= Pointwise positivity does not ensure weak existence
{title2=$\theta(0)=1,\quad\theta(x)=\min(x^2,1)\ (x\ne0)$}
There need not be a <weak solution of a stochastic differential equation> $dX_t=\sqrt{2\theta(X_t)}dB_t$, $X_0=0$, even when $\theta$ is a bounded <Borel measurable function> strictly positive everywhere. For the displayed example, any solution would be a <continuous local martingale> with $[X]_t=2\int_0^t\theta(X_s)ds\leq2t$, hence a <square-integrable> <martingale>. The <occupation-times formula> applied to $\mathbf1_{\{0\}}$ gives $0=\int_0^t\mathbf1_{\{X_s=0\}}d[X]_s=2\int_0^t\mathbf1_{\{X_s=0\}}ds$. Since $\theta(x)\leq x^2$ away from zero, $\mathbb EX_t^2\leq2\int_0^t\mathbb EX_s^2ds$. The <Gronwall inequality> makes $X\equiv0$, a contradiction. Thus a <diffusion amplitude> positive at a single problematic point cannot repair arbitrarily slow neighbouring motion. A local positive lower bound, as in the <Brownian time change for a variable diffusion speed>, rules out this example.
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