Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 34 / 5

Before $\tau$, put $a_s=\sigma (Y_s)-\sigma (X_s)$ and $d_s=b_2(Y_s)-b_1(X_s)$, so $d{Z}_s=a_{s}d{B}_s+d_{s}ds$. For $Z_s\ge 0$, the Lipschitz and drift-order hypotheses give $|a_s|\le K{Z}_s$ and $d_s\ge b_1(Y_s)-b_1(X_s)\ge -L{Z}_s$. The derivatives of $f_{ϵ}(z)=(z+ϵ{)}^{-1}$ are $f_{ϵ}^{\prime }=-(z+ϵ{)}^{-2}$ and $f_{ϵ}^{\prime \prime }=2(z+ϵ{)}^{-3}$. Itô's formula givesThe drift integrand is bounded above byThe stochastic integrand has magnitude at most $K{Z}_s/(Z_s+ϵ{)}^2\le K/(4ϵ)$, so its integral on a finite horizon is a square-integrable martingale with mean zero. Global Lipschitz coefficients give finite second moments of the SDE solutions on finite horizons, which also justify the drift expectation; alternatively localize first and use boundedness of $f_{ϵ}$ and Fatou. Since $Z_{t\wedge \tau }\ge 0$, taking expectations and replacing the stopped integral by the larger positive full-time bound yields the reciprocal barrier proof of scalar diffusion comparison:

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.