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Replace the square-root coefficient near zero by smooth globally Lipschitz cutoffs that vanish below half the cutoff level. With a common Brownian driver, local pathwise uniqueness makes the solutions agree until the larger cutoff level. The exit times increase as the levels decrease. The compatible integrands $1_{\{s\le T_n\}}\sqrt{Z_s}$ have increasing squares and expected time-integrals at most $zt$ by the nonnegative local martingale bound. Their limit is square-integrable on every finite horizon. The Itô isometry gives a continuous global limiting stochastic integral; at a finite limiting exit time its value is zero, and it remains absorbed afterward.

Globally Lipschitz drift and diffusion coefficients have linear growth and give a nonexplosive strong stochastic solution for a prescribed initial state and Brownian motion. The solution satisfies pathwise uniqueness. Local versions apply after stopping inside compact regions.

First stop the solution on leaving $[-n,n]$, at ${\tau }_n$, and set $F_n(t)=\mathbb{E}{\sup }_{u\le t}|X_{u\wedge {\tau }_n}{|}^2$. The stochastic integral representation, the Doob L2 maximal inequality and the Cauchy-Schwarz inequality give, for $0\le t\le 1$,Here stopping ensures square integrability before the calculation, and $8{\sigma }^2+2t{b}^2\le 8({\sigma }^2+b^2)$ explains the constant. The Gronwall inequality gives $F_n(t)\le e^{8At}-1$. For globally Lipschitz functions as coefficients, the global existence theorem for stochastic differential equations with Lipschitz coefficients supplies a nonexplosive solution. Letting $n$ increase and using Fatou's lemma proves the slightly stronger maximal second-moment bound under linear growth:

For coefficients that are only locally Lipschitz functions, let ${\pi }_n(x)=\max (-n,\min (x,n))$ and replace the coefficients by $\sigma ({\pi }_n(x))$ and $b({\pi }_n(x))$. These are globally Lipschitz functions, agree with the originals on $[-n,n]$, and obey the same linear growth condition for an SDE, since $|{\pi }_n(x)|\le |x|$. The globally defined solutions agree until their common exits from $[-n,n]$ by pathwise uniqueness. The uniform preceding estimate and Markov's inequality implyPatch the solutions before their increasing exit times. Their limiting lifetime exceeds $1$ almost surely by this bound. Thus a pathwise unique strong solution exists throughout $[0,1]$ even for locally Lipschitz coefficients.

The existence and pathwise uniqueness theorem for a stochastic differential equation states that locally Lipschitz drift $b$ and diffusion $\sigma$ on an open domain give a pathwise unique strong solution of a stochastic differential equation up to a maximal lifetime $\xi$. The initial value is measurable with respect to the initial sigma-algebra, and the filtration satisfies the usual conditions for a filtration. On ${\mathbb{R}}^d$, if $T_n$ are exits from increasing balls, then $T_n\uparrow \xi$; a finite lifetime entails escape from every compact subset. The coefficients need not satisfy a global growth bound for this local result.

One construction truncates the coefficients outside each compact region to globally Lipschitz coefficients, solves the resulting global equations, and patches their solutions until the exits. Pathwise uniqueness makes solutions from larger truncations agree on the smaller stopped interval. Their union is the maximal local solution of a stochastic differential equation, and any extension would contradict the exhaustion of the domain by the exit times.

Let $X,Y$ be two solutions with the same Brownian driver and the same initial value. Write $D=X-Y$, and take a common Lipschitz constant $L$ for both coefficients. Stop when $|X|+|Y|$ reaches $n$. Before this stopping time ${\tau }_n$, all difference integrands below are bounded. The Itô formula givesThe stopped stochastic integral is a true martingale on every finite horizon. Lipschitz continuity therefore yieldsSince $D_0=0$, the Gronwall inequality makes the left side zero. Let $n\to \infty$; the continuous global solutions are bounded on each compact time interval, so their stopping times exhaust it. Equality first at every rational time and then by continuity gives indistinguishability. This proves pathwise uniqueness for globally Lipschitz drift and diffusion coefficients. The argument does not require the initial value to have a global second moment: after stopping, the difference is bounded and initially zero.

A weak solution of a stochastic differential equation consists of a choice of probability space, filtration, driving Brownian motion and adapted solution with the specified initial law. The driving noise and the space are part of what may be chosen.

A strong solution of a stochastic differential equation is constructed for a prescribed Brownian motion and initial value, as a nonanticipating measurable functional of that data; for deterministic initial value it is adapted to the usual augmentation of the Brownian filtration. No further independent randomness is required.

