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Lévy's characterization of Brownian motion states that a real continuous local martingale $M$ with $M_0=0$ and quadratic variation $⟨M{⟩}_t=t$ is a standard Brownian motion with respect to its filtration. Equivalently, its increments are independent of the past and have centered normal distribution with variances equal to elapsed times.

Let $H_s=\mathrm{sgn}({\sup }_{r\le s}{B}_r-1)$. The running maximum is continuous and adapted, so $H$ is bounded and progressively measurable process. It has a predictable version for integration; changing its value at the single switching time does not change its stochastic integral. Thus $X=\int HdB$ is a continuous local martingale starting at zero. Since $H_s^2=1$,Lévy's characterization of Brownian motion yields $X$ is standard Brownian motion. The sign depends on the entire past through the running maximum, but the unit quadratic-variation rate is exactly what the characterization needs.

Itô's formula identifies the infinitesimal generator on twice continuously differentiable functions:Subtract the drift coefficient integral from that formula to obtainStopping when $|X|$ leaves a bounded interval makes the integrand bounded on each finite time interval. Therefore $M^f$ is a continuous local martingale. In fact $f^{\prime }\in L^{\infty }$ and the lognormal distribution solution has finite second moments on finite intervals, so here it is square integrable on each such interval as well.

The quadratic covariation of two stochastic integrals against the same Brownian motion is the integral of the product of their integrands. ConsequentlyThis is the quadratic covariation, which agrees with the pathwise quadratic covariation for these continuous local martingales.

Condition on the Brownian path up to $t\le T$. The independent normal increment $B_T-B_t$ has $\mathbb{E}\cos (B_T-B_t)=e^{-(T-t)/2}$ and zero expected sine. HenceApply Itô's formula to this expression. Its time derivative and second spatial derivative cancel, leavingTherefore the required representation isThe bounded integrand is square integrable, and uniqueness follows from the Itô isometry. The drift-containing decomposition in part (b) alone would not be the required representation as a constant plus one stochastic integral.

A sufficient condition is that $x\mapsto \sqrt{2\theta (x)}$ is globally Lipschitz. A more familiar, stronger condition implying this is that $\theta$ is globally Lipschitz and $\theta \ge \epsilon >0$. Both conditions repair the existence issue in part (i).

To connect this to the suggested stochastic differential equation, apply the coordinate and coordinate-product cutoff tests to any solution of the martingale problem. As before they giveDefine, using stochastic integrals against the coordinates of $X$,These integrals exist even without a global positive lower bound: their quadratic variations are exactly $t$. Indeed,The Lévy characterization of multidimensional Brownian motion identifies $W$ as standard $d$-dimensional Brownian motion. By associativity of stochastic integration,The diffusion matrix $\sqrt{2\theta (x)}I$ is globally Lipschitz under the stated condition, and boundedness of $\theta$ gives bounded growth. The global existence theorem for stochastic differential equations with Lipschitz coefficients gives a unique strong solution of a stochastic differential equation and uniqueness in law. Every solution to the martingale problem has just been represented as a weak solution of a stochastic differential equation of this form, so they all have the same law. Conversely, the Itô formula applied to any $f\in C_c^{\infty }$ gives a true martingale, since its stochastic integrand is bounded. Thus the martingale problem is well posed under this sufficient condition.

The stopped representation from part (a)(ii) and the Itô isometry give the exact second-moment identityThe cross term with the initial value is zero because the stopped stochastic integral has zero mean. This is the useful finite-time computation; evaluating the remaining occupation integral explicitly is unnecessary for proving integrability of $T$.

For $|x|\le L$,Combining the second-moment identity with part (i) yieldsLet $t\to \infty$ and use monotone convergence:In particular $T<\infty$ almost surely, a conclusion obtained without assuming finite exit in the stopping argument.

One can also evaluate the limiting second moment exactly. At exit $M_T$ is either $e^{-L}$ or $e^L$, soBy bounded convergence theorem and part (i), $\mathbb{E}{M}_T=1$. ThereforeThe same bound also holds at every finite time, by applying $(y-e^{-L})(y-e^L)\le 0$ to $y=M_{t\wedge T}$ and taking expected values. It improves the preceding bound to $\mathbb{E}T\le c_L^{-1}(e^L+e^{-L}-2)$ if desired.

The native PDF repeats $d{W}_y$ in its list of independent noise terms, whereas its third stochastic differential equation uses $d{W}_z$. We use the intended three independent coordinate Brownian motions $W_x,W_y,W_z$.

Put $\mathbf{X}=(X,Y,Z)$, $r=|\mathbf{X}|$ and $S=r^2$. Before the origin is reached, Itô's formula givesThe $6$ is the sum of three quadratic variation terms, each $2dt$. The drift contribution is $-2\alpha dt$ because the numerator is $|\mathbf{X}{|}^2$. The radial stochastic integral can be written using a scalar Brownian motion $B$:Thus $S/2$ is a squared Bessel process of dimension $\delta =3-\alpha$ until it reaches zero. If this equation is taken globally with the usual nonnegative squared Bessel process continuation, taking expected values givesThe stochastic integral has zero expected value: stopping first at large levels bounds the integrands, yields a finite first moment for $S$, and then makes $\mathbb{E}{\int }_0^{t}S(u)du$ finite, so the unrestricted integral is a square-integrable martingale.

Boundary qualification. The punctured domain in the question does not itself specify that continuation. The boundary classification of a Bessel process says that the origin is inaccessible from $r_0>0$ when $\delta \ge 2$, equivalently $\alpha \le 1$. In this parameter range the original stochastic differential equation is global in the stated domain and the boxed result holds as written.

For $1<\alpha <3$, we have $0<\delta <2$: the origin is reached in finite time with probability one. One can also see accessibility directly from the radial generator of an SDE diffusionFor $r<b$, the harmonic function $1-(r/b{)}^{\alpha -1}$ is the probability of reaching zero before $b$. Finite-time access follows by using the nonnegative functionin the stopped Itô formula; it bounds the expected value of the time to reach zero or $b$. Letting $b\to \infty$ gives almost sure finite hitting of zero. The coefficients of the original vector stochastic differential equation are undefined there, so its stated punctured-domain version has no prescribed long-time evolution in this range.

The natural continuing radial squared Bessel process gives $6-2\alpha$, but stopping and holding the particle at the origin would give a different answer. Indeed, with $T_0$ the first hit of zero, Itô's formula for the stopped process givesSince $T_0<\infty$ almost surely, dominated convergence applied to $(t\wedge T_0)/t\le 1$ givesTherefore $6-2\alpha$ is the usual continued-radial answer; for $1<\alpha <3$ a boundary convention is necessary. It is not a uniquely defined limit for the punctured-domain model alone.

At $\alpha =0$ the transverse coordinates are independent Brownian motions with diffusion coefficient $D=1$. The cylinder is infinitely long, so the axial coordinate cannot affect its first contact time. By rotational invariance of Brownian motion, the mean first-passage time $u$ depends only on the transverse radius $\rho =\sqrt{x^2+y^2}$.

The Kolmogorov backward equation for a mean first-passage time is $Lu=-1$. Here $L={\partial }_x^2+{\partial }_y^2$, so the radial boundary value problem isThe absorbing boundary condition for a diffusion is imposed on the circular cross-section. At $\rho =0$, the solution must extend smoothly over the axis. Multiplying by $\rho$ and integrating givesSmoothness at the axis excludes the logarithm term, and $u(R)=0$ fixes $C_0=R^2/4$. ConsequentlyThis also verifies the finiteness assumption behind the Kolmogorov backward equation. Let $T$ be the first contact time and apply the stopped Itô formula to the bounded nonnegative function $u$ just obtained. Its bounded derivatives on the disk make the stopped stochastic integral a martingale, givingBy monotone convergence, $T$ has finite expected value; continuity and dominated convergence then make the final term vanish as $t\to \infty$. Hence $u({\hat{r}}_0)=\mathbb{E}T$. In particular, the mean first-passage time is $R^2/4$ on the axis and tends to zero at the adsorbing wall.

For continuous local martingales $M,N$ and a sequence in which each term is a partition of an interval whose mesh tends to zero, the sumsconverge in the sense of uniform convergence on compacts in probability to a continuous increasing process. To identify the limit, put $C=[M]+[N]$ and choose Radon-Nikodym derivativesIf $(U,V)$ is a centered bivariate normal distribution with covariance matrix $\left(\begin{array}{cc}a& c\\ c& b\end{array}\right)$, thenLocalizing, representing the pair as stochastic integrals against a two-dimensional Brownian motion, and approximating the integrands by bounded predictable step processes proves the convergence. The step-process case follows from the weak law of large numbers for independent Gaussian increments; the Burkholder-Davis-Gundy inequality controls the approximation error. Since $|\mathbb{E}[UV]|\le \mathbb{E}|UV|\le \sqrt{\mathbb{E}{U}^2\mathbb{E}{V}^2}$,

A semimartingale is the sum of a local martingale and an adapted finite-variation process. This is the broad class of integrators for which the Itô stochastic integral is defined.

A stochastic differential equation specifies infinitesimal drift and random diffusion through a stochastic integral equation.

A theorem interchanging a parameter integral with an Itô integral, under appropriate measurability and integrability hypotheses. For a bounded deterministic integrand on a finite parameter-time rectangle, the square-integrability conditions hold; triangular domains can be handled by an indicator of the domain.

For $t<1$ this stochastic integral solves $d{X}_t=d{B}_t-X_t(1-t{)}^{-1}dt$. The Itô product rule also gives $X_t=B_t-(1-t){\int }_0^t{B}_s(1-s{)}^{-2}ds$. The process is centered Gaussian, with covariance $\min (s,t)-st$, and has a continuous extension $X_1=0$. Time reversal of a Brownian bridge proves endpoint continuity without assuming it at the start.

The usual conditions are completeness and right continuity of a filtration: ${\mathcal{F}}_0$ contains all subsets of null events in the ambient probability space, and ${\mathcal{F}}_t={\bigcap }_{s>t}{\mathcal{F}}_s$. They allow the standard continuous-time martingale, stopping time and stochastic integration theorems to be used without repeated augmentation qualifications. An absolute continuity of measures change preserves old null sets but can introduce new ones; the usual completion under the new measure may therefore be understood when needed.

For a nonnegative classical solution $V$ of a finite-horizon Hamilton-Jacobi-Bellman equation with nonnegative running reward $r$, the process $V(X_t,t)+{\int }_0^{t}r(X_s,a_s)ds$ is a local supermartingale under every admissible control. The Itô formula gives nonpositive drift because the chosen control cannot exceed the supremum in the equation. Stop where the stochastic integral becomes a true martingale, take expectations and apply the Fatou lemma. Nonnegativity prevents loss of a lower bound when removing the stops, giving the verification upper bound. A maximizing control attains the bound when its stopped reward processes are uniformly integrable.

For $d\sigma =-\sigma dt/\tau +\sqrt{\kappa }dW$ with ${\sigma }_0=0$, the explicit stochastic integral is ${\sigma }_t=\sqrt{\kappa }{\int }_0^t{e}^{-(t-u)/\tau }d{W}_u$. The Itô isometry on the overlap of the two integration intervals gives the displayed covariance. The second exponential records the zero initial condition; removing it gives the stationary Ornstein-Uhlenbeck process covariance.