$. The tower property then gives the corresponding identity conditional on any earlier time.

A bounded previsible process has ${\varphi }_i$ measurable at time $i-1$ and $|{\varphi }_i|\le C$. Its martingale transform $V$ is adapted and integrable, since each increment has absolute expectation at most $C(\mathbb{E}|M_i|+\mathbb{E}|M_{i-1}|)$. Moreover,Thus $V$ is a martingale, with $V_0=0$.

For the Itô integral, begin with a bounded adapted step integrand $H_s=H_i$ on $(t_i,t_{i+1}]$. Its integral is $I={\sum }_i{H}_i(W_{t_{i+1}}-W_{t_i})$. Each summand has zero conditional mean. For $i<j$, conditioning the cross product on ${\mathcal{F}}_{t_j}$ makes its expectation zero. The conditional second moment of a Brownian increment is $t_{i+1}-t_i$, henceApproximate the predictable integrand $g(s,W_s)$ in $L^2(dsd\mathbb{P})$ by such step processes. The same identity makes their integrals Cauchy in $L^2$ and defines the limit; both its first and second moments converge. This gives the requested mean-zero result and the Itô isometry:

For the Ornstein-Uhlenbeck process, the deterministic integrating factor has no quadratic covariation with $X$. The Itô formula therefore gives $d(e^{\kappa t}{X}_t)=\kappa \theta e^{\kappa t}dt+\sigma e^{\kappa t}d{W}_t$. Integrating and dividing by $e^{\kappa t}$ yieldsThe mean-zero integral and Itô isometry giveAt $\kappa =0$ these formulas are read by continuity: $X_t=x+\sigma W_t$, with mean $x$ and variance ${\sigma }^{2}t$.

For the CIR model, let $C_0=c_0\ge 0$ and use the usual positive parameters. Before a possible visit to zero, the derivatives of $\sqrt{c}$ are $1/(2\sqrt{c})$ and $-1/(4c^{3/2})$. Applying the Itô formula gives the Square root of a CIR diffusion:The PDF's claim under $2ab={\sigma }^2$ is false: that choice leaves the nonzero drift ${\sigma }^2/(8Z_t)-a{Z}_t/2$, which is not affine in $Z_t$. The actual interior cancellation condition is $4ab={\sigma }^2$. Even then the principal square root is nonnegative and cannot be an unrestricted Gaussian OU process.

To give the intended corrected construction, let a signed process solve $d{Y}_t=-a{Y}_{t}dt/2+(\sigma /2)d{B}_t$, with $Y_0=\sqrt{c_0}$. The Itô formula givesWriting ${\tilde{W}}_t={\int }_0^t\mathrm{sgn}(Y_s)d{B}_s$, with either sign chosen at zero, produces a Brownian motion; the diffusion term is $\sigma |Y_t|d{\tilde{W}}_t$. Thus $Y^2$ is a CIR model with $ab={\sigma }^2/4$, while $Z=|Y|$ is reflected at zero. The reflection contributes local time of a semimartingale, so the interior SDE alone is not a global unrestricted OU equation. Its second moment is $c_0{e}^{-at}+{\sigma }^2(1-e^{-at})/(4a)$, agreeing with the CIR mean for that corrected parameter choice.

The requested mean under the parameters actually printed is still available without the false OU assertion. Taking expectation in the original CIR equation gives $m^{\prime }=a(b-m)$; the stochastic integral has zero mean after the usual moment/localization justification. Consequently, for the printed choice as well as general admissible parameters,

For $f(x)=x$, the continuous martingale is $M_t^f=B_t$. Since $B_t$ has the centered normal distribution of variance $t$,For $f(x)=x^2$, the continuous martingale is $B_t^2-t$. The Gaussian fourth moment givesThis verifies both cases. The time integral in the printed identity is understood with its omitted $ds$.

Write $M^{(n)}=M^{f_n}$. The derivatives of the approximations areThus $|g_n|\le 1$. The Itô formula gives $M_t^{(n)}={\int }_0^t{g}_n(B_s)d{B}_s$. This use of the formula is valid despite the two second-derivative discontinuities: mollify the $C^1$ function $f_n$, whose derivative is Lipschitz. Its smooth derivatives converge boundedly, while the second derivatives are bounded by $n$ and converge except at $\pm 1/n$. For every $s>0$, the normal distribution of $B_s$ assigns those points zero probability. Dominated convergence for the drift integral and the Itô isometry for the stochastic integral pass the smooth formula to $f_n$. Equivalently, this identifies the martingale in the assumed decomposition and permits its cross-covariances to be calculated.

