  Modern Algorithmic Trading ("Algo") allows institutional investors and
traders to liquidate or establish big security positions in a fully automated
or low-touch manner. Most existing academic or industrial Algos focus on how to
"slice" a big parent order into smaller child orders over a given time horizon.
Few models rigorously tackle the actual placement of these child orders.
Instead, placement is mostly done with a combination of empirical signals and
heuristic decision processes. A self-contained, realistic, and fully functional
Child Order Placement (COP) model may never exist due to all the inherent
complexities, e.g., fragmentation due to multiple venues, dynamics of limit
order books, lit vs. dark liquidity, different trading sessions and rules. In
this paper, we propose a reductionism COP model that focuses exclusively on the
interplay between placing passive limit orders and sniping using aggressive
takeout orders. The dynamic programming model assumes the form of a stochastic
linear-quadratic regulator (LQR) and allows closed-form solutions under the
backward Bellman equations. Explored in detail are model assumptions and
general settings, the choice of state and control variables and the cost
functions, and the derivation of the closed-form solutions.
