Generalized Ornstein-Uhlenbeck Processes and Extensions
Generalized Ornstein-Uhlenbeck Processes and Extensions
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广义 Ornstein-Uhlenbeck 过程和扩展
DOI：
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发表时间：
2011
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通讯作者：
Anita Behme
中科院分区：
文献类型：
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作者：
Anita Behme
广义Ornstein-Uhlenbeck过程满足二元Levy过程$(U_t，L_t){tgeq0}$的随机微分方程$dV_t=V_(t-)Du_t+dL_t$.在本文中，对于给定的二元Levy过程$(U，L)$，给出了严格存在的充要条件
得到了随机微分方程dVt=Vt-，dUt+dLt$的定常解。可能会出现非因果的解决方案。确定了定态解的形式，并证明了它在分布上是唯一的，只要它存在。对于非因果解，给出了$U$和$L$关于相应的扩展滤子保持半鞅的一个充分条件。分析了定态解的分布特性。特别地，根据过程$(U，L)，得到了期望函数和自相关函数，并在几种感兴趣的情况下描述了尾部行为。在$U$有大小为$-1$跳跃的情况下，给出了解的定律(绝对)连续的充要条件。最后，定义了一个多元广义Ornstein-Uhlenbeck过程，它由一个Levy过程$(X_t，Y_t)_{tgeq0}$驱动，其中$(X_t，Y_t)在RR^{d imes d}imes RR^d，，dgeq 1，$中。证明了这个过程$(V_T)_{tgeq0}$可解另一个Levy过程$(U_t，L_t)_{tgeq0}$的随机微分方程$dV_t=du_t V_{t-}+dL_t$，它以$(X，Y)$的形式给出。在关于$Ce(X)$极限行为的一些附加条件下，得到了严格平稳解存在的充要条件。
The generalized Ornstein-Uhlenbeck process $V_t$ fulfills the stochastic differential equation $dV_t = V_{t-} dU_t + dL_t$ for a bivariate Levy process $(U_t,L_t)_{tgeq 0}$. In this thesis, for a given bivariate Levy process $(U,L)$, necessary and sufficient conditions for the existence of a strictly
stationary solution of the stochastic differential equation $dV_t = V_{t-} , dU_t + dL_t$ are obtained. Noncausal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for $U$ and $L$ to remain semimartingales with respect to the corresponding expanded filtration is given. Distributional properties of the stationary solutions are analysed. In particular the expectation and autocorrelation function are obtained in terms of the process $(U,L)$ and in several cases of interest the tail behaviour is described. In the case where $U$ has jumps of size $-1$, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given. Finally, a multivariate generalized Ornstein-Uhlenbeck process driven by a Levy process $(X_t,Y_t)_{tgeq 0}$, with $(X_t,Y_t)in RR^{d imes d} imes RR^d,, dgeq 1,$ is defined. It is shown that this process $(V_t)_{tgeq 0}$ solves the stochastic differential equation $dV_t = dU_t V_{t-} + dL_t$ for another Levy process $(U_t,L_t)_{tgeq 0}$ in $RR^{d imes d} imes RR^d$, which is given in terms of $(X,Y)$. Under some extra conditions on the limit behaviour of $cE(X)$, necessary and sufficient conditions for the existence of strictly stationary solutions are deduced.