 Solution

The Gronwall inequality applied to part (a) givesContinuity implies $Z_{\tau }=0$ on $\{\tau <\infty \}$. Thus on $\{\tau \le t\}$ the stopped reciprocal equals $1/ϵ$. Positivity givesLet $ϵ\downarrow 0$, using $z_0>0$, to obtain $\mathbb{P}(\tau \le t)=0$. Taking the union over integer $t$ proves $\tau =\infty$ almost surely. Hence the strict order preservation for scalar Lipschitz diffusions conclusion iswhich is stronger than the requested weak inequality. The single-event argument prevents a gap between fixed-time ordering and simultaneous ordering at all times.