 Solution

The increasing quadratic variation has a limit in $[0,\infty ]$. Since $X_t$ has a finite limit, the exponentials have limits too, with value zero when the bracket is infinite. Expanding exponents provesbecause the coefficient of $X_t$ on the right isand the bracket coefficient is $-1/2$. Taking limits preserves the identity, including the zero case.

Let $A=\mathbb{E}[\mathcal{E}(X{)}_{\infty }]$, $B=\mathbb{E}[\mathcal{E}(rX{)}_{\infty }]$, and $D=\mathbb{E}[e^{r{X}_{\infty }/2}]$. The Holder inequality first yieldsWhen $D$ is finite, apply the Jensen inequality to the concave function $z\mapsto z^{2/(r+1)}$:Since $D>0$, rearrangement gives the terminal scaling inequality for stochastic exponentialsHere $2(r^2-1)/(r+1)=2(r-1)$. If the last exponential moment is infinite, the bound has no positive content; the finite-moment form is the one used below.