On Stock Trading Using a PI Controller in an Idealized Market: The Robust Positive Expectation Property Shirzad Malekpour 1 , James A. Primbs 2 and B. Ross Barmish 3 Abstract— 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. I. INTRODUCTION Over the last several years, a new line of research has been developing which is aimed at using classical linear feedback control concepts in a stock trading context; e.g., see [1]-[20]. Perhaps the most salient feature of the most recent papers in this direction is the fact that the feedback rules defining the strategies are “model-free.” That is, instead of determining the investment level I (t) based on some parameterized model of stock prices which might be estimated over time, the controller is performance driven; i.e., I (t) is adaptively updated based on the cumulative profits and losses g(t) up to time t without concern for prediction of future prices; e.g., see [1]-[6]. To illustrate, in the case of static “long” linear feedback, the amount invested I (t) at time t is I (t)= I 0 + Kg(t) 1 Shirzad Malekpour is a graduate student working towards his doctoral dissertation in the Department of Electrical and Computer Engineering, Uni- versity of Wisconsin, Madison, WI 53706. E-mail: smalekpour@wisc.edu. 2 James A. Primbs is a faculty member in the Department of Systems Engineering, University of Texas, Dallas, TX 75080. Supported under NSF grant ECE-1160795. E-mail: james.primbs@utdallas.edu. 3 B. Ross Barmish is a faculty member in the Department Of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706. Supported under NSF grant ECE-1160795. E-mail: barmish@engr.wisc.edu. where I 0 > 0 is the initial investment and K ≥ 0 is the feedback gain. It is also possible to include short selling in this formulation by allowing I (t) < 0; e.g., consider I 0 < 0 and K ≤ 0 above. Working in this type of setting, the main objective in this paper is to generalize the “robust positive expectation” result given in [1] and [2]. More specifically, in these papers, using a combination of two linear feedbacks, one long and one short, the so-called Simultaneous Long-Short (SLS) controller is seen to have a remarkable property: In an ide- alized market with benchmark prices that follow Geometric Brownian Motion (GBM) with non-zero drift μ, non-zero feedback gain K and volatility σ, irrespective of the sign and magnitude of μ and the magnitude of σ, the SLS static linear feedback controller leads to a trading gain with positive expected value E[g(t)] for all t> 0. It is established that E[g(t)] = I 0 K [ e Kμt + e −Kμt − 2 ] which is readily seen to be positive for all t> 0. We call this the robust positive expectation property. In the sequel, we generalize the static feedback analysis to handle the case when the controller includes dynamics. We consider a classical Proportional-Integral (PI) controller I (t)= I 0 + K P g(t)+ K I t ∫ 0 g(τ )dτ, noting that we use differentiator-free dynamics to avoid prob- lems associated with price signals which typically include high-frequency components. Our goal is to derive expres- sions for the mean and variance of g(t). Working with a long-short version of the PI controller above, our main result is that the robust positive expectation result E[g(t)] > 0 still holds except for the trivial break-even case with either both feedback gains (K P ,K I ) = (0, 0) or drift μ =0. Our analysis of the PI trading strategy involves a num- ber of technical issues: First, we show that when dealing with E[g(t)], the stochastic equations for trading can be reduced to a classical second order differential equation with parameters being simple functions of I 0 , μ, K P and K I . Hence, in addition to positivity of the expectation, many aspects of the behavior of E[g(t)] > 0 such as damping, overshoot and oscillations become straightforward to study. The paper also considers a number of additional items such as the analysis of variance, monotonicity properties of the expected value of g(t) and a suggestion for further research 52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy 978-1-4673-5716-6/13/$31.00 ©2013 IEEE 1210