随机微分方程解的比较定理与解的轨道唯一性

随机微分方程解的比较定理与解的轨道唯一性 ON THE COMPARISON THEOREM AND PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS 许明浩 Xu Minghao first-author 本文论论了如下型随机微分方程dx t =σ（t,x t ）dB t +b（t,x t ）dt （1） 此处Bt是Brown运动,建立了关于（1）的解的比较定理,在此基础上,并利用常微分方程的积分不等式,得到了保证（1）的解轨道唯一的充分条件。后来又讨论了随机微分方程组 dx=∑（t,x t ）dB t +b（i）（t,x t ）dt （i=1,2,…,n） （2）也得到了保证方程组（2）的解轨道唯一的充分条件。 In this paper, We consider the following stochastic differential equation dx t =σ（t, x t ）dB t +b（t, x t ）dt, （1） where B t is a Brownian motion and obtain the following conclusion: Suppose that σ（t, x）and b（t, x）are continuous on [0,∞)×R and the following conditions are satisfied: ⅰ) there exists a strictly increasing function ρ（u） on [0, ∞) and a square integrabla function φ（t） on [0, T] for all T > 0, such that and that |σ（t, x）-σ（t, y）|≤φ（t）σ（|x-y|） for all x, y∈R and all t∈[0,∞); ⅱ) there exists a continuous function g（t,u） on [0,∞)×[0,∞), such that |b（t, x）-b（t, y）|≤g（t,|x-y|） for all x, y∈R and all t∈[0,∞), where g（t, 0）≡0, g（t, u） is nondeereasing and concave in u for each fixed t, and that the following initial value problem du/dt=g（t, u）, u（o）=0 （2） has a unique solution u（t）≡0 on [0, ∞), then the pathwise uniqueness of solutions for equation （1） is established. In this paper, we also investigate the following stochstic differential system dx t i =σ j i （t, x t ）dB t i +b i （t, x t ）dt （i=1, 2,…, n） （3） and obtain some conditions of the pathwise uniqueness of solutions for system （3）. 1985-02-01 2021-04-01 2