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Say $Q(t)$ is the exchange rate at time $t$. It's the price in domestic currency of one unit of foreign currency and converts foreign currency into domestic currency.

The model for the dynamics of this exchange rate is:

$$\frac{dQ(t)}{Q(t)}=\mu_Q dt+\sigma_Q dB(t)$$

Then the reverse exchange rate, $R(t)$, would be the price in foreign currency of one unit of domestic currency modeled by:

$$R(t)=\frac{1}{Q(t)}$$

My question is, how would I derive $dR(t)$?

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With the help of Itô's lemma, you can show that

$$df(Q)=f'(Q)dQ+\frac{1}{2}f''(Q)dQ^2 \; .$$

Putting $R=f(Q)=1/Q$ and using

$$\frac{dQ(t)}{Q(t)}=\mu_Q dt+\sigma_Q dB(t)$$

you should get

$$dR = \frac{\sigma_Q^2-\mu_Q}{Q}dt-\frac{\sigma_Q}{Q}dB$$

or equivalently

$$\frac{dR(t)}{R(t)} = (\sigma_Q^2-\mu_Q)dt-\sigma_Q dB(t) \; .$$

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