Stochastic processes is one of the key operations research tools for analysis of complex phenomenon. This paper has a unique application to the study of mean changing models in stock markets. The idea is to enter and exit stock markets like Apple Computer and the broad S&P500 index at good times and prices (long and short). Research by Chopra and Ziemba showed that mean estimation was far more important to portfolio success than variance or co-variance estimation. The idea in the stochastic process model is to determine when the mean changes and then reverse the position direction. This is applied to Apple Computer stock in 2012 when it rallied dramatically then had a large fall and Apple Computer and the S&P500 in the 2020 Covid-19 era. The results show that the mean changing model greatly improves on a buy and hold strategy even for securities that have has large gains over time but periodic losses which the model can exploit. This type of model is also useful to exit bubble-like stock markets and a number of these in the US, Japan, China and Iceland are described. An innovation in this paper is the exit entry long short feature which is important in financial markets.
