  Optimal control models for limit order trading often assume that the
underlying asset price is a Brownian motion since they deal with relatively
short time scales. The resulting optimal bid and ask limit order prices tend to
track the underlying price as one might expect. This is indeed the case with
the model of Avellaneda and Stoikov (2008), which has been studied extensively.
We consider here this model under the condition when the underlying price is
mean reverting. Our main result is that when time is far from the terminal, the
optimal price for bid and ask limit orders is constant, which means that it
does not track the underlying price. Numerical simulations confirm this
behavior. When the underlying price is mean reverting, then for times
sufficiently far from terminal, it is more advantageous to focus on the mean
price and ignore fluctuations around it. Mean reversion suggests that limit
orders will be executed with some regularity, and this is why they are optimal.
We also explore intermediate time regimes where limit order prices are
influenced by the inventory of outstanding orders. The duration of this
intermediate regime depends on the liquidity of the market as measured by
specific parameters in the model.
