扩散过程的随机序及其在偏微分方程上的应用（英文）

Stochastic Ordering of Diffusion Process and Application on Partial Differential Equations

随机序用于比较分布函数的中心位置和分散程度,而这两个特征反映了两个变量或随机过程的大小关系.由于随机过程的不确定性性质,其随机序的研究相对较为困难.因此,本文旨在分析扩散过程随机序关系,以随机微分方程为媒介,利用条件期望的性质,直接证明了扩散过程的强序、增凸序、增凹序及Laplace-Stieltjes转移序的性质.然后将随机序方法应用到扩散过程的Fokker-Planck方程中,验证了一类偏微分方程解的弱比较定理.

Stochastic ordering could be used to compare both the location and the dispersion of distribution function which reflect the magnitude relation between two random variables or stochastic processes. The latter is more complex since its properties are more complicated. Therefore, the aim of this paper is analysing the relations of stochastic orderings of diffusion processes. Focused on comparing the magnitude and variability of diffusion processes, some properties are demonstrated about strong ordering, increasing convex ordering, increasing concave ordering and Laplace-Stieltjes transform ordering of diffusion processes which are defined as stochastic differential equations respectively. Then the results are applied to partial differential equation and the weak comparison theorem is proved since the densities of diffusion processes satisfies Fokker-Planck equations.