Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 28J / d / Solution

For equal rates, the change of numeraire from the Euro bank account to the Dollar account has density process $Z_t=Y_t/Y_0$, a strictly positive mean-one martingale under $Q_E$ by part (b). Conditional change of measure givesThus the reciprocal is a genuine martingale under $Q_D$, directly from the Euro-measure martingale property, without recalculating its stochastic differential equation.