Local strong existence and uniqueness up to exit for locally Lipschitz SDEs

For each $n$, choose scalar piecewise-linear cutoffs ${\eta }_n(r)$ and ${\theta }_n(r)$ with ${\eta }_n=1$ on $[0,n]$, ${\eta }_n=0$ on $[n+1,\mathrm{\infty })$, ${\theta }_n=0$ on $[0,(2n{)}^{-1}]$, and ${\theta }_n=1$ on $[n^{-1},\mathrm{\infty })$. Put ${\chi }_n(x)={\eta }_n(\mathrm{\mid }x\mathrm{\mid }){\theta }_n(\mathrm{dist}(x,U^c))$ on $U$, extend it by zero outside $U$, and use ${\theta }_n\equiv 1$ when $U={\mathbb{R}}^d$. The norm and distance functions are Lipschitz, so ${\chi }_n$ is Lipschitz, equals one on $K_n$, and has compact support inside $U$.

Define $b_n={\chi }_{n}b$ and ${\sigma }_n={\chi }_n\sigma$ on $U$, extended by zero outside $U$. On a compact neighborhood of their support the original coefficients are bounded and locally Lipschitz; a finite subcover gives a uniform local Lipschitz constant there. Their products with ${\chi }_n$ are therefore locally Lipschitz, including at the support boundary where the cutoff vanishes. A compactly supported locally Lipschitz map on ${\mathbb{R}}^d$ is globally Lipschitz: use a finite cover and a Lebesgue number for nearby pairs, and boundedness divided by that Lebesgue number for pairs farther apart. Thus $b_n,{\sigma }_n$ are globally Lipschitz and have linear growth.

By [F2], each truncated equation with coefficients $b_n,{\sigma }_n$ and initial value $\xi$ has a unique global strong solution $Z^n$. If $K_n=\mathrm{\varnothing }$, take its exit time to be zero; otherwise stop at the first exit ${\tau }_n$ from $\mathrm{int}{K}_n$. For $k\ge n$, stop $Z^n,Z^k$ at the first time either leaves $\mathrm{int}{K}_n$. Before that stop both states lie in $K_n$, where both truncations equal $b,\sigma$.

Let $D=Z^n-Z^k$ up to the common stop $\tau$, and put $v(t)=\mathbb{E}{\sup }_{s\le t}\mathrm{\mid }{D}_{s\wedge \tau }{\mathrm{\mid }}^2$, finite by [F2]. Write a finite-energy matrix integral coordinatewise as $M_s^i={\sum }_{j=1}^m{\int }_0^s{G}_u^{ij}d{W}_u^j$. Since $\mathrm{\mid }{\sum }_j{a}_j{\mathrm{\mid }}^2\le m{\sum }_j\mathrm{\mid }{a}_j{\mathrm{\mid }}^2$ by [F6], summing the scalar bounds [F4] gives $\mathbb{E}{\sup }_{s\le t}\mathrm{\mid }{M}_s{\mathrm{\mid }}^2\le 4m\mathbb{E}{\int }_0^t\mathrm{\parallel }{G}_u{\mathrm{\parallel }}_F^{2}du$. Apply this to $G_u=1_{\{u\le \tau \}}(\sigma (u,Z_u^n)-\sigma (u,Z_u^k))$; boundedness on $K_n$ gives finite energy. The common coefficient Lipschitz constant $L_n$ on $K_n$, [F5], and $D_0=0$ now give $v(t)\le (2T{L}_n^2+8m{L}_n^2){\int }_0^{t}v(s)ds\mathrm{.}$ With $C_n=2T{L}_n^2+8m{L}_n^2$, the continuous majorant $V(t)=C_n{\int }_0^{t}v(s)ds$ satisfies $v\le V$ and $V(t)\le C_n{\int }_0^{t}V(s)ds$, so [F7] gives $V=v=0$. Hence the truncated solutions agree until either exits $K_n$; continuity makes those exit times equal, and $Z^k$ stopped at ${\tau }_n$ equals $Z^n$ stopped there.

The stopped family $X^{(n)}:=Z_{\cdot \wedge {\tau }_n}^n$ is compatible by step 4.1 and satisfies the stopped integral equation because each cutoff equals the original coefficients on $K_n$. The exit times increase, since the compact sets are nested and the paths agree up to each earlier exit. Since ${\bigcup }_n\mathrm{int}{K}_n=U$, its lifetime $\zeta ={\lim }_n{\tau }_n$ and common values before $\zeta$ give a local strong solution in the sense of [F1]. For any two local solutions, their $K_n$-stopped path suprema are bounded by $\mathrm{\mid }\xi \mathrm{\mid }+n$ and hence are square-integrable; the same difference estimate therefore proves equality up to the minimum of their lifetimes. AC is exactly [A1].