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I've been thinking about the following problem.

Consider an agent who starts out with \$1 and on any given day $t$, is given the opportunity to invest in an asset with expected return $\mu_t$ and volatility $\sigma_t$. The assets returns are independent for assets offered on different days.

Let's say there's a fixed holding period $\tau$ at which the asset must be sold.

I would like to formulate this as a utility maximization problem, where at each time step the agent decides how much (if anything) to invest in the offered asset. I would like to use some sort of approximate dynamic programming.

Is this a well-known problem? Or is it possible to simplify it into something that has a well-known solution?

I think it's quite applicable to Private Equity for instance, where an investor is pitched a different company every day and then has a 10-year holding period.

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