Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2017 / 1 / a

Apply Itô's formula to $f(u,s)=e^{-\alpha u}s$. Its second stock derivative is zero, and the two drift terms cancel, givingIntegrating and multiplying by $e^{\alpha T}$ solves the linear stochastic differential equation:The Itô integral has a deterministic integrand, so it is a centered Gaussian random variable independent of the current filtration ${\mathcal{F}}_t$. Its variance follows from the Itô isometry. With $\tau =T-t$,The second line is also the limit as $\alpha \to 0$. The normal law asserted here is conditional on ${\mathcal{F}}_t$; an arbitrary random initial value need not give an unconditional normal distribution.