Maximal local solution of a stochastic differential equation

A maximal local solution solves a stochastic differential equation until its lifetime, usually the first exit from every compact subset of the region where its coefficients are locally Lipschitz. If the lifetime is finite, the path approaches the boundary of that region or becomes unbounded.

If the drift and diffusion coefficients are locally Lipschitz on an open domain, then for every initial point there is a pathwise unique strong solution up to a maximal lifetime. That lifetime is the limit of the exit times from an increasing sequence of compact subsets of the domain.

The linear growth condition bounds the drift norm and diffusion-matrix norm by $C(1+|x|)$ for one constant $C$. Together with globally Lipschitz coefficients it gives global strong existence and pathwise uniqueness, preventing finite-time explosion. Linear coefficient maps satisfy it even when they are not bounded.

For $dX=\sigma (X)dB+b(X)dt$, $X_0=0$, and ${\sigma }^2+b^2\le A(1+X^2)$, the Doob L2 maximal inequality and the Cauchy-Schwarz inequality imply, for $t\le 1$, $F(t)\le 8A{\int }_0^t(1+F(s))ds$, where $F(t)=\mathbb{E}{\sup }_{s\le t}|X_s{|}^2$. The Gronwall inequality yields $F(t)\le e^{8At}-1$. Apply the calculation first to stopped paths; truncating locally Lipschitz functions and using this uniform bound gives exit probabilities at most $(e^{8A}-1)/n^2$, proving nonexplosion on the unit time interval.