Topics in stochastic processes and mathematical finance: counterparty risk valuation and hedging, Markov consistency and Markov copulae, and dynamic performance assessment indices

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BieleckiDMS-1211256 The investigator and his colleague continue development of new mathematical methods for risk management in complex stochastic dynamical systems, such as financial markets. They consider problems in three areas. The first is mathematical modeling for financial risk management, with application to hedging, valuation, and management of counterparty credit risk (CCR). Here they develop mathematical tools for evaluating and managing counterparty risk embedded in a large variety of over-the-counter contracts. The valuation and hedging of CCR in credit default swaps and interest rate swaps, which are essential for the financial industry, are studied and new analytical tools are developed for this purpose. The second area involves applications of stochastic analysis to investigation of dependence between Feller-Markov processes. The investigators study dynamic "copula" problems with regard to Markov processes: given N one-dimensional Markov processes, construct a multivariate Markov process such that each component is also Markovian in its own filtration, and such that its law agrees with the law of the original Markov process. The third area deals with mathematical modeling of dynamic performance assessment indices with applications to conic finance. New applications of convex analysis, probability, and L0-module theory are developed to study dynamic Performance Assessment Indices. Dynamic performance assessment indices are measures of performances of a given activity in a random environment. They apply this theory to conic finance for the purpose of computing acceptable bounds for arbitrage-free bid and ask prices in illiquid markets. The investigator and his colleague develop methods to analyze and manage risk in financial markets. By viewing markets as stochastic systems whose behavior changes in time, they can bring to bear concepts from dynamical systems, probability, and stochastic systems. They aim to take into account the provisions of recent legislation, such as the Dodd-Frank act of 2011, and the provisions of the Basel III regulations. Outcomes of the project contribute to increasing the efficiency and competitiveness of financial institutions (both government and private), by effectively controlling the counterparty risk and, by extension, the systemic risk in financial markets. The study of dynamic performance measures and assessment indices in particular provides new tools beyond the classical Value at Risk or Sharpe Ratio for measuring the risk and performance of a given financial institution (or portfolio). Results are useful for market participants, including regulators and government agencies. The project includes the training of graduate students in stochastic analysis and its application to financial markets.