Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 25 / 4 / d / Solution

Under $\mathbb{Q}$, $X_T=x+W_T^{\mathbb{Q}}$ has density ${\phi }_T(z-x)=(2\pi T{)}^{-1/2}\exp (-(z-x{)}^2/(2T))$. The reciprocal Radon-Nikodym derivative depends only on $X_T$:Consequently the probability density function under the original measure isIt integrates to one because ${\mathbb{E}}_{\mathbb{Q}}\cosh X_T=e^{T/2}\cosh x$. Completing the square also gives the useful mixture distribution formThus the terminal law is a mixture of $N(x+T,T)$ and $N(x-T,T)$ with the displayed positive weights. The diffusion with hyperbolic tangent drift density also exhibits the Doob h-transform with $h(z)=\cosh z$.