Stability of Regime-Switching Jump Diffusions
Stability of Regime-Switching Jump Diffusions
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DOI：
10.1137/080738301
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发表时间：
2010-06
期刊：
影响因子：
--
通讯作者：
G. Yin;F. Xi
中科院分区：
文献类型：
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作者：
G. Yin;F. Xi
研究了一类切换跳扩散过程的稳定性。所考虑的过程可以被认为是由随机开关器件调制的跳跃扩散过程。我们研究的动机源于通信系统，柔性制造和生产计划，金融工程和经济学中的广泛应用。本文考虑的两分量过程$（X（t），\alpha（t））$的一个显著特征是转换过程$\alpha（t）$依赖于$X（t）$。本文主要研究了切换跳跃扩散的长时间行为，即稳定性。首先回顾正则性和稳定性的定义。其次，它表明，在适当的条件下，底层系统是正规的或没有有限的爆炸时间。为了研究平凡解（或平衡点0）的稳定性，考虑了在0邻域内可线性化（在x变量中）的系统。得到了稳定性和不稳定性的充分条件。然后，几乎必然稳定性检查通过处理的李雅普诺夫指数。由于漂移和扩散系数相关的最大和最小特征值，稳定性条件为稳定性和不稳定性提供了一个缺口。为了弥补这一差距，利用变换技术得到了稳定性的充分必要条件。
This work is concerned with the stability of a class of switching jump-diffusion processes. The processes under consideration can be thought of as a number of jump-diffusion processes modulated by a random switching device. The motivation of our study stems from a wide range of applications in communication systems, flexible manufacturing and production planning, financial engineering, and economics. A distinct feature of the two-component process $(X(t),\alpha(t))$ considered in this paper is that the switching process $\alpha(t)$ depends on the $X(t)$ process. This paper focuses on the long-time behavior, namely, stability of the switching jump diffusions. First, the definitions of regularity and stability are recalled. Next it is shown that under suitable conditions, the underlying systems are regular or have no finite explosion time. To study stability of the trivial solution (or the equilibrium point 0), systems that are linearizable (in the $x$ variable) in a neighborhood of 0 are considered. Sufficient conditions for stability and instability are obtained. Then, almost sure stability is examined by treating a Lyapunov exponent. The stability conditions present a gap for stability and instability owing to the maximum and minimal eigenvalues associated with the drift and diffusion coefficients. To close the gap, a transformation technique is used to obtain a necessary and sufficient condition for stability.