= Bernoulli stochastic differential equation
{c}
{title2=$dX=\alpha X\,dB+X^\delta\,dt$}
On the positive half-line, divide by the <stochastic exponential> $F_t=e^{\alpha B_t-\alpha^2t/2}$. The <Itô product rule> reduces the equation to the random <Bernoulli differential equation> $d(X/F)/dt=F^{\delta-1}(X/F)^\delta$. For $\delta\ne1$ this gives $X_t=F_t[x_0^{1-\delta}+(1-\delta)\int_0^tF_s^{\delta-1}ds]^{1/(1-\delta)}$, until the bracket vanishes; for $\delta=1$, $X_t=x_0F_te^t$.
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