Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 38 / 1 / a / Solution

Write $B=\beta$ for the driving Brownian motion throughout; the two symbols in the displayed formulas refer to the same process. For $t<1$ the deterministic integrand is square-integrable on $[0,t]$, sois well defined. The Itô product rule, with zero quadratic covariation against the deterministic factor, givesFor the other form, apply the same Itô product rule to $B_t/(1-t)$:Multiplying by $1-t$ gives the stochastic integral representation of a Brownian bridgeThe drift coefficient is singular only at the terminal endpoint, so this calculation is performed on compact subintervals of $[0,1)$.