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Hey I have problem with understanding change of numeraire technique. For example we have

$dr^d(t)=\kappa_1(\theta_1(t)-r^d(t))dt+\sigma_1 dW_1$ (under measure $Q^1$ associated with domestic bank account)

$dr^f(t)=\kappa_2(\theta_2(t)-r^f(t))dt+\sigma_2 dW_2$ (under measure $Q^2$ associated with domestic bank account)

where $W_1$ - Wiener process under $Q^1$, $W_2$ - Wiener process under $Q^2$ and $dW_1dW_2=\rho dt$ (here the first question - these two Wiener processes are written under different measures but we can write this correlation because when we write one of them under another measure then only drift term will change and it doesn't have any impact on correlation?)

And now I want to write $r^f$ process under $Q^1$ measure because I want to simulate these processes. How to do it? I know that we must to define FX rate process. Can we write this process directly under $Q^1$ measure in the form (1 unit of foreign currency = $X$ times unit of domestic currency): $$dX(t)=[r^d(t)-r^f(t)]dt+vdW^X(t)$$ where now $dW^XdW_1=\rho_{X,d}dt$ and $dW^XdW_2=\rho_{X,f}dt$.

If so far everything is OK, how can we find dynamics of $r^f$ under $Q^1$? Maybe there exists books/papers where everything is calculated step by step?

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    $\begingroup$ Hint: Always try to identify your tradable assets. If $B$ is the money market account in EUR and $X$ is the exchange rate to USD, and you want to express everything in the USD risk neutral measure, then notice that $XB$ is a tradable asset in USD. Use Ito's formula on $XB$ given the dynamics of $X$ and $B$ and then see what transform you need to do to make the USD asset $XB$ drift less. $\endgroup$
    – user34971
    Commented Nov 3, 2021 at 10:54
  • $\begingroup$ Can we apply Ito's formula for $XB$ if $X$ and $B$ are written under different measures? Because $X$ is under measure $Q^1$ and foreign bank account $B$ is under measure $Q^2$. $\endgroup$
    – Mr.Price
    Commented Nov 3, 2021 at 18:02

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