Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 38 / 5 / a / Solution

Multiply the linear stochastic differential equation by $e^{-at}$. The Itô product rule givesThe integrand is deterministic, so the Itô integral has a centered normal distribution. Its variance, by the Itô isometry, isHenceAt $a=0$ the correct continuous-limit formula is $N(z,b^{2}t)$. A zero variance, for example when $b=0$, denotes the deterministic distribution. For $a<0$ the variance is still positive because both numerator and denominator in its quotient are negative. This is the explicit Ornstein-Uhlenbeck solution, allowing either sign of the linear drift coefficient.