Generalization of affine feedback stock trading results to include stop-loss orders

The main question we would like to address in this paper is as follows: Given a geometric Brownian motion (GBM) as the underlying stock price model, what is the cumulative distribution function (CDF) for the trading profit or loss, call it g(t), when an affine feedback control strategy with stop-loss order is considered? Moreover, is it possible to obtain a closed-form characterization for the desired CDF for g(t) so that a theoretician or practical trader might be benefited from it? The answers to these questions are affirmative. In this paper, we provide a closed-form expression for the cumulative distribution function for the trading profit or loss. In addition, we show that the affine feedback controller with stop-loss order indeed generalizes the result without stop order in the sense of distribution function. Some simulations and illustrative examples are also provided as supporting evidence of the theory.