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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.

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.

For any initial point, rotate its vector through the Brownian angle:This is a nonanticipating function of the prescribed Brownian path and has the required initial value. Equivalently, $Z_t=X_t+i{Y}_t=(x_0+i{y}_0)e^{i{B}_t}$. The Itô formula gives $d{Z}_t=i{Z}_{t}d{B}_t-Z_{t}dt/2$, whose real and imaginary parts are exactly the equations in part c. Therefore the displayed formula is the strong solution of a stochastic differential equation, and uniqueness from part d makes it the only one.

The Brownian rotation in the plane preserves radius: $X_t^2+Y_t^2=x_0^2+y_0^2$. Nonzero starting points move around their initial circle, and the origin remains fixed.

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.

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

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 printed assertion is false as a supremum bound uniform over shrinking bandwidths. The supplied hint proves a pointwise stochastic order with constants uniform in $x$. Taking the supremum inside the probability changes the problem: a growing number of spatial windows produces a logarithmic cost. We first establish the valid consequence of the hint, then give a counterexample satisfying the printed assumptions.

For $I=I_{x,h}=[x-h,x+h]$, let $p=\mu (I)$, $A_T={\int }_0^T{1}_I(X_t)dt$ and $N_T={\int }_0^T{1}_I(X_t)\sigma (X_t)d{W}_t$. The Invariant distribution of an Itô diffusion has a probability density function bounded above and bounded away from zero on the required compact interval. Choosing $h_0<M/4$ therefore gives constants $c_0,C_0>0$ such that $c_{0}h\le p\le C_{0}h$, uniformly for $|x|\le M/2$ and $0<h<h_0$.

For the centered function $f=1_I-p$, $\int |f|d\mu =2p(1-p)\le 2p$. Outside $[-M,M]$, $f=-p$. The given second-moment estimate and the Chebyshev inequality yieldOn the event $A_T\ge Tp/2$, the drift coefficient contributes an average whose error is at most $R{h}^{\alpha }$, by Hölder continuity. The Itô isometry and stationarity give $\mathbb{E}{N}_T^2\le \Vert \sigma {\Vert }_{\infty }^{2}Tp$. Another use of the Chebyshev inequality provesThis also accounts for the zero-denominator convention through the first exceptional event. The bound proves pointwise $R{h}^{\alpha }+O_{\mathbb{P}}((Th{)}^{-1/2})$, uniformly in the location's tail probabilities. It supplies no bound for the probability of a supremum over all locations.

For an explicit counterexample, take $\sigma \equiv 1$ and the bounded globally Lipschitz functionIt has Lipschitz constant $R=1$, exponent $\alpha =1$, and satisfies the inward-drift conditions with $M=2$, $\gamma =2$. The stochastic differential equation has a unique strong solution of a stochastic differential equation. Its Invariant distribution of an Itô diffusion has densityThe exponent is zero on $[-1,1]$, equals $-(|x|-1{)}^2$ when $1<|x|<2$, and equals $3-2|x|$ outside $[-2,2]$. This verifies integrability, positivity and a constant density ${\rho }_0=Z^{-1}$ on the central interval. Start the Itô diffusion in this Invariant distribution of an Itô diffusion to obtain the required stationary process.

Set $h=T^{-1/3}$ and choose $N=\lfloor (4h{)}^{-1}\rfloor$ windows with centers $x_j=-1/2+(4j-2)h$, $1\le j\le N$. Their intervals $I_j=[x_j-h,x_j+h]$ are disjoint and lie in $[-1,1]$, where the drift is zero. WriteThen ${\hat{b}}_T(x_j,h)=M_j(T)/A_j(T)$ whenever $A_j(T)>0$. The same occupation bound as above gives $\mathbb{E}(A_j(T)-q{)}^2\le C_{1}T{h}^2$ for $T\ge 1$. With $\delta =T^{-1/6}$, a union bound and the Chebyshev inequality implyThe quadratic variations are $⟨M_j⟩=A_j$, and the cross quadratic covariations vanish since the windows are disjoint. Each clock tends to infinity almost surely: for any fixed window, the occupation estimate at times $2^k$, followed by the Borel-Cantelli lemmas, gives $A_j(2^k)/(2^k\mu (I_j))\to 1$. The Knight theorem for orthogonal martingales therefore represents $M_j(t)=B_j(A_j(t))$ using independent standard Brownian motions $B_1,\dots ,B_N$. This theorem is applied to the finite collection for each $T$; no growing-dimensional central limit theorem is being assumed.

On the preceding clock event, compare each time-changed Brownian motion with its value at deterministic time $q$. The Brownian reflection principle and a union bound show that, for every fixed $\epsilon >0$,This bound does not require the Brownian motions to be independent of their clocks. In particular, ${\max }_j|M_j(T)-B_j(q)|/\sqrt{q}\to 0$ in probability. The variables $B_j(q)/\sqrt{q}$ are independent $N(0,1)$ random variables. Their Gaussian maximum exceeds $\sqrt{\log N}$ with probability tending to one: for $r\ge 1$, integration of the normal density over $[r,r+1/r]$ gives $\mathbb{P}(|Z|>r)\ge c{r}^{-1}{e}^{-r^2/2}$, and henceSince $A_j(T)\le (1+\delta )q$ on the clock event, it follows that, for some $c_3>0$,But $\sqrt{Th}Rh=1$ for this bandwidth. Thus the error after subtraction of $Rh$ and multiplication by $\sqrt{Th}$ is not bounded in probability, contradicting the printed stochastic order. The valid pointwise bound above is provable; the uniform shrinking-bandwidth assertion is not. This counterexample shows why a uniform rate needs at least a $\sqrt{\log (1/h)}$ enlargement of the noise scale. If $h$ is instead held fixed and constants may depend on $h$, this counterexample does not contradict that different interpretation.

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.

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.

Since $B_{T_a}=0$, applying Itô formula to $B_t^3$ after $T_a$ givesBefore $T_a$, both sides vanish. Since $T_a$ is a stopping time determined by $B$, this is a strong solution of a stochastic differential equation.

Taking $a=0$ gives $X_t=B_t^3$, whereas any $a>0$ gives a solution that remains zero until $T_a$; these differ with positive probability while using the same Brownian motion and initial value. Therefore pathwise uniqueness fails.

A strong solution of a stochastic differential equation is adapted to the completed filtration of a prescribed Brownian motion $B$ on a prescribed probability space and satisfiesalmost surely. A weak solution of a stochastic differential equation may choose the filtered probability space, Brownian motion, and adapted process as part of the solution; only the displayed integral equation and the prescribed initial law are required.

A strong solution of a stochastic differential equation is an adapted process $X$ on a prescribed filtered probability space carrying a prescribed Brownian motion $W$, satisfyingalmost surely. A weak solution of a stochastic differential equation consists of a filtered probability space, a Brownian motion, and an adapted process on that space satisfying the same integral equation; the space and driving Brownian motion are part of what may be chosen.