On arbitrage possibilities via linear feedback in an idealized Brownian Motion stock market

This paper extends the so-called Simultaneous Long-Short (SLS) linear feedback stock trading analysis given in [2]. Whereas the previous work addresses a class of idealized markets involving continuously differentiable stock prices, this work concentrates on markets governed by Geometric Brownian Motion (GBM). For this class of stock price variations, the main results in this paper address the extent to which a positive trading gain g(t) > 0 can be guaranteed. We prove that the SLS feedback controller possesses a remarkable robustness property that guarantees a positive expected trading gain E[g(t)] > 0 in all idealized GBM markets with non-zero drift. Additionally, the main results of this paper include closed form expressions for both g(t) and its probability density function. Finally, the use of the SLS controller is illustrated via a detailed numerical example involving a large number of simulations.