Optimal Control of Trading Algorithms: A General Impulse Control Approach

We propose a general framework for intraday trading based on the control of trading algorithms. Given a set of generic parameterized algorithms (which have to be specified by the controller ex-ante), our aim is to optimize the dates $(\tau_i)_i$ at which they are launched, the length $(\delta_i)_i$ of the trading period, and the value of the parameters $({\cal E}_i)_i$ kept during the time interval $[\tau_i,\tau_i + \delta_i)$. This provides the financial agent a decision tool for selecting which algorithm (and for which set of parameters and time length) should be used in the different phases of the trading period. From the mathematical point of view, this gives rise to a nonclassical impulse control problem where not only the regime ${\cal E}_i$ but also the period $[\tau_i,\tau_i+ \delta_i)$ have to be determined by the controller at the impulse time $\tau_i$. We adapt the weak dynamic programming principle of Bouchard and Touzi [SIAM J. Control Optim., 49 (2011), pp. 948-962] to our context to provide a characterization of the associated value function as a discontinuous viscosity solution of a system of partial differential equations with appropriate boundary conditions, for which we prove a comparison principle. We also propose a numerical scheme for the resolution of the above system and show that it is convergent. We finally provide a simple example of application to a problem of optimal stock trading with a nonlinear market impact function. This shows how parameters adapt to the market.