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I wanted to find some more information of this topic, but I found very little.
I might be interested in optimizing a stock investment portfolio. Maybe I could use beta or some other common risk measure to weight each stock, when I apply it to real data. However, here is the theory part:

There is an urn containing balls of two colors, corresponding to the two investments.
The goal is to play so as to maximize your expected return.

We assume the division of the portfolio into two investments $(X_{n},1-X_{n})$. At each time a ball is drawn and replaced, and the corresponding investment is monitored: the first is monitored with probability $X_{n}$ and the second one with probability $1-X_{n}$.
If the monitored investment rise above some threshold, then a portion $\gamma_{n}$ of the other investment is relocated into that investment.

Moreover,

If we define $T_{n}$ recursively by $\frac{T_{n}}{T_{n+1}} = 1-\gamma_{n}$, this is a time-dependent Polya urn process, with $a = T_{n+1} - T_{n}$, modified so that the reinforcement only occurs if the chosen investment exceeds the threshold. If $\gamma_{n}= \frac{1}{n}$ then $a_{n} \equiv 1$ and one obtains a diagonal Polya urn.

How should I proceed? Any suggestion about materials/paper to read about this topic? Applications?

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As a Polya urn is often referred to as the rich get richer, it maybe possible to use his urn to get richer.

It is known that these urns can be related to the Dirichlet-multinomial distribution. That being the case, i suggest looking at this link, http://jmlr.org/papers/volume18/10-231/10-231.pdf, to see if it will answer your question. It too deals with time-dependent polya urns

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