Uniqueness in law means that any two weak solutions of a stochastic differential equation with the same initial distribution have the same law as solution processes. Pathwise uniqueness means that two solutions on the same filtered space, driven by the same Brownian motion and with the same initial value almost surely, are indistinguishable: $\mathbb{P}(X_t={\tilde{X}}_t\text{ for every }t)=1$. The latter concerns equality of paths on a common space, whereas uniqueness in law concerns equality of probability distributions.

Both coefficient functions are linear, hence globally Lipschitz. Their norms are bounded by a constant times $|(x,y)|$, so they satisfy the linear growth condition for an SDE. The existence and pathwise uniqueness theorem for a stochastic differential equation gives a unique global strong solution of a stochastic differential equation for every deterministic starting point $(x_0,y_0)\in {\mathbb{R}}^2$. Linear growth prevents finite-time explosion.

The theorem gives pathwise uniqueness, and strong existence with pathwise uniqueness gives uniqueness in law. Thus every initial value is permitted, and both forms of uniqueness hold. The coefficients need not be bounded for this theorem; their global Lipschitz and linear-growth properties are sufficient.

A strong solution of a stochastic differential equation is constructed for the prescribed driving Brownian motion on the prescribed stochastic basis, adapted to the augmented filtration generated by that driver and the initial data, and satisfies the integral equation almost surely. For this equation it must be continuous, nonnegative and satisfy $Z_t=z+{\int }_0^t\sqrt{Z_s}d{B}_s$. Pathwise uniqueness means that any two solutions on the same filtered space, with the same initial value and the same driving Brownian motion, are indistinguishable.

The standard global existence theorem for stochastic differential equations with Lipschitz coefficients cannot be applied to the square-root branching diffusion, because $\sigma (x)=\sqrt{x}$ is not Lipschitz at zero:It is locally Lipschitz on $(0,\infty )$, which gives a solution only up to its possible approach to zero by that theorem. Thus the missing hypothesis is the Lipschitz bound at the absorbing boundary, not any failure of continuity. Failure of this sufficient criterion does not establish failure of strong existence or uniqueness; part (c) constructs strong existence directly.

For fixed $\epsilon >0$, the derivative of ${\sigma }_{\epsilon }$ is zero on $(-\infty ,\epsilon /2]$, bounded on the compact transition interval $[\epsilon /2,\epsilon ]$, and equals $1/(2\sqrt{x})\le 1/(2\sqrt{\epsilon })$ for $x\ge \epsilon$. Thus ${\sigma }_{\epsilon }$ is globally Lipschitz, with at most linear growth. The global existence theorem for stochastic differential equations with Lipschitz coefficients supplies a global strong stochastic solution and pathwise uniqueness for the modified equation.

The solution stays nonnegative. More precisely, if $z>\epsilon /2$, it cannot cross $\epsilon /2$: on reaching this level the constant continuation solves the equation, and pathwise uniqueness forces that continuation. If $z\le \epsilon /2$, it is constant from the outset. In either case $Z^{\epsilon }$ is a nonnegative local martingale, so $\mathbb{E}{Z}_s^{\epsilon }\le z$.

For ${\epsilon }^{\prime }<\epsilon <z$, both coefficients agree with $\sqrt{x}$ on $[\epsilon ,\infty )$. Stop the two solutions when either first reaches $\epsilon$, and also at a common upper bound. On the resulting compact state interval, the coefficient is Lipschitz. The Itô isometry applied to their difference, followed by the Gronwall inequality, gives zero expected squared difference. Remove the upper bound using the nonexplosion of each modified solution. The two solutions agree until the first of these lower exits, and by continuity both have value $\epsilon$ there, so their exits at that level coincide. In particular,up to indistinguishability. If $\epsilon \ge z$, the first exit time is zero and the same assertion is immediate from their initial values.

Now take a decreasing sequence $0<{\epsilon }_n<z$ with ${\epsilon }_n\downarrow 0$. Write $Z^n=Z^{{\epsilon }_n}$ and ${\tau }_n=T_{{\epsilon }_n}$. Compatibility gives ${\tau }_n\le {\tau }_{n+1}$; indeed, a continuous path cannot reach the smaller level before reaching the larger one. Let $\tau ={\lim }_n{\tau }_n$, a stopping time. On $[0,\tau )$ the compatible solutions define an adapted continuous process $Z$ by setting $Z_t=Z_t^n$ whenever $t\le {\tau }_n$. We still need to justify the finite limiting endpoint and the integral equation there, rather than assume they exist.

