The Asymptotic Behavior Of Markov Operators And Economic Systems

This thesis consists of two parts. The asymptotic behaviors of Markov operators and somerelated problems in economical system are discussed respectively.The density evolution of discrete dynamical system is introduced in Chapter 2. Someimportant concepts such as ergodic, mixing, exact, etc., and some remarkable conclusions suchas Birkho? individual ergodic theorem, the description of ergodicity, mixing and exactness,and the asymptotic stability of discrete dynamical system are reviewed.Two typical examples are studied by orbit evolution and by density evolution in Chapter3. The first example is about the chaotic behaviors and the statistical stability of a family ofLorenz maps. The second example is known as the weak repellor paradox. One can see thesimilarity and di?erence between the density evolution and orbit evolution from the examples.The asymptotic behaviors of Markov operators are studied in Chapter 4. Instead ofglobal conditions, some local conditions are introduced to study the mean convergence, weakconvergence and strong convergence of the iterates of Markov operators. If a doubly stochas-tic operator(Frobenius-Perron operator) P satisfies the local lower set condition, then P isindividually weak(strong) stable, and if a Markov operator P satisfies the local lower func-tion condition, then P is individually strong stable. When P has a unique stationary den-sity, one can obtain corresponding global stability. The asymptotic behaviors of compositeMarkov(Frobenius-Perron) operators are also studied via the local conditions.Part 2 consists of some examples related to economical system.The first example is about the stability of Solow economic growth model under the pertur-bation of white noise. It is proved that the Solow economic growth model is asymptoticallystable under the perturbation of white noise, that is, for any initial density, the evolution ofthe system will approach to a stationary density.The second example is about the evolution of Eguiluz and Zimmermann's model of infor-mation transmission, fat tail and herding e?ect in a financial market. A finite state irreducibleMarkov chain is used to modelling the evolution of the state of the system. The stationarystate of the Markov chain is stable. So the results obtained by simulation are credible. Somenumerical results are also given.The third example is about bank crisis. The transition of debts and assets are modelledby a Markov chain according to the structure of the bank system. The stability of the banksystem under an instant big shock is considered.Risk analysis is considered in the last two chapters. Some concepts and important conclu-sions concerning risk in economics with uncertainty, insurance and finance are reviewed. Afamily of coherent risk measures are given, which incorporate the objective uncertainty andthe subjective bias of the decision maker.