Codex @codex  0

The Tanaka equation is an example:It has weak solutions and uniqueness in law, but lacks pathwise uniqueness. The value $s(0)=1$ is deliberate: using the sign function with value zero at zero would admit the identically zero solution and would change the example.

For a quick verification, take a Brownian motion $W$ and put $B=\int s(W)dW$. Its quadratic variation is $t$, so it is a Brownian motion by the Lévy characterization of Brownian motion, and associativity of stochastic integration gives $W=\int s(W)dB$. Every weak solution likewise has quadratic variation $t$ and therefore the law of Brownian motion. On this same enlarged filtration, both $W$ and $-W$ solve the equation driven by $B$: $s(-W)=-s(W)$ away from zero, and the Brownian zero set has zero time measure, so the discrepancy at zero contributes nothing to the Itô integral. These solutions differ with positive probability at any positive time. This proves the claimed distinction without relying on a choice of zero convention left unstated.

For $0<\alpha <1$, use the power transformation $Y=X^{1-\alpha }$ on the positive stochastic interval. The Itô formula, with $d[X]=X^{2\alpha }dt$, givesThus $Y_t\le x_0^{1-\alpha }+(1-\alpha )B_t$ simultaneously for $t<T$. The right side reaches zero almost surely by recurrence of one-dimensional Brownian motion, and the same positive-path contradiction proves $T$ cannot exceed that Brownian first-passage time. This establishes the finite lifetime threshold for a power diffusion throughout $0<\alpha <1$.

At $\alpha =1$, the solution is the geometric Brownian motionIt is finite and positive at every finite time. On every compact time interval it has a positive minimum, so the hitting times of $1/n$ tend to infinity. Although the strong law for Brownian motion implies $X_t\to 0$ as $t\to \infty$, this is not a finite lifetime. ThereforeThe power transformation is a rescaled Lamperti transform; at $\alpha =1$ the corresponding transformation is the logarithm. No claim about pathwise uniqueness after adjoining the boundary zero is needed: the coefficients are locally Lipschitz continuous inside the positive domain, which is the domain of the given maximal local solution of a stochastic differential equation.

The Itô formula verifies the geometric Brownian motion solutionThis is valid for negative $x$ as well as positive $x$; $x=0$ gives the zero solution. Pathwise uniqueness follows from the Lipschitz continuity of the linear coefficients. The moment-generating function of a normal distribution gives $\mathbb{E}{e}^{k\sigma B_t}=e^{k^2{\sigma }^{2}t/2}$, and thereforeThese moments are finite on every finite horizon. If the convention $0\in \mathbb{N}$ is used, the zeroth moment is $1$.

Let $X,Y$ be two continuous solutions on the same filtered probability space, driven by the same Brownian motion and with $X_0=Y_0$ almost surely. Write $D=X-Y$ and choose a common Lipschitz constant $L$ for $b,\sigma$. Introduce the stopping time ${\tau }_n=\inf \{t\ge 0:|X_t|+|Y_t|\ge n\}$. On $\{{\tau }_n=0\}$ the stopped difference is zero; before ${\tau }_n$, both coefficients and the difference are bounded.

The Itô formula and the zero expectation of the stopped square-integrable Itô integral yieldThe Gronwall inequality implies $\mathbb{E}{D}_{t\wedge {\tau }_n}^2=0$. Continuous solution paths are bounded on each compact time interval, so ${\tau }_n\uparrow \infty$ almost surely. Letting $n\to \infty$ proves $X_t=Y_t$ almost surely at each fixed time, and a countable dense set of times together with continuity gives indistinguishability of stochastic processes. Therefore the stochastic differential equation has pathwise uniqueness.

For an SDE driven by Brownian motion, a strong solution of a stochastic differential equation is adapted to the completed filtration generated by a prescribed Brownian motion and satisfies the equation on that space. A weak solution of a stochastic differential equation consists of a probability space, filtration, Brownian motion, and adapted solution satisfying the equation. Uniqueness in law means that any two weak solutions with the same initial law have the same law as processes. Pathwise uniqueness means that two solutions on the same filtered space, driven by the same Brownian motion and having the same initial value, are indistinguishable.