For this purpose use the compatible predictable integrandsTheir squares increase pointwise with $n$: on the smaller interval compatibility makes the values equal, and outside it the old integrand is zero. Their limit $H$ is predictable and equals $\sqrt{Z_s}$ before $\tau$, and zero after $\tau$, apart from irrelevant single-time endpoints. For every finite $t$,The monotone convergence theorem gives $\mathbb{E}{\int }_0^t{H}_s^{2}ds\le zt$. Moreover, compatibility makes $(H-H^n{)}^2=H^2-(H^n{)}^2$ away from endpoints, so its expected time-integral tends to zero. The Itô isometry, and the Doob L2 maximal inequality if uniform convergence on a finite time interval is desired, therefore give a continuous stochastic-integral limitBefore ${\tau }_n$, the integrand agrees with $\sqrt{Z^n}$, which is the original square-root coefficient there. Locality of the stochastic integral yields $L_{t\wedge {\tau }_n}=Z_{t\wedge {\tau }_n}^n$. A countable intersection in $n$ and path continuity give this equality simultaneously for all times.

On $\{\tau <\infty \}$, every ${\tau }_n$ is finite and $L_{{\tau }_n}={\epsilon }_n$. Continuity of $L$ thus gives $L_{\tau }=0$. The limiting integrand is zero after $\tau$, so $L$ remains zero afterward. On $\{\tau =\infty \}$, every finite time is covered by one of the compatible positive solutions. Consequently $L$ is nonnegative everywhere and $H_s=\sqrt{L_s}$ for almost every $s$, including after absorption. The cutoff construction of an absorbed square-root diffusion has producedAll modified solutions, stopping times and integrands were constructed from the same prescribed Brownian motion; hence this is a strong stochastic solution. The integral is square-integrable on every finite horizon by the bound $zt$, which also rules out a hidden finite-time explosion in the patching argument.

Apply the Itô formula to $Y_t=e^{B_t+t/2}$. The time derivative contributes $Y_{t}dt/2$, and the quadratic-variation correction contributes another $Y_{t}dt/2$. ThusBoth the drift $b(y)=y$ and diffusion coefficient $\sigma (y)=y$ are globally Lipschitz. The global existence theorem for stochastic differential equations with Lipschitz coefficients gives a unique strong stochastic solution for a fixed driving Brownian motion and initial value; in particular pathwise uniqueness holds. The explicit solution is strictly positive.

Write $N_t={\int }_0^t{Y}_{s}d{B}_s$. This is a continuous local martingale withThe strong law for Brownian motion gives $B_t/t\to 0$ almost surely, so $\log Y_t/t\to 1/2$, $Y_t\to \infty$, and $H_t\to \infty$. The Dambis-Dubins-Schwarz theorem therefore supplies a Brownian motion $\beta$ in the inverse-clock filtration with $N_t={\beta }_{H_t}$ for all $t$. Finally $d{H}_s=Y_s^{2}ds$, hence $Y_{s}ds=Y_s^{-1}d{H}_s$. Integrating the stochastic equation givesThe infinite clock lifetime verifies that this Brownian motion is defined for every nonnegative clock time without an additional extension.

Fix $x=X_0>0$ and $a>1/2$. For the construction take $0<\epsilon <x$, the range relevant to localization at zero, and defineThis drift is bounded by $a/\epsilon$ and globally Lipschitz with constant at most $a/{\epsilon }^2$. Together with the constant diffusion coefficient $1$, the global existence theorem for stochastic differential equations with Lipschitz coefficients gives a unique strong stochastic solution $U^{\epsilon }$ offor the prescribed driving Brownian motion. Let $T_{\epsilon }^{\epsilon }=\inf \{t:U_t^{\epsilon }=\epsilon \}$ and choose $X_t^{\epsilon }=U_{t\wedge T_{\epsilon }^{\epsilon }}^{\epsilon }$ as the stopped continuation. Before this time the drift is exactly $a/X_t^{\epsilon }$. Uniqueness here means uniqueness up to the stopping time; the equation imposes no restriction on arbitrary continuations after it.

For $0<\delta <\epsilon <x$, the two equations have the same coefficients while both solutions exceed $\epsilon$. By local pathwise uniqueness they coincide until that level is reached. Thus the first hit of $\epsilon$ by $X^{\delta }$ is $T_{\epsilon }^{\epsilon }$, and its later first hit of $\delta$ satisfiesPatch along a deterministic sequence ${\epsilon }_n\downarrow 0$: for $t<T:={\lim }_n{T}_{{\epsilon }_n}^{{\epsilon }_n}$, choose $n$ with $t<T_{{\epsilon }_n}^{{\epsilon }_n}$ and set $X_t=X_t^{{\epsilon }_n}$. Compatibility makes the definition independent of $n$. It gives a continuous adapted positive process satisfying the original equation on every compact time interval before $T$. The same compatibility holds for other threshold choices, so the lifetime is the limit as $\epsilon \downarrow 0$, not a property of the chosen sequence.

