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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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This will be the inverse process

$$\frac{1}{S_t}$$

Applying Itô's formula the dynamics are then given by

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

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