If $m\ge n$, then $|g_m-g_n|\le 1_{\{|x|<1/n\}}$. The Itô isometry and Doob L2 maximal inequality therefore give, for every fixed $T$,The Gaussian density is bounded by $(2\pi s{)}^{-1/2}$, soConsequently the processes are Cauchy in the complete space of continuous random paths with norm $(\mathbb{E}{\sup }_{s\le T}|Z_s{|}^2{)}^{1/2}$. They have a continuous-path limit $M$ there. Limits obtained for different integer horizons agree on overlaps; choose these versions consistently. Each $M_t$ is adapted, and passing to the $L^2$ limit in $\mathbb{E}[M_t^{(n)}\mid {\mathcal{F}}_s]=M_s^{(n)}$ shows that $M$ is a continuous martingale. Letting $m\to \infty$ in the preceding bound proves the requested convergence, with an explicit rate:The same Itô isometry identifies $M_t={\int }_0^t\mathrm{sgn}(B_s)d{B}_s$, with $\mathrm{sgn}(0)=0$.

There is a genuine factor-of-two error in the last printed conclusion. The stated $f_n$ has second derivative $n{1}_{\{|x|<1/n\}}$ almost everywhere, and the generator in the decomposition is one half of the second derivative. Hence the exact identity isBecause ${\sup }_x|f_n(x)-|x||\le 1/(2n)$ and $f_n(0)=1/(2n)$, this already proves uniform $L^2$ convergence on bounded intervals to $L_t=|B_t|-M_t$. It does not by itself prove almost-sure convergence of the full sequence, so we supply that additional step.

Take $n_k=k^2$. For each fixed integer $T$ and each $\epsilon >0$, the maximal-error bound and Markov's inequality implyThe first Borel-Cantelli lemma, applied to rational positive $\epsilon$ and integer $T$, gives uniform almost-sure convergence of this subsequence on every bounded interval. Therefore $L_{k^2}\to L$ uniformly there almost surely.

For $k^2\le n\le (k+1{)}^2$, the indicator windows are nested. The ratio-one interpolation for shrinking-window occupation integrals givesBoth ratios tend uniformly to one as $k\to \infty$. The two bounds converge uniformly to the continuous nonnegative process $L$, so the entire sequence does too. Thus the correct normalized conclusion isand with the factor actually printed in the question the limit isThese are the occupation approximations for Brownian local time. In the Tanaka formula convention, $L_t$ is the local time of a semimartingale at zero. It is not identically zero: $\mathbb{E}{L}_t=\mathbb{E}|B_t|=\sqrt{2t/\pi }>0$ for $t>0$, since $M$ starts at zero and is a martingale. Thus the discrepancy cannot be removed by treating the two limits as equal. The standard normalization is also given in Lalley's notes on the Itô calculus and local time.

Put ${\mathcal{L}}_{r}V(s)=\frac{1}{2}{\sigma }^2{s}^2{V}^{\prime \prime }(s)+rs{V}^{\prime }(s)$ and define the nonnegative reserve rate $a(s)=rV(s)-{\mathcal{L}}_{r}V(s)$. The obstacle problem gives $V\ge g$ and $a\ge 0$. The American-option superhedge with a funded reserve invests the local surplus in the bond rather than consuming it.

For initial wealth $x\ge V(S_0)$, setand choose the stock and bond holdingsThus $D_t\ge 0$ and $F_t\ge V(S_t)\ge g(S_t)$, pathwise at every time. The Itô formula under the original drift givesAdding these equations yieldsThis is a self-financing strategy, and its nonnegative wealth makes it an admissible trading strategy. Continuity of the stock and local regularity of $V$ ensure local integrability of the holdings. The construction does not require $\mu =r$.

For a classical solution, the usual Itô formula applies directly. The smooth fit solution below is $C^1$ and piecewise $C^2$, with locally absolutely continuous first derivative. The generalized Itô formula applies with its almost-everywhere second derivative; the absence of a derivative jump means no boundary local time of a semimartingale term. This is the usual regularity interpretation of the perpetual American option obstacle equation.

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.

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 $L_t^0(B)$.

One-dimensional reflected Brownian motion on $[0,\infty )$ has the same law as $|B_t|$. The Tanaka formula represents it as a Brownian motion plus a nondecreasing local time of a semimartingale that grows only at zero.

At a regular exercise boundary for a diffusion optimal stopping problem, value matching is commonly accompanied by matching first derivatives. This makes the value $C^1$ across the boundary and eliminates a derivative-jump local time of a semimartingale term in the generalized Itô formula. It does not assert that second derivatives agree, or that smooth fit holds for every possible stopping problem.

For a continuous semimartingale $X$,where $L^0(X)$ is the local time of a semimartingale at zero.