To prove $T=\infty$, put $\nu =2a-1>0$ and $s(y)=y^{-\nu }$. Its generator vanishes. On each finite horizon, Itô's formula makes $s(X_{t\wedge T_{\epsilon }^{\epsilon }}^{\epsilon })$ a true martingale, because its stochastic integrand is bounded by $\nu {\epsilon }^{-\nu -1}$. Positivity therefore impliesSince $\{T\le t\}$ is contained in each $\{T_{\epsilon }^{\epsilon }\le t\}$, letting $\epsilon \downarrow 0$ gives $\mathbb{P}(T\le t)=0$. Taking a countable union over integer $t$ proves $T=\infty$ almost surely. This is the truncation construction of a positive Bessel strong solution.

For two positive solutions driven by the same Brownian motion, localize at their first hits of $\epsilon$. The cutoff equations and pathwise uniqueness make them equal up to those times. On every finite interval both positive continuous paths have positive minima, so decreasing $\epsilon$ removes the localization and proves indistinguishability. ThusThe resulting global solution also supplies the stopped solution for any threshold $\epsilon \ge x$, completing the literal all-threshold existence request. The monotonicity used to construct the zero-boundary lifetime concerns the thresholds $\epsilon <x$ tending to zero; upper-threshold hitting times have the opposite ordering. Strong adaptation is preserved because every cutoff solution is strong and the patch uses their stopping times in the given filtration.

Uniqueness in law means that any two weak stochastic solutions with the same prescribed initial distribution have the same distribution as random elements of the continuous-path space. Their probability spaces and driving Brownian motions may differ.

Pathwise uniqueness means that if two solutions are defined on the same filtered probability space, driven by the same Brownian motion, and have the same initial variable almost surely, thenThis is indistinguishability of stochastic processes, a statement about coupled sample paths. It is stronger than requiring only equality of their distributions.

On $I=g(\mathbb{R})$ define $h(y)=g^{\prime }(g^{-1}(y))$. Differentiation givesThe mean value theorem proves Lipschitz continuity of $h$ on $I$, with constant $2B$.

If an endpoint of $I$ is finite, the Lipschitz bound gives a finite limiting value of $h$ there. That value must be zero. Otherwise $h$ would be bounded below by a positive number near the endpoint, andwould make $g^{-1}$ approach a finite limit, contradicting its tending to $+\infty$ or $-\infty$. The zero extension of a scale diffusion coefficient at finite endpoints therefore gives a globally Lipschitz function $\bar{h}$ on $\mathbb{R}$: set it to zero beyond each finite endpoint and retain $h$ on $I$.

Use the following standard global existence theorem for stochastic differential equations with Lipschitz coefficients: globally Lipschitz drift and diffusion coefficients give a nonexplosive, pathwise unique strong stochastic solution for every prescribed initial state and driving Brownian motion. Global Lipschitz continuity also gives the required linear-growth bound. Apply it toIt remains to check that $Y$ stays in $I$, so the inverse transform is defined.

Before the first boundary time, put $X=g^{-1}(Y)$. Since $(g^{-1}{)}^{\prime }=1/h$ and $(g^{-1}{)}^{\prime \prime }=-h^{\prime }/h^2$, the Itô formula, stopped inside compact subintervals of $I$, yieldsFor each finite $T$, up to that boundary time,A finite endpoint of $I$ would require $X$ to diverge, which this bound excludes. Therefore $Y$ stays in $I$ at every finite time, and $X=g^{-1}(Y)$ is a global strong stochastic solution.

Finally, any two solutions driven by the same $W$ transform into solutions of the globally Lipschitz $Y$ equation. Its pathwise uniqueness makes the transforms, and hence their inverses, indistinguishable. Thus

The reference Brownian motion must start at $x$, and both laws must use the same stopped-path rule. Under the reference probability measure let $R_s=x+W_s$, let $\tau$ be its exit time from $(a,b)$, and put $T=t\wedge \tau$ and $c=(\delta -1)/2$. The Girsanov density for a stopped Bessel process, as a functional of the stopped reference path, isUse the predictable process ${\theta }_s=c{R}_s^{-1}{1}_{\{s\le \tau \}}$, with any harmless choice at the endpoint. On $[0,t]$, $\int {\theta }_s^{2}ds\le c^{2}t/a^2$. The Novikov condition holds; under the density $D$, the Girsanov theorem makes ${\tilde{W}}_s=W_s-{\int }_0^s{\theta }_{u}du$ a Brownian motion. Before exit, $d{R}_s=c{R}_s^{-1}ds+d{\tilde{W}}_s$, which is the required stochastic differential equation. The drift is a locally Lipschitz function on $(0,\infty )$, so pathwise uniqueness and uniqueness in law hold up to this exit time.

