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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.

Use one $d$-dimensional Brownian motion $W$ to couple the diffusion processesThe global existence theorem for stochastic differential equations with Lipschitz coefficients supplies a unique nonexplosive strong stochastic solution. Let $x$ be the deterministic drift trajectory. Subtraction and the Gronwall inequality give, on every finite horizon $t$,This also follows directly from the integral equations with continuous forcing, so the coupling can be chosen simultaneously for all $\epsilon$.

For completeness derive the needed Feynman-Kac formula with its potential sign. For fixed $t$, apply the Itô formula and the Itô product rule toIts drift is the exponential factor times $-u_t^{\epsilon }+({\epsilon }^2\Delta /2+b\cdot \nabla )u^{\epsilon }+c{u}^{\epsilon }$, which vanishes by the partial differential equation. It is a local martingale. Boundedness of $u^{\epsilon }$ on the finite time slab and boundedness of $c$ make $Y$ bounded there, so it is a true martingale. Its endpoint expectations giveThe deterministic trajectory on $[0,t]$ is compact. Uniform pathwise convergence puts all sufficiently small-noise paths in a fixed compact neighbourhood of that trajectory. Continuity of $c$ therefore implies ${\sup }_{s\le t}|c(X_s^{\epsilon })-c(x_s)|\to 0$; global uniform continuity of $c$ is not required. Continuity of $f$ gives convergence of the terminal factor. Finally,The dominated convergence theorem proves the zero-noise limit with a bounded potential:At $t=0$ this is the given initial condition. All boundedness arguments concern a fixed finite horizon; no uniform bound over infinite time is needed.

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.

The drift $b(z)=\tanh z$ has derivative $b^{\prime }(z)=1/{\cosh }^{2}z$, so $|b^{\prime }(z)|\le 1$. Hence it is globally Lipschitz. The diffusion coefficient $\sigma (z)=1$ is globally Lipschitz as well, and both coefficients satisfy a linear growth bound.

The global existence theorem for stochastic differential equations with Lipschitz coefficients states that globally Lipschitz coefficients with linear growth give, for each deterministic initial point, an adapted continuous strong solution of a stochastic differential equation on every finite interval, with pathwise uniqueness and no finite-time explosion. Applying this theorem gives a unique strong solution for every $X_0=x\in \mathbb{R}$, satisfying

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

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.

Write the diffusion and drift coefficients asBoth are continuously differentiable, withThese derivatives are bounded. For example $2|x|\le 1+x^2$ bounds both absolute values by one. The mean value theorem therefore makes both coefficients globally Lipschitz. They are also bounded, so satisfy the linear growth condition for an SDE.

The global existence theorem for stochastic differential equations with Lipschitz coefficients consequently givesIn particular, there is no finite explosion time. The solution has continuous paths, so all subsequent level-hitting and stopping arguments apply.

Let $K={\sup }_{x}f(x)<\infty$ and let $C$ be a Lipschitz constant for $f$. Sinceboth coefficients $f$ and $-f^2/2$ are bounded and globally Lipschitz. The global existence theorem for stochastic differential equations with Lipschitz coefficients gives a unique global strong solution of a stochastic differential equation.

The same Itô formula cancellation as before givesFor $S=T_{-a}\wedge T_b$, the stopped exponential is bounded between $e^{-a}$ and $e^b$. Continuity and strict positivity of $f$ giveThe Itô isometry therefore yieldsThe monotone convergence theorem proves $\mathbb{E}S<\infty$. The bounded optional stopping argument is now identical to part (b)(iii), and givesPassing from level hitting to strict exceedances as in part (b)(iv),No global positive lower bound on $f$ was needed: compact-interval positivity suffices for each exit argument. The law is independent of the particular $f$ because the scale function is always $s(x)=e^x$: its infinitesimal generator satisfiesThis is the exponential scale function for a drift tied to diffusivity. Changing $f$ changes how quickly the path is traversed, but leaves these boundary probabilities and the maximum law unchanged.