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I would like to generate conditional correlated random variables. I have a correlation matrix between normal variables, and these variables are modeled through SDEs.

What are the algorithms to generate such random values? One example I found is; http://www.mathworks.com/help/toolbox/econ/interpolate.html

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  • $\begingroup$ What software do you wish to use for this? $\endgroup$
    – Michelle
    Feb 11, 2012 at 6:05
  • $\begingroup$ I can use any software. As I point out Matlab does this, but I would like to learn the numerical method behind this method; ex how to do Brownian bridge on multivariate distrubution $\endgroup$
    – adam
    Feb 11, 2012 at 22:01

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You can generate correlated gaussian variables by computing the Cholesky decomposition of the variance matrix and multiplying independent gaussian variables with it. There was a similar question on Quant Stackexchange a few days ago.

To simulate the solution of a stochastic differential equation (SDE), you can just discretize it, i.e., consider it as a difference equation (this is called the "Euler scheme") and use the correlated brownian motion you want. In some cases, this is too imprecise, but it is possible to derive a second-order approximation of the SDE from Ito's formula (this is called the "Milstein scheme"; it is an analogue of the Runge-Kutta method). If you are using R, this is implemented (for instance) in the sde package and described in the accompanying book.

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  • $\begingroup$ Yes I know with Cholesky you can simulate correlated random variables. Here the issues is after having a sample of N variable by length T samples, how can I interpolate/conditional sample between the already produced samples $\endgroup$
    – adam
    Feb 11, 2012 at 22:00
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    $\begingroup$ If it is just a brownian motion, and not a general SDE, you can multiply the sample you already have by the inverse of the Cholesky matrix: this gives you a sample from N independent brownian motions. You can then interpolate them separately (these are independent brownian bridges), and multiply the result by the Cholesky matrix. $\endgroup$ Feb 11, 2012 at 23:53

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