For a version involving just the endpoint and an ordinary integral, the Itô formula for $\log R$ givesThusThe stopped density is measurable from the stopped path, so it is also the Radon-Nikodym derivative of the stopped-path laws. The stopping times in this formula are evaluated on the reference path, rather than taken from a different realization of $X$. A zero-start reference Brownian motion would have a different deterministic initial value and could not dominate this law.

A weak solution of a stochastic differential equation consists of a filtered probability space, an adapted continuous process $X$, and a Brownian motion $B$ relative to that filtration, with specified initial distribution, such that for every finite $t$ the coefficient integrals are finite andIn one dimension the integrability conditions are ${\int }_0^t\sigma (X_s{)}^{2}ds<\infty$ and ${\int }_0^t|b(X_s)|ds<\infty$. The noise integral is the Itô integral. A solution on a prescribed stochastic basis with prescribed $B$ is distinguished from this weak construction.

A strong solution of a stochastic differential equation uses the prescribed $B$ and initial variable, and is adapted to their augmented natural filtration; thus no additional randomness is needed. For deterministic $X_0=x$, each $X_t$ is determined by the driving Brownian motion through time $t$. Merely being adapted to a larger filtration is insufficient for this definition.

Uniqueness in law means that any two weak stochastic solutions with the same initial distribution have the same law of $X$ on continuous path space. Pathwise uniqueness means that two solutions on one stochastic basis, with the same $B$ and almost surely equal initial values, are indistinguishable stochastic processes. The latter compares paths under the same noise, rather than just comparing their probability distributions.

The Girsanov theorem construction proves a weak stochastic solution, but does not by itself establish a strong stochastic solution. It constructs $X=x+W$ first, and then constructs a new driving Brownian motion $B=W-\int b(X_s)ds$. Adaptation of $X$ to the original filtration of $W$ does not by itself prove adaptation to the smaller natural filtration of $B$. Inverting this relation requires an additional result; uniqueness in law alone is insufficient in general.

For the particular coefficients in part (b), however, the literal claim that a strong solution may fail is incorrect. The strong existence theorem for additive-noise SDEs with bounded measurable drift gives strong existence and pathwise uniqueness for bounded Borel state-dependent $b$ and unit diffusion. Consequently every weak realization here is determined by its driving noise and initial value. This theorem is deeper than the Girsanov theorem argument and no proof is needed here. Examples with path-dependent drift or with a nonconstant diffusion coefficient do not contradict it.

The valid distinction is therefore: the argument in part (b) has established weak existence and uniqueness in law; strong existence requires an additional theorem, and in this class that theorem is available. A primary reference for the correction is On strong solutions and explicit formulas for solutions of stochastic integral equations.

Specify an initial value or initial probability distribution. A strong solution of a stochastic differential equation uses a prescribed Brownian motion on a prescribed filtered probability space. It is continuous, adapted to the completed filtration generated by that Brownian motion and the initial value, and satisfiesalmost surely for every $t$, with ${\int }_0^t(|b(X_s)|+\sigma (X_s{)}^2)ds<\infty$ almost surely. The given initial variable is independent of future Brownian motion increments. Some courses allow a larger prescribed filtration in their definition of a strong solution of a stochastic differential equation; the Brownian-generated convention states explicitly what absence of extra randomness means.

A weak solution of a stochastic differential equation consists of a filtered probability space, a Brownian motion relative to its filtration, and a continuous adapted process $X$ satisfying the same integrability and integral equation. Here the space and noise are part of the unknown; the filtration may contain randomness beyond the initial variable and the driving path.

Pathwise uniqueness means that on any common filtered probability space, any two solutions with the same driving Brownian motion and the same initial value are indistinguishable. Uniqueness in law means that all weak solutions with the same initial probability distribution have the same law as path-valued random variables, even on different spaces. Pathwise uniqueness compares shared-noise paths; uniqueness in law compares path distributions. Neither uniqueness notion by itself asserts existence.