Tatyana V. Nonlinear dynamic models in the problems of financial mathematics and the theory of risk.

The dissertation is devoted to construction of dynamic models for financial market and finding of the existence condition for system optimal control. The two algorithms for interest rate modeling on securities market are suggested; the both methods give explicit solutions of nonlinear stochastic differential equations. Applications of those methods are shown for cases of equations, the most used in interest rate modeling on financial market: Cox-Ingerssoll-Ross equation, Hull-White (extended Vasicek) and Cox-Ingersoll-Ross equation (extended Vasicek). Financial strategies are also investigated and portfolios of stocks and bonds are simulated. The two methods to find optimal moments of switching between portfolios of securities are developed. Those methods enable the investor operating on financial market to receive of maximal profit at some capital investments. The homogeneous controlled Markov chains as models of financial strategies are investigated, and the existence conditions of the optimal strategy are obtained. The stochastic differential equations with fractional Wiener process and field are investigated. The existence theorems of optimal control for those equations are proved. Key words: dynamic model, interest rate, financial strategy, portfolio of stocks and bonds, moments of switching, stochastic equations, homogeneous Markov chains, optimal control, fractional Wiener process.