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Abstract
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.
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A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue. Methodol Comput Appl Probab 2014. [DOI: 10.1007/s11009-014-9405-8] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Blom J, De Turck K, Mandjes M. Rare Event Analysis of Markov-Modulated Infinite-Server Queues: A Poisson Limit. STOCH MODELS 2013. [DOI: 10.1080/15326349.2013.838511] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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