Interesting question

Usually people will express as dw1dw2=rho*dt or in the direct form dw1*rho+sqrt(1-rho^2)*dw2.

But I am not sure what you mean.

I think your statement is interesting. Correlation is a yes or no question. Not "if...then, yes or no". Just like something that what is the sufficient and necessary condition for me to be a human being ? Do you think that make sense ?

thanks for your answer！
actually， my question is “Assume you have a Brownian motion on stock S. Assume you have another Brownian motion on stock L: Assume you want to impose that both S and L are correlated, what could be a necessary and su¢ cient condition you need to impose on the Brownian motions so you canguarantee that S and L are correlated?”

Do you know any smple answer for this question? thanks

in this case, I should say, dSdL=rho*dt. rho is the correlation coefficient. I have no other thinkings coming up to mind. People usually write in this way.