Optimal control of trading algorithms: a general impulse control approach
Résumé
We propose a general framework for intra-day trading based on the control of trading algorithms. Given a generic parameterized algorithm, we control the dates $(\tau_i)_i$ at which it is launched, the length $(\delta_i)_i$ of the trading period and the value of the parameters $(\Ec_i)_i$ kept during the time interval $[\tau_i,\tau_i\p \delta_i[$. This gives rise to a non-classical impulse control problem where not only the regime $\Ec_i$ but also the period $[\tau_i,\tau_i\p \delta_i[$ has to be determined by the controller at the impulse time $\tau_i$. We adapt the {\sl weak dynamic programming principle} of Bouchard and Touzi (2009) to our context and provide a characterization of the associated value function as a discontinuous viscosity solution of a system of PDEs 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 an example of application to a problem of optimal stock trading with a non-linear market impact function.
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