= Explosion threshold for a Bernoulli stochastic differential equation
{title2=$\tau=\inf\left\{t:\int_0^tF_s^{\delta-1}ds=\frac{x_0^{1-\delta}}{\delta-1}\right\}\quad(\delta>1)$}
The explicit <Bernoulli stochastic differential equation> solution remains positive and finite on every finite interval when $0<\delta\le1$. For $\delta>1$ it diverges when the displayed integral reaches its threshold. At $\alpha=0$ this occurs deterministically. At $\alpha\ne0$, 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 $\alpha\ne0$.
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