Dynamic Asset Pricing On Stochastic Control

In the study of continuous time financial market modeling, the theories and methods of stochastic control have been one of important tools. On the basis of modern finance development, the situation of financial mathematics study have been explained. Especially, together with our study in optimal consumption/investment, option pricing, dynamic risk measure etc., we deal with the related concepts and results.The problem of optimization from consumption and/or terminal wealth is a classical one. Compared with the classical problem of consumption and investment, we proceed to study three new problems. First, the investor has anticipation for future, while the his investment has been constrained and ho has higher interest rate for borrowing. Subsequently, the large investor has an effect on the market, and he also has anticipation for future. Finally, higher interest rate for borrowing and dividend has been taken into account.Regarding case one, the investor possesses information about the terminal value of the Brown motion, possibly distorted by 'noise'. We adopt the technique from the so-called enlargement of filtration, to deal with our problem. General existence results are established for optimal portfolio /consumtion policy. Equivalent condition for optimality are obtained, and explicit solution leading to feedback formulae are derived for special utility functions and for deterministic coefficients.Regarding case two, we examine optimal consumption and investment problem for a 'large investor', who possesses information about the terminal values of the components of the Brown motion, possibly distorted by 'noise'. Existence of optimal policies is established using martingale and duality techniques under general assumptions on the securities' price processes and the investor's preferences. Explicit solutions are provided for .specific cases involving an agent with logarithmic utilities.Regarding case three, a consumption/portforlion with borrowing is considered. The investor invests a riskless arid a risk stock with dividend payment. On different utilities, asymptotic behavior of optimal policies is studied for either very small or very large wealth.The problem of valuation for American options is extended to deal with both constrained portfolio and different interest rates for borrowing and lending. In the presence of the constraints and different interest rates, a single arbitrage-free price is replaced by an entire interval of arbitrage-free prices. We characterize arbitrage-free interval. The analysis involves martingale theory, optimal stopping,stochastic control problem and convex analysis. As for the general incomplete financial market, the upper-and lower-hedging prices of arbitrage-free interval are obtained.The quisimartingales decomposition has been proved. Furthermore, this decomposition has been applied to the problem of hedging American and European contigent claims in thesetting of incomplete financial markets.In the financial market system, the measure and hedging of financial risk are important. Considering a financial market with risky stocks and riskless bond, we describe the stochastic model of stock prices with stochastic volatility. In the context of our model, we propose measuring risk as smallest expected weighted loss. Thus, the above-mentioned is transformed into one of risk control problems. Furthermore, the existence of optimal risk control strategy is proved. By duality approach, we characterize the optimal control law.