论文标题
Markov改编的广义Ornstein-Uhlenbeck过程和风险理论的应用
Markov-modulated generalized Ornstein-Uhlenbeck processes and an application in risk theory
论文作者
论文摘要
我们通过在连续时间内嵌入Markov调制的随机复发方程来得出Markov改编的广义Ornstein-Uhlenbeck过程。事实证明，所获得的过程是由双变量马尔可夫添加过程驱动的特定随机微分方程的独特解。我们在驱动马尔可夫添加过程方面明确介绍了这种随机微分方程及其解决方案。此外，我们为马尔可夫修饰的广义ornstein-uhlenbeck过程提供了必要和充分的条件，并证明其固定分布是通过Markov添加过程的特定指数功能给出的。最后，我们提出了马尔可夫调制的广义Ornstein-Uhlenbeck过程作为随机投资的Markov调制风险模型。这将Paulsen的风险过程推广到了马尔可夫开关环境。我们在此风险模型中得出了一个公式，该公式以马尔可夫添加过程的指数函数的分布来表达破坏概率。
We derive the Markov-modulated generalized Ornstein-Uhlenbeck process by embedding a Markov-modulated random recurrence equation in continuous time. The obtained process turns out to be the unique solution of a certain stochastic differential equation driven by a bivariate Markov-additive process. We present this stochastic differential equation as well as its solution explicitely in terms of the driving Markov-additive process. Moreover, we give necessary and sufficient conditions for strict stationarity of the Markov-modulated generalized Ornstein-Uhlenbeck process, and prove that its stationary distribution is given by the distribution of a specific exponential functional of Markov-additive processes. Finally we propose an application of the Markov-modulated generalized Ornstein-Uhlenbeck process as Markov-modulated risk model with stochastic investment. This generalizes Paulsen's risk process to a Markov-switching environment. We derive a formula in this risk model that expresses the ruin probability in terms of the distribution of an exponential functional of a Markov-additive process.
