I am currently stuck in an assignment in mathematics on the modelling of foreign exchanges. Any help is welcome!
Consider a model for the exchange rate between two countries $f$ (foreign) and $d$ (domestic), and denote by $e_t$ the price in domestic currency of one unit of the foreign currency. Assume that the short rate in the two countries is constant and denote it by $r^d$ respectively by $r^f$. $B_{t}^{d} = e^{r^{d}t} (B^{f}_{t} = e^{r^{f}t})$ denote the price of the domestic (foreign) savings account. The exchange-rate dynamics under $P$ is given by $de_{t} = \mu e_{t}dt + \sigma e_{t}dW_{t}$.
Denote by $Q^{d}$ the domestic martingale measure, that is the martingale measure that corresponds to the numeraire $B^{d}$. Show that under $Q^{d}$ the drift of $e_{t}$ is equal to $r^{d} −r^{f}$. Hint: consider the drift of the asset $S^{f}_{t} := e_{t}B^{f}_{t}$ (the value of the foreign savings account denominated in domestic currency).