The Stochastic Orderings Of Ito Process And Its Applications

Ito process described as the important model system in the financial system, the econometrics, statistical physics and biology, the existing research focuses on the stability of the solution and orbit analysis, but study on the theory of comparison theorem which plays an important contribution to the stochastic dominance was relatively rarely. Compare Theorem of SDE establishes pathwise almost surely dominance. That is, when a random process with probability 1 is greater than or equal to another random process, this is what we usually refer to the comparison theorem. As we all know, stochastic orders as the foundation of the dominant theory of stochastic processes has a wide range of applications in economic utility theory, insurance and reliability. Thus, there are some similarity in comparison theorem of stochastic differential equations and stochastic ordering theory to solve some issues with the same purpose. This paper aims to combine both the theoretical basis to study the nature of stochastic ordering of Ito process.This paper is divided into three parts: the first part give some conclusions which of great importance in the theory of stochastic ordering and stochastic differential equations and will play a key role in our study, including Comparison Theorem of stochastic differential equations, methods to determine the stochastic order relationship between randoms and the coupling theorem of stochastic ordering; The second part is the introduction of Ito process's stochastic ordering, we first study the stochastic order of simple Ito processes, including simple process, geometric Brownian motion, Langevin equation and the liner equation ,mainly based on the nature of stochastic order may permit; The third part is the stochastic ordering of general Ito's process, due to the solution is not explicit, therefore, using the coupling theorem of stochastic ordering and corresponding construction method and the the comparison theorem of stochastic differential equation to prove; The fourth part on the basis of the existing conclusions, first applied stochastic order of Ito process to the partial differential equations, the transfer probability density function of Ito process's solution meets the Fokker-Plank equation, so we will use the conclusion that the initial stochastic order decides the final under some conditions and then we get a general comparison theorem of partial differential equations, followed by a brief description of the application of Ito process's stochastic ordering in the stock investment and game options.