Locally Lipschitz Itô SDE on an open state domain

Assume AC. Fix a finite horizon $T>0$, finite dimensions $d,m\ge 1$, and an open set $U\subseteq {\mathbb{R}}^d$. Use a complete usual filtered probability space and an adapted standard $m$-dimensional Brownian motion whose increments after time $s$ are independent of ${\mathcal{F}}_s$, as in $d$-dimensional Brownian motion and Continuous Brownian Ito processes. Let $b:[0,T]\times U\to {\mathbb{R}}^d$ and $\sigma :[0,T]\times U\to {\mathbb{R}}^{d\times m}$ be Borel maps. Require that they are bounded on every compact subset of $[0,T]\times U$ and locally Lipschitz in the state uniformly in time: for every compact $K\subset U$ there is $L_K<\mathrm{\infty }$ such that, for every $t\in [0,T]$ and $x,y\in K$, $\mathrm{\mid }b(t,x)-b(t,y)\mathrm{\mid }+\mathrm{\parallel }\sigma (t,x)-\sigma (t,y){\mathrm{\parallel }}_F\le L_K\mathrm{\mid }x-y\mathrm{\mid }\mathrm{.}$ Let $\xi$ be ${\mathcal{F}}_0$-measurable, take values in $U$ almost surely, be independent of the future Brownian increments, and satisfy $\mathbb{E}\mathrm{\mid }\xi {\mathrm{\mid }}^2<\mathrm{\infty }$.

For $n\ge 1$, set $K_n=\{x\in U:\mathrm{\mid }x\mathrm{\mid }\le n,\mathrm{dist}(x,U^c)\ge n^{-1}\}\mathrm{.}$ When $U={\mathbb{R}}^d$, interpret $\mathrm{dist}(x,U^c)=+\mathrm{\infty }$. If $U^c\mathrm{\ne }\mathrm{\varnothing }$, the distance function is $1$-Lipschitz, so each $K_n$ is closed; it is bounded, hence compact by Heine--Borel. For $U={\mathbb{R}}^d$, each $K_n$ is a closed ball and is compact as well. As $n$ increases, the radius bound increases and the distance threshold decreases, so the sets are nested. Every $x\in U$ belongs to $\mathrm{int}{K}_n$ for all sufficiently large $n$: choose $n>\mathrm{\mid }x\mathrm{\mid }$ and, when $U^c\mathrm{\ne }\mathrm{\varnothing }$, $n^{-1}<\mathrm{dist}(x,U^c)$. Thus their interiors exhaust $U$. $\mathrm{\mid }d(x,A)-d(y,A)\mathrm{\mid }\le d(x,y)$, so the distance to a fixed nonempty set is $1$-Lipschitz Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line

A local strong solution up to exit is the compatible family of stopped adapted continuous processes $X^{(n)}$ and stopping times ${\tau }_n$ Continuous-time stopping times and stopped sigma-algebras such that ${\tau }_n=\inf \{t\in [0,T]:X_t^{(n)}\mathrm{\notin }\mathrm{int}{K}_n\}\wedge T,X_{t\wedge {\tau }_n}^{(n)}=\xi +{\int }_0^{t\wedge {\tau }_n}b(s,X_s^{(n)})ds+{\int }_0^{t\wedge {\tau }_n}\sigma (s,X_s^{(n)})d{W}_s,$ with the equation holding indistinguishably and with $X^{(n)}$ constant after ${\tau }_n$. Empty early $K_n$ simply give ${\tau }_n=0$. Compatibility means that for $m\ge n$, $X^{(m)}$ stopped at ${\tau }_n$ equals $X^{(n)}$ up to indistinguishability. The maximal lifetime is $\zeta ={\lim }_n{\tau }_n$; for $t<\zeta$, the common value of $X_t^{(n)}$ for sufficiently large $n$ defines the maximal local solution $X_t$.

This definition records a local integral equation; it does not assert that a solution exists or that $\zeta =T$. Those are separate theorem obligations. The local coefficient conditions allow superlinear growth; global Lipschitz hypotheses will be imposed only on the coefficient truncations used in the existence proof.