Stochastic Methods in Finance

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The objectives of the proposed research are to develop the mathematical models and theory for financial applications and to apply the results to financial markets. Most finance theory has the implicit assumption that traders can buy or sell as many shares of securities as they wish by immediate transactions. But in actual markets, this assumption is not satisfied, and thus the subject "liquidity risk" has received wide attention. Liquidity risk is the additional risk in the market due to the timing and size of a trade. The price process may depend on the activities of traders, especially the trading volume. The optimal trading strategy for the investor and the optimal liquidation problems need to be investigated from the aspects of practice as well as mathematics.