On a continuous analogue of the stochastic difference equation Xn = ρ X n–1 + Bn
On a continuous analogue of the stochastic difference equation Xn = ρ X n–1 + Bn
复制标题
随机差分方程的连续模拟 Xn = ρ X n–1 + Bn
DOI：
10.2307/1426557
复制
发表时间：
1980
影响因子：
1.2
通讯作者：
S. Wolfe
中科院分区：
文献类型：
--
作者：
S. Wolfe
设 Bi, B 2 ,••• 为独立同分布随机变量序列，设 X, 为独立于 B 的随机变量；对于 n ~ 1，令 p 为常数，使得 0 < p < 1 并令 Xl' X 2 ,•• 为由关系式 X, = pXn l + B n 递归定义的另一个随机变量序列。在本文中，我们让{B(t): O~t<oo}是一个具有独立、同质增量的随机过程，并定义另一个随机过程{Xir): 0 ~ t ~ oo}，它与随机过程{B(t): 0 ~ t < oo}具有相同的关系，因为随机变量序列Xj, X 2 , ••• 代表Bj, B 2 , •••• 可以看出，X(t)当且仅当 E[Iog+ IB(1)\] < 00 时，收敛于随机变量 X t~ +00，此时 X 具有 L 类分布函数。还得到了其他几个相关结果。给出了离散时间随机过程可嵌入连续时间随机过程的充要条件。
Let Bi, B 2 , ••• be a sequence of independent,_ identically distributed random variables, let X, be a random variable that is independent of B; for n ~ 1, let p be a constant such that 0 < p < 1 and let Xl' X 2 , ••• be another sequence of random variables that are defined recursively by the relationships X, = pXn l + B n • It can be shown that the sequence of random variables Xj, X 2 , ••• converges in law to a random variable X if and only if E[Iog+ IBll]<oo. In this paper we let {B(t): O~t<oo} be a stochastic process with independent, homogeneous increments and define another stochastic process {Xir): 0 ~ t ~ oo} that stands in the same relationship to the stochastic process {B(t): 0 ~ t < oo} as the sequence of random variables Xj, X 2 , ••• stands to Bj, B 2 , •••• It is shown that X(t) converges in law to a random variable X as t~ +00 if and only if E[Iog+ IB(1)\] < 00 in which case X has a distribution function of class L. Several other related results are obtained. A necessary and sufficient condition is given for a discrete-time stochastic process to be embeddable in a continuous-time stochastic process.