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To create a correlated random time series of returns you can always use a cholesky décomposition (if you have a positive definite covariance matrix ) .A simple manual implementation of it is described here : https://www.quantstart.com/articles/Cholesky-Decomposition-in-Python-and-NumPy/ For technical understanding you can check the Wikipedia page here,which ...

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You known some, but not all the correlations. Try to assume that the unknown correlations are 0. If this causes your correlaion matrix not to be positive definite (which you'll need), then you'll need to tweak or make up some more non-zero correlations. However you probably won't need it. Then just see my answer here.

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I don't think the claim that "Lévy alpha-stable distributions are better descriptors of returns" is universally accepted. While Mandelbrot (and others before him) has correctly identified non-normality of returns in financial time series, he wasn't really equipped at the time (1963) to pursue its real nature. Appropriate models appeared only much ...

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I have done that. The distribution if there are no dividends, mergers or bankruptcy and if liquidity costs are ignored is $$\Pr(r|r^*;\gamma)=\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{r^*}{\gamma}\right)\right]^{-1}\frac{\gamma}{\gamma^2+(r-r^*)^2}.$$ It has no expected value. You can find a reduced form discussion here https://youtu.be/R3fcVUBgIZw. If you ...

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