Markov-modulated Ornstein-Uhlenbeck processes
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2014
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| _version_ | 1866914824905031680 |
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| author | Huang, Gang Jansen, Marijn Mandjes, Michel Spreij, Peter De Turck, Koen |
| author_facet | Huang, Gang Jansen, Marijn Mandjes, Michel Spreij, Peter De Turck, Koen |
| contents | In this paper we consider an Ornstein-Uhlenbeck (OU) process $(M(t))_{t\geqslant 0}$ whose parameters are determined by an external Markov process $(X(t))_{t\geqslant 0}$ on a finite state space $\{1,\ldots,d\}$; this process is usually referred to as Markov-modulated Ornstein-Uhlenbeck (MMOU). 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 $t=t_1,\ldots,t_K.$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1412_7952 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Markov-modulated Ornstein-Uhlenbeck processes Huang, Gang Jansen, Marijn Mandjes, Michel Spreij, Peter De Turck, Koen Probability 60F05, 60F17 In this paper we consider an Ornstein-Uhlenbeck (OU) process $(M(t))_{t\geqslant 0}$ whose parameters are determined by an external Markov process $(X(t))_{t\geqslant 0}$ on a finite state space $\{1,\ldots,d\}$; this process is usually referred to as Markov-modulated Ornstein-Uhlenbeck (MMOU). 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 $t=t_1,\ldots,t_K.$ 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. |
| title | Markov-modulated Ornstein-Uhlenbeck processes |
| topic | Probability 60F05, 60F17 |
| url | https://arxiv.org/abs/1412.7952 |