The exchange rate between a domestic currency money market and a foreign currency money market can be expressed as $$ dQ(t) = (r_d - r_f)Q(t)dt + \sigma Q(t)d\tilde{W}(t) $$ where $r_d$ is the interest rate for the domestic market, and $r_f$ for foreign.
In my head, I believe that the exchange rate should decrease when the domestic interest rate goes up, indicating the domestic currency is strengthening. For example if the Fed were to increase rates, then $EUR/USD$ should decrease, given that the ECB doesn't do much. So, if $EUR/USD$ was 1.14 yesterday, it should be below 1.14 today.
To my understanding, the SDE for $Q(t)$ doesn't seem to reflect this fact. It seems that $Q(t)$ would increase if $r_d$ were to go up. I would like to resolve this contradiction, so any help would be appreciated.