Applied Research Of Stochastic Analysis In Dynamic System And Finance

For the stochastic dynamic system and optimal portfolio selection in financial markets with random disturbances, stochastic analysis is an important tool of research. In particular, for the population dynamic system,the random noises in the nature will interfere the stable of population dynamics. But because of the underlying random nature of financial markets, it is such an important tool. This dissertation uses some theories and techniques of stochastic analysis to investigate the dynamical properties of a predator-prey model in which the predators and preys disperse among n patches with stochastic perturbation, and the optimal investment outcomes of general form of utility functions called hyperbolic absolute risk aversion(HARA) utility functions with stochastic markets.In the first chapter, some background knowledge and the present progess of predator-prey dynamic system and optimal portfolio selection are introduced. At the same time, the main work of this paper and some basic theory are presented.In the second chapter, the predator-prey system with the dispersal in predators and preys with stochastic perturbation is investigated. We show that there is a unique global and positive solution with any given positive initial value,furthermore, the property of ultimate boundedness was obtained.In addition, we find out the sufficient conditions for the extinction to the preys and the whole system.Finally,the conclusions are verified by numerical simulation.In the third chapter,we consider the optimal investment outcomes of general form of utility functions called hyperbolic absolute risk aversion(HARA) utility functions with stochastic markets. We derive the optimal terminal portfolio values using the equivalent static program by a variational method. We also found expectations of investment outcomes and probabilities of investment outcomes which are more than Der T.The explicit optimal strategies for the hyperbolic absolute risk aversion utility functions are obtained by the integration representation of a martingale.