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.
