Applications Of The Boltzmann Kinetic Theory In Econophysics

The Boltzmann equation is an important model in the kinetic theory of gases. Inrecent years, it is a popular research direction to study the problems in economics ap-plying the theories and methods of Boltzmann equation. It is meaningful to study theportfolio distribution for the reason that the agents and the firms usually combine alter-native investments into proper portfolios to diversify the risk. At the beginning of thisthesis, we establish a multi-dimensional model with non-debt condition to character-ize the time evolution of the portfolio distribution. It is difcult to solve the equationvia wild sums due to the characteristic functions in the kernels of the model, whichmake the structure complicated. So we prove the existence and uniqueness of the mildsolution of this model by contractive-mapping principle. Furthermore, we obtain themoment estimates and some related properties.The most important part of the thesis is the study concerning the long-time be-haviors of the solutions. Because it is very helpful for investors to know as much aspossible the large time tendency of the investment in the financial market. Due to thecomplexity of the kernel and less symmetric structure in our model, the solution onlyconserves the zeroth order moment, which is quite diferent from the classical Boltz-mann equation. So it is difcult to obtain the nontrivial stationary state directly evenin the sense of weak convergence. Therefore, we consider the time scaled solutionand discuss the asymptotic limit. The main result shows that the model can be ap-proximated by a multi-dimensional Fokker-Planck equation in the sense of long timebehaviors under some constraints, from which we can see the significance to study themulti-dimensional model. We should note that the multi-dimensional Fokker-Planckequation we finally obtained is not simply a sum of those in one-dimensional modelsbut depends on the relations between every two risky assets. In the proof of this the-orem, we first prove the L1-weak compactness of {gr(w,t)}r,t. This result is obtainedby the Dunford-Pettis criterion combining the L2estimate of gr(w,t) together with the moment estimate and zeroth order conservation. It is necessary to prove the L1-weakcompactness of {gr(w,t)}r,tbecause one can only get the limit equation in a distribu-tional sense without it. Besides, we choose a more sophisticated method to weaken thehypothesis for the convergence to zero of1(r),2(r).