# Time Interval of Existence of an SDE solution with Locally Lipschitz Drift

Consider the stochastic ODE $$dX = F(X)dt + dB$$ where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "almost surely positive" random time.

My question is, can we find a measure 1 set and a deterministic constant $C$, such that on this measure 1 set the random time $T$>$C$? This seems stronger than saying "$T$ is almost surely positive" random time.

In general, the answer to your question is no. For instance, consider $$d X(t) = X(t)^2 dt + \sigma d B(t) \;, \quad X(0) = x_0 \;.$$ In this case, the drift $F(x) = x^2$ is locally Lipschitz since its derivative is locally bounded. When $\sigma =0$, we obtain the ODE $$\dot{x}(t) = x(t)^2 \;, \quad x(0) = x_0$$ with solution $$x(t) = \frac{x_0}{1-x_0 t}$$ which is well-defined for $0 \le t < 1/x_0$. Thus, for any $C>0$ and $x_0 < 1/C$, the ODE solution exists and is unique over $[0,C]$.
The situation is different for the SDE: in any time interval, there is a nonzero (but possibly tiny) probability that $X(t)$ reaches a state where the drift dominates. From this point forward, with high probability, the SDE solution evolves like the ODE solution but with a random initial condition. Hence, the solution is defined up to a random time.
This phenomenon is illustrated in the figure below with $x_0 = 1/10$ and $C=10$. In red is the ODE solution for comparison. In different shades of grey are ten independent realizations of $X(t)$.