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What techniques have you tried? If none, you could start with looking into Granger Causality @wiki by, perhaps, using this Bivariate Granger Causality R tool You have to be quite careful in how you interpret the results, as there are constraints on what the possible factors of influence can be (when testing for Granger Causality).


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You can use Ito's product rule $d(X \, Y) = dX \, Y + X \, dY + dX \, dY$. In your case, you have $$ dS_{1,t} = S_{1,t} \left( r_1 dt + \sigma_1 dW_{1,t} \right) $$ and $$ dS_{2,t} = S_{2,t} \left( r_2 dt + \sigma_1 dW_{2,t} \right) $$


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By all means, if $dS_i = \mu(S^i_t)\mathrm dt + \sigma(S^i_t)\mathrm dW^i_t$ for $i=1,2$ and you know explicit formulas for $S_i$ then their product satisfies the SDE you derive using the Ito lemma for a product. Could you elaborate, on what is the issue you have with the drift term?



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