The aim of the research to be done is to develop new mathematical methodologies and models in the areas of stochastic analysis and financial engineering, for the propose of solving complex and important problems related to financial risk management and to financial decision making. Due to the current world-wide market turbulence, and taking into account the known causes that triggered the credit crunch, particular emphasis will be put on dynamic valuation and hedging of credit derivatives, with focus on credit default swaps and basket credit derivatives, as well as other complex financial instruments such as inflation-indexed swaps. This research will frame the following two main research areas: mathematical modeling for the purpose of financial risk management, with applications, among others, to hedging, valuation, and management of portfolio credit risk, and applications of stochastic analysis to studying of dependence between stochastic processes. Although the forthcoming research will yield fundamental advances within the areas of stochastic analysis and financial mathematics, it is anticipated that it will lead to new and practical tools that eventually become widely used in the financial industry and other applied disciplines, so to mitigate and to appropriately control the involved risks.
This research will be of fundamental importance for several reasons. First, the troubled credit derivative industry will benefit directly from it, as new methods and techniques based on solid mathematical and financial advances will be developed for the purpose of valuing and managing basket credit derivatives, such as basket swaps, cash collateral debt obligations, asset backed securities, etc. This industry suffered great losses since the spring of 2007 mainly because the nature of such complex hybrid derivatives as structured credit derivatives was not really well understood, especially on the level of risk management (hedging of risk exposure). The research to be done will result in developing new robust dynamic models for these securities and provide reliable dynamic hedging strategies associated to these models, which consequently will give a better and broader understanding of an important part of the modern financial markets. Second, new applications of stochastic analysis will be developed, in particular with regard to dynamic acceptability indices. Dynamic acceptability indices are unitless measures of performances of a given cashflow, which will be studied from abstract probability point of view. The obtained results will be beneficiary for all market participants including regulators and government agencies, and will lead to a better understanding of market efficiency. Third, valuation and hedging of credit default swaps (CDS) as well as valuation and hedging of credit default swaptions, which are essential for the financial industry, will be specifically emphasized and new analytical tools will be developed for this purpose. Fourth, if true, then an analog of Sklar's theorem for probability measures on canonical spaces of stochastic processes will be an important extension of the classical Sklar's theorem. 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. Fifth, the researchers will study the dynamic "copula" problem with regard to Markov processes: for a given multidimensional Markov process, with each coordinate being also Markovian, what conditions need to be satisfied by the pseudo-differential operator corresponding to the (extended) infinitesimal generator of the multidimensional process, given that the pseudo-differential operator corresponding to the (extended) infinitesimal generator of each coordinate are known. Sixth, 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 equity options, basket credit derivatives, etc.).
