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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.

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.

On the right-hand side, ${\mathbb{E}}_x$ means that the reference Brownian motion starts at $B_0=x$; on the left it means the stochastic differential equation starts at $X_0=x$. Since $h$ is smooth and compactly supported, $h^{\prime }$ is bounded and globally Lipschitz continuous. Define the stochastic exponentialThe Novikov condition holds on every finite horizon $T$, because $\mathbb{E}\exp (\frac{1}{2}{\int }_0^T{h}^{\prime }(B_s{)}^{2}ds)\le \exp (T\Vert h^{\prime }{\Vert }_{\infty }^2/2)$. Thus $D$ is a true martingale. Under $d\mathbb{Q}=D_{T}d\mathbb{P}$, the Girsanov theorem makes ${\hat{B}}_t=B_t-x-{\int }_0^t{h}^{\prime }(B_s)ds$ a standard Brownian motion for $t\le T$. Consequently $B$ under $\mathbb{Q}$ solves the prescribed stochastic differential equation; the existence and pathwise uniqueness theorem for a stochastic differential equation identifies its law with that of $X$.

The Itô formula gives ${\int }_0^t{h}^{\prime }(B_s)d{B}_s=h(B_t)-h(x)-\frac{1}{2}{\int }_0^t{h}^{\prime \prime }(B_s)ds$. Hence, for any bounded measurable $f$,The finite-horizon argument applies separately to every $t\ge 0$.

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