Martingales and Complete Market

Abstract

The earlier assumption on the share market analysis is based on the markov property assumption. This model, to some extent, helps us to predict the risk involved in a particular industry and thereby helps us to settle down with maximum gain under prescribed limiting property. A special type of stochastic process which is based on conditional expectations as sequence of random variables called Martingales has become a better tool to study continuous trading. This type of perspective helps one to have two types of options one on the sampling and the other on the stopping process. The market can be regarded as a complete one in the sense that lower risk on one is compensated by higher profit on the other for investors. The study of this series has been taken up by Harrison and Pliska and we find that the martingale theory plays an important role in optional sampling and optional investment on various shares yielding low and high returns. We explain in detail the different stages of improvement on this model and their implications on the consumers satisfaction.