Estimation Using the Martingale Assumption

The martingale assumption is used to derive the conditional distribution of the underlying state of the stock, where the state is a buy or sell, characterized by a state transition probability matrix. The state can be inferred from the stock returns, and is not directly observable. The martingale characterization of returns reflects the 'rational expectation' of the market, while the transition probabilities reflect the 'behavioral economics' of investors. These complementary aspects are combined to derive the state conditional probabilities and their likelihood ratios