  In this paper, we propose a minimal model beyond geometric Brownian motion
that aims to describe price actions with market inefficiency. From simple
financial theory considerations, we arrive at a simple two-variable hidden
Markovian time series model, with one of the variable entirely unobserved.
Then, we analyze the simplest version of the model, using path integral and
Green's function techniques from physics. We show that in this model, the
inefficient market price is trend-following when the standard deviation of the
log reasonable price ($\sigma$) is larger than that of the log market price
($\sigma'$), and mean-reversing when it is smaller. The risk premium is
proportional to the difference between the current market price and the
exponential moving average (EMA) of the past prices. This model thus provides a
theoretical explanation how the EMA of the past price can directly affect
future prices, i.e., the so-called ``Bollinger bands" in technical analyses. We
then carry out a maximum likelihood estimate for the model parameters from the
observed market price, by integrating out the reasonable price in Fourier
space. Finally we analyze recent S\&P500 index data and see to what extent the
real world data can be described by this simple model.
