Martingale measures in the market with restricted information

The formula of Black and Scholes for the valuation of options has led to the great development of mathematical finance. Mathematical finance is attracting more and more attention of researchers. Some useful work has been done, but the majority of discussions are based on perfect markets. A perfect market includes the following conditions: (1) many buyers; (2) many sellers; (3) individual trades do not affect the market; (4) the units of goods sold by different sellers are the same; (5) there is perfect information, that is, all buyers and sellers have complete information on the price being asked and offered in other parts of themarket; (6) there is perfect freedom of entry to and exit from themarket. Real financial markets are imperfectmarkets. In fact there are some investors different to general investors in the financial market. Because of their conditions, for example, they live in the country, the investors cannot know all market information such as some invest policies, construction plans, and so on, which are known by general investors. They might only know price information of risky assets. These make the investor’s information incomplete. It is well known that hedging market risk and capturing arbitrage opportunity are closed to market information. So it conforms to financial application to discuss financial markets under different information. There are several recent papers dealing with restricted information in finance. Schweizer [15] presents risk-minimizing hedging