The proposed research will be organized into two principal research areas: 1) Development of mathematical theory for risk adjusted portfolio optimization and fixed income investment management, incorporating factor modeling of security returns, and 2) Development of mathematical theory for credit risk valuation and hedging. There are three fundamental aspects of the first research area: 1A) The concept of risk adjusted control (optimization) criteria for investment problems with finite and infinite planning horizons. More specifically, we shall continue to develop theory and applications for mean-variance stochastic control. This kind of control problems is appealing to financial industry as it extends to more realistic dynamic framework the classical approach of Harry Markowitz (1990 Nobel Prize in Economics). 1B) Development of control methodologies for the fixed income investment problems with finite planning horizon coinciding with the maturity of the underlying bond. We shall attempt to overcome certain singularity problems arising in this context, which will lead to derivation of applicable fixed income investment strategies. 1C) Explicit consideration given to statistical estimation issues. Optimal control methodologies are rarely used in the financial industry, largely because of statistical difficulties associated with the estimation of constant drift coefficients in diffusion process models of individual securities. Practitioners typically channel their energies into forecasting security returns based upon exogenous factors such as interest rates and firm- specific accounting measures. The proposed research will serve to reduce the gap between theory and practice by developing optimization models, which explicitly incorporate exogenous factors. By explicitly modeling the dependence of the assets on factors, it will be possible to obtain more realistic models, to better understand the statistical estimation difficulties, and to be in a position to apply adaptive control methods. The fundamental aspects of the second research area are: 2A) Explicit consideration given to possibility of credit migrations of defaultable contingent claims. This will be done in the context of conditionally Markov chains both with regard to a single defaultable claim as well as with regard to several such claims. The latter situation is very important for applications to basket credit derivatives. 2B) Development of mathematical theory for hedging of certain classes of credit derivatives that are vital for financial industry. This will be done in the context of martingale representations associated with conditionally Markov chains. 2C) Development of analytical tools for computation of certain class of functionals of conditional Markov processes. This will build upon the classical Feynman-Kac characterizations, and will find applications for valuation and hedging of some fundamental (basket) credit derivatives such as default swaps.
Although the proposed research will require fundamental advances within the areas of applied mathematics, probability, and financial economics, it is anticipated that the proposed research will lead to new and practical tools that eventually become widely used in the financial industry, and possibly in the insurance industry as well. The main reason for this expectation is that the proposed research addresses the need of the two industries for quantitative methodologies that would enable financial and insurance decision makers to properly manage certain categories of risks. The major implication of incidence of risks in financial/insurance decision- making is the possibility of financial losses. In general, neither the possibility of financial losses, nor their occasional severity, can be completely eliminated. However, workable tools, such as pricing and hedging strategies, are sought for controlling some of the risks underlying financial and insurance industries, so that both the possibility and severity of losses are kept to minimum. The proposed research will provide mathematical basis for development of such tools.
