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Multivariate generalized Ornstein–Uhlenbeck processes

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  • Behme, Anita
  • Lindner, Alexander

Abstract

De Haan and Karandikar (1989) [7] introduced generalized Ornstein–Uhlenbeck processes as one-dimensional processes (Vt)t≥0 which are basically characterized by the fact that for each h>0 the equidistantly sampled process (Vnh)n∈N0 satisfies the random recurrence equation Vnh=A(n−1)h,nhV(n−1)h+B(n−1)h,nh, n∈N, where (A(n−1)h,nh,B(n−1)h,nh)n∈N is an i.i.d. sequence with positive A0,h for each h>0. We generalize this concept to a multivariate setting and use it to define multivariate generalized Ornstein–Uhlenbeck (MGOU) processes which occur to be characterized by a starting random variable and some Lévy process (X,Y) in Rm×m×Rm. The stochastic differential equation an MGOU process satisfies is also derived. We further study invariant subspaces and irreducibility of the models generated by MGOU processes and use this to give necessary and sufficient conditions for the existence of strictly stationary MGOU processes under some extra conditions.

Suggested Citation

  • Behme, Anita & Lindner, Alexander, 2012. "Multivariate generalized Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1487-1518.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1487-1518
    DOI: 10.1