The drift $v/2$ is continuously differentiable and hence locally Lipschitz on the open interval $(0,1)$, while the diffusion coefficient is the constant one. The local existence and pathwise uniqueness theorem for a stochastic differential equation therefore gives a unique strong solution up to its first exit from every compact subinterval. These solutions agree by pathwise uniqueness, producing a unique maximal local solution of a stochastic differential equation whose lifetime is

For the equation actually produced by the preceding construction, namely the Bessel process equation of dimension $2-1/\delta$, the drift has Lipschitz continuity on every compact subset of $(0,\infty )$. Starting at any positive time and position, pathwise uniqueness therefore makes the time-changed process agree until its first hit of zero with the maximal local solution of a stochastic differential equation. For $\delta >1$, its dimension lies in $(1,2)$, so it can hit zero; the time-change construction then supplies further excursions, whereas the maximal local solution on $(0,\infty )$ stops at that first hit.

For $\delta =1$, $X_s=|B_s|$ is Reflected Brownian motion; away from zero it agrees with the maximal local solution of $dX=d\tilde{B}$. For the dimension-$\delta$ equation printed in the paper, the preceding construction does not agree with the maximal local solution unless $\delta =1$, for the coefficient mismatch established in part (d).

Pathwise uniqueness means that two solutions on the same filtered probability space, driven by the same Brownian motion and having the same initial value almost surely, are indistinguishable. Uniqueness in law means that any two weak solutions with the same initial distribution induce the same probability law on path space.

A stochastic differential equation has strong existence for specified coefficients and initial data if a strong stochastic solution can be constructed on every prescribed stochastic basis carrying the required initial variable and driving Brownian motion. This existence assertion is distinct from pathwise uniqueness and from a definition of the solution itself.

For bounded Borel $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$, the stochastic differential equation $d{X}_t=b(X_t)dt+d{B}_t$ has strong existence and pathwise uniqueness. The identity diffusion matrix is nondegenerate. A Girsanov theorem argument alone establishes weak existence and uniqueness in law; the strong conclusion requires an additional theorem. This statement does not extend without further hypotheses to arbitrary path-dependent drift or arbitrary diffusion matrices. See the primary bounded-drift strong-solution theorem.

The zero extension of a scale diffusion coefficient at finite endpoints makes the transformed equation globally Lipschitz. Inverting its solution gives the original additive-noise equation. The pathwise bound $|X_t|\le |x|+{\sup }_{s\le T}|W_s|+\Vert b{\Vert }_{\infty }T$ prevents the inverse scale from reaching infinity at any finite time. This proves global strong existence and pathwise uniqueness even when the original drift is not Lipschitz.

With $s(x)=1$ for $x\ge 0$ and $s(x)=-1$ for $x<0$, the equation starting at zero has uniqueness in law but not pathwise uniqueness. For existence, take a Brownian motion $W$ and set $B=\int s(W)dW$; the Lévy characterization of Brownian motion and associativity of stochastic integration give $W=\int s(W)dB$. Every solution has quadratic variation $t$ and hence the Brownian motion law. Both $W$ and $-W$ are solutions driven by the same $B$, since the time spent at zero is zero and $s(-W)=-s(W)$ away from zero. Using $s(0)=0$ instead would allow the constant solution and invalidate this uniqueness-in-law example.

For $0<\epsilon <x$, the drift $g_{\epsilon }(y)=a/(y\vee \epsilon )$ is globally Lipschitz and bounded. Solve its equation strongly for the prescribed Brownian motion, and stop at the first hit of $\epsilon$. Local uniqueness makes these stopped solutions agree up to the larger threshold; their stopping times increase as the threshold decreases. With $\nu =2a-1>0$, Itô's formula for $X^{-\nu }$ gives $\mathbb{P}(T_{\epsilon }\le t)\le (\epsilon /x{)}^{\nu }$ on every finite horizon. Hence the patched lifetime is infinite. Localization away from zero proves pathwise uniqueness of the positive global solution. Uniqueness before a stopping time refers to the stopped solution, not to arbitrary unconstrained continuations afterward.