On Some Mathematical Problems From Asset Pricing

This thesis focuses on some mathematical problems from asset pricing. It consists of four parts.In the first part, we consider option pricing. We provide a partial differential equation for European options on a stock whose price process follows a general geometric Riemannian Brow-nian motion. The existence and the uniqueness of solutions to the partial differential equation are investigated. A proper Riemannian metric on R can make the distribution of the stock return rates induced by our model have the character of leptokurtosis and fat-tail; besides, it can also explain price bias and volatility smile.In the second part, we discuss heterogeneous beliefs bubbles. Under the assumption that the dividends of an asset satisfy a CIR model, we obtain a closed formula for the minimal equilibrium price by the method of viscosity solutions. Then the minimal heterogeneous beliefs bubbles are determined explicitly.In the third part, we rigorously analyze entry and exit decisions of a project. Instead of supposing that the costs are constant in classical research, we assume that they are linear with respect to the price of the commodity produced by the project. Under this assumption, we obtain a condition which guarantees that investing in the project is worthless; besides, the project may be terminated when the commodity price is greater than a certain value. In contrast, there are no such results provided that the costs are constant. Moreover, we offer an explicit solution of entry and exit decisions if the project is worthy to be invested in.In the fourth part, we consider the equivalence of dynamic and static asset allocations in the case where the price of the risk asset (a stock, for example) is driven by a Poisson process. By restricting utility functions and trading strategies and using the variation method, we obtain a necessary and sufficient condition for the equivalence of dynamic and static asset allocations. Then we provide a simple sufficient condition for the equivalence.