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I would like to ask if there exist any mathematical proof or model which addresses how the Kelly criterion can be applied to find portfolio weights when the stocks are correlated.

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Luenberger's book has a discussion on growth-optimal (i.e. Kelly) portfolios, also for the multivariate case with correlated assets.

@BOOK{Luenberger1998,
  title        = {Investment Science},
  publisher    = {Oxford University Press},
  year         = 1998,
  author       = {David G. Luenberger}
}
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  • $\begingroup$ That books gets very mixed reviews on Amazon and is not cheap. Do you have it and, if so, do you recommend it. I won't blame you if I get it and end up not liking it :). Thanks. $\endgroup$
    – mark leeds
    May 23, 2020 at 15:14
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This post provides a model for the Kelly Criterion under no leverage and no short constraints, and yields the following quadratic program:

$$\max_f g = r + \sum_{i=1}^n f_i(\mu_i - r) - \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n f_i f_j \hat{\Sigma}_{ij}$$

$$\textrm{s.t.} \sum_{i=1}^n f_i \leq 1$$

$$f_i \in [0, 1]$$

They also have code examples in Python showing how to solve it with Pyomo and IPOpt.

They cite this paper, which looks like it has a derivation of this model.

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