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I'd tackle this using some handwavery application of the bivariate conditional normal distribution to the geometric brownian motions: For a multivariate normally distributed random variable $X$, partitioned into two blocks $X_1,X_2$, i.e. $X\equiv\left(X_1,X_2\right)^T$ with zero mean and covariance matrix $$ \Sigma\equiv\begin{pmatrix} \Sigma_{11} & \...


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