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This is an example of minimum price variation (also known as the minimum price increment or the minimum price fluctuation). All public quotes for US equities are displayed to the nearest penny. (Hidden quotes may be entered at sub-penny increments.) US stock indices follow this convention and thus quote to the nearest penny. The oil listing is odd indeed. ...

Regarding Joshua's inspired answer, I'm still not sure how you guarantee that scaling gives you the exact high and low values. I suppose that you could simulate until you get a result that is close enough. But that could be hard when, e.g., Open is near High and far from Low. An alternative solution is to construct a Brownian Bridge between Open and High, ...

Following references from the answer provided by @Richard, we see that the optimality condition for a continuous process in general (and therefore an OU process in particular) is covered in Section 2 concluding on page 6 of Thompson 2002, where he also represents the solution in terms of the Hamilton-Jacobi-Bellman equations. If you change the limits of the ...

you find theoretical results for the Ornstein-Uhlenbeck process if you search for "pairs trading". In pairs trading it is assumed that the ratio of the pair is mean reverting. Then one often models this ratio as Ornstein–Uhlenbeck process. You find something on page 11 here Further theoretical results that might be of interest can be found here. All these ...

I think a good way to think about your problem is the example of finding an optimal VWAP trading strategy. You basically have a finite point in time by which you must have performed your transaction and you trade a similar asset than the one you are considering, one with the same underlying assumptions of mean-reversion (I make such assumption in the same ...