# Optimal Executions for Minimizing Slippage

There has been a considerable body of work for finding trading strategies that minimize the slippage wrt arrival price. For instance, the following are on of the most well known papers:

[1] Robert Almgren, Neil Chriss, "Optimal execution of portfolio transactions"

[2] Robert Almgren, "Optimal execution with nonlinear impact functions and trading-enhanced risk"

[3] Mauricio Labadie, Charles-Albert Lehalle, "Optimal trading algorithms and self-similar processes: a p-variation approach"

One criticism I have for these papers is the optimal trading quantities are independent on the price realization. More precisely, the number of shares to trade at time $t$ do not depends on the price I can observe at that moment which is an important piece of information.

Is there any academic work where the price realization is incorporated in the decision process?

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You are right, these work use deterministic control. Framework using stochastic control exist:

• Bouchard, B., Dang, N.-M., Lehalle, C.-A., 2011. Optimal control of trading algorithms: a general impulse control approach. SIAM J. Financial Mathematics 2 (1), 404-438. URL http://epubs.siam.org/doi/abs/10.1137/090777293?af=R
• Kharroubi, I., Pham, H., Jun. Optimal portfolio liquidation with execution cost and risk. SIAM J. Finan. Math., 1(1), 897–931. (35 pages) URL http://arxiv.org/abs/0906.2565
• Guéant, O., Lehalle, C.-A., Fernandez-Tapia, J., 2012. Optimal Execution with Limit Orders. SIAM Journal on Financial Mathematics 13 (1), 740-764. http://arxiv.org/abs/1106.3279
• Bayraktar, E., Ludkovski, M., Jun. 2012. Liquidation in limit order books with controlled intensity. Mathematical Finance. URL http://arxiv.org/abs/1105.0247

For market making you have few papers too.

Below is a picture of the first paper in the list. You can see on the left the locally traded volume (in red) and on the right 3 mean trajectories (conditioned by the presence of a trend).

It is a very good idea to use stochastic control; of course it is more CPU consuming than deterministic control. In between you have the Almgren and Lorenz paper:

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