This research project will continue development of new methodologies and models in the areas of stochastic analysis, and stochastic methods in finance and financial engineering, for the purpose of solving complex problems in financial decision making and risk management. Particular emphasis will be put on applications to valuation and hedging of credit derivatives, with focus on credit default swaps and basket credit derivatives, as well on applications to valuation and hedging of convertible bonds. One goal of the project is therefore development of a sound mathematical theory of basket credit derivatives and convertible bonds and related issues of hedging, valuation, and management of credit risk and convertible risk. This in particular will require new results in representation for stochastic processes, as well as new results for modeling of stochastic dependence between random processes. Thus, another goal of the project will be to use stochastic analysis in finite and infinite dimensions for studying of dependence between stochastic processes. In particular, a new theory of semimartingale copulae and Markov copulae will be worked out and applied. Moreover, new applications of stochastic analysis will be developed, in particular with regard to martingale representations and backward stochastic differential equations with reflections (also driven by jump martingales).
This research project will be of fundamental importance for several reasons, both from the applications point of view as well as from the purely theoretical perspective. First, the booming credit derivative industry will benefit from it, as development of tractable mathematical tools for the purpose of valuing and managing of basket credit derivatives, such as basket swaps, collateralized debt obligations, and credit indices, will provide the industry with new methodologically sound procedures. Likewise, the convertible bond industry will benefit from this research as our new decomposition results specified for the regime switching market model should provide a better quantitative tools for this industry, which suffered great losses in the Spring of 2005 -- possibly because the nature of such complex hybrid derivatives as convertible bonds was not really well understood. In addition, valuation and hedging of credit default swaps, which is essential for the finance industry, will be specifically emphasized and new analytical tools will be developed for this purpose. On theoretical side, project will also be of fundamental importance for several reasons. First, if true, then an analog of Sklar's theorem for the case of probability measures on canonical spaces of stochastic processes will be an important extension of the classical theorem of A. Sklar (1959), which was proved for real valued random variables. Perhaps, an analog of Sklar's theorem for probability measures on some general vector spaces (such as Polish spaces) will be derived in the process. Second, for those semimartingale processes for which their local characteristics determine their laws it will be important to study the following question: what is the class of multivariate (vector valued) semimartingales with given univariate local characteristics. Given the strategic importance of basket products for financial industry there will be a practical importance of studying of the above problems in view of potential applications, such as valuation and hedging of basket derivatives (basket options, basket credit derivatives, etc.).
