On stock trading using a PI controller in an idealized market: The robust positive expectation property

In a number of recent papers, a new line of research has been unfolding which is aimed at using classical linear feedback control in a model-free stock trading context. The salient feature of this approach is that no model for stock price dynamics is used to determine the dollar amount invested I(t). Instead, the investment level is performance driven and generated in a model-free manner via an adaptive feedback on the cumulative gains and losses g(t). One of the main results obtained to date is paraphrased as follows: Under idealized market conditions with stock prices governed by a non-trivial Geometric Brownian Motion (GBM), a combination of two static linear feedbacks, one long and one short, leads to a positive expected value for the trading gain g(t) for all t > 0. Since this holds independently of the parameters underlying the GBM process, it is called the “robust positive expectation” property. Working in this same GBM setting, the main objective in this paper is to generalize this result from static to dynamic feedback. To this end, we consider a Proportional-Integral (PI) controller for the investment function I(t). Subsequently, we reduce the stochastic trading equations for the expectation of g(t) to a classical second order system and use the closed- form solution to prove that the robust positive expectation property still holds. We also consider a number of other issues such as the analysis of the variance of g(t) and the monotonic dependence of g(t) on the feedback gains. Finally, we provide simulations showing how the PI controller performs in a real market with prices obtained from historical data.