Advanced stochastic methods in mathematical finance and related fields

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The project focuses on developments of comprehensive methods of Stochastic Analysis which are closely related to Mathematical Finance, Mathematical Statistics and Actuarial Science. A solid part of the project is devoted to processes with long-range dependence (Fractional Brownian Motion, Fractional Levy Processes). Developing theory of these processes we apply it to parameter estimation problems in statistical models containing a long-range dependence component. We derive a number of properties for the likelihood estimates of the drift parameter (consistency, asymptotic normality, etc) as well as for the so-called Hurst index of self-similarity. We provide also a systematic study of financial markets with long-range dependence. Developing option pricing theory for these markets we derive pricing and hedging formulas for standard, past-dependent and barrier options. A part of the project deals with the pricing of equity-linked life insurance contracts, a well-developing area of Mathematical Finance and Actuarial Science. Extending partial hedging methods and techniques to defaultable markets and markets with long-range dependence we apply these methods to pricing of such long-term finance/insurance policies. Another part of the project is devoted to multidimensional regression models and multidimensional models for returns. We propose a very general semimartingale model for such modeling which includes many models studied before. To provide an adequate study of the least squares estimates we develop a techniques of operator-valued stochastic exponentials. We develop also a method of polynomial extensions of multidimensional probability distributions to get a better fitting for returns. The project proposes to study optional semimartingales as possible techniques applicable in Mathematical Finance and Filtering theory. The project can be characterized as fundamental research aimed at producing innovative ideas, comprehensive techniques and results that will be important for theory and practice.