# FX Rate dynamics

Let's suppose USD/EUR price in USD follows a GBM with $$dS_t = rS_tdt + \sigma S_tdW_t$$ What process does EUR/USD follow in EUR?

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$$\frac{1}{S_t}$$
$$d\frac{1}{S_t}=\frac{-1}{S_t^2}dS_t+\frac{1}{S_t^3}dS_tdS_t$$ some simple algebra then leads to $$d\frac{1}{S_t}=\frac{1}{S_t}(\sigma^2 -r)dt+\frac{1}{S_t}\sigma dW_t$$