Explosion threshold for a Bernoulli stochastic differential equation ($\tau =\inf \left\{t:{\int }_0^t{F}_s^{\delta -1}ds=\frac{x_0^{1-\delta }}{\delta -1}\right\}(\delta >1)$)

The explicit Bernoulli stochastic differential equation solution remains positive and finite on every finite interval when $0<\delta \le 1$. For $\delta >1$ it diverges when the displayed integral reaches its threshold. At $\alpha =0$ this occurs deterministically. At $\alpha \ne 0$, Brownian motion has positive probability of making the integral exceed any threshold on a fixed interval, so global existence is not almost sure. The distinction between positive explosion probability and certain explosion matters: the negative linear drift of $\log F$ makes its total positive-power exponential integral finite almost surely when $\alpha \ne 0$.