Markov-modulated Ornstein–Uhlenbeck processes
Markov-modulated Ornstein–Uhlenbeck processes
复制标题
DOI：
10.1017/apr.2015.15
复制
发表时间：
2014-12
影响因子：
1.2
通讯作者：
Gang Huang;Marijn Jansen;M. Mandjes;P. Spreij;K. D. Turck
中科院分区：
文献类型：
--
作者：
Gang Huang;Marijn Jansen;M. Mandjes;P. Spreij;K. D. Turck
本文考虑有限状态空间{1，.上的Ornstein-Uhlenbeck过程（M（t））t≥0，其参数由外Markov过程（X（t））t≥0确定。. ., d};该过程通常被称为马尔可夫调制的奥恩斯坦-乌伦贝克。我们使用随机积分理论来确定M（t）的均值和方差的显式表达式。然后，我们建立了一个偏微分方程组（PDE）的拉普拉斯变换的M（t）和状态X（t）的背景过程，共同为时间epoch t = t1，. . .,塔卡然后我们使用这个偏微分方程建立一个递归，产生M（t）的所有矩及其平稳对应;我们还找到了M（t）和M（t + u）之间的协方差表达式。然后，我们建立了一个功能的中心极限定理M（t）的情况下，某些参数的潜在的马尔可夫过程的缩放，结合调制马尔可夫过程被加速;有趣的是，特定的缩放导致截然不同的限制过程。最后，我们考虑的情况下，一个单一的马尔可夫过程调制多个马尔可夫过程的文件。
Abstract In this paper we consider an Ornstein–Uhlenbeck (OU) process (M(t))t≥0 whose parameters are determined by an external Markov process (X(t))t≥0 on a finite state space {1, . . ., d}; this process is usually referred to as Markov-modulated Ornstein–Uhlenbeck. We use stochastic integration theory to determine explicit expressions for the mean and variance of M(t). Then we establish a system of partial differential equations (PDEs) for the Laplace transform of M(t) and the state X(t) of the background process, jointly for time epochs t = t1, . . ., tK. Then we use this PDE to set up a recursion that yields all moments of M(t) and its stationary counterpart; we also find an expression for the covariance between M(t) and M(t + u). We then establish a functional central limit theorem for M(t) for the situation that certain parameters of the underlying OU processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, specific scalings lead to drastically different limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple OU processes.