Taking into account the correlation in Barrier options on a Basket

In a Barrier option (where the contract cancels when the underlying hits the barrier) I succesfully found the way to compute the probability of a single underlying touching the barrier (with constant volatility). The problem comes when I have an average basket of underlyings $B_t$ $$B_t = \sum_{i=1}^n \omega_iS_t,\qquad \sum_{i=1}^n\omega_i=1\ .$$ I don't know how to properly calculate the probability of the basket $B_t$ touching the barrier. I think I need to take into account the correlations $\langle dW_t^{i},dW_{t}^{j}\rangle=\rho_{ij}$ between assets but i just don't know how to do it.

I'm considering constant volatility and constant correlation thoughout an arbitrary time step.

Extra: Same question if the basket is Best Of/Worst Of type.

I'm not expecting a complete developement of the answer but I would find very helpfull any indications or references. Thanks a lot.

Edit: Yes, i'm considering that Stock prices follow geometric Brownian motions. I also forgot to say that the prices at the end of the time interval are known, so the probabilities are actually conditioned probabilitites.

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I guess the stocks in your basket, are each following a geometric browian motion, is that right ? Even in that case you can't get closed form formulas, and you have to use approximations. Regards – TheBridge Nov 23 '11 at 18:37
Is it important for you that prices at the end of the time interval are known? That is a strange question from a industry point of view - because if prices at the end of the time interval are known then probably we have complete information about the path of the basket up to that time? Perhaps you mean that prices at the start of the time interval are known? – zoom Dec 7 '11 at 9:48
@user561749 This is to calculate the touching probabilities in a Monte Carlo simulation. In that context you know the price at a beginning and at the end of a cerain timestep but you don't know what happened in between. – FKaria Dec 7 '11 at 10:25
@FKaria: Thanks for your explanation. Now I understand the relevance of your question. One workaround is of course to only check the barrier at the discrete timesteps, but this is sometimes a crude approximation I guess. – zoom Dec 7 '11 at 12:41
I think that $\langle dW_t^{i},dW_{t}^{j}\rangle=\rho_{ij}dt$ – Jase Dec 7 '12 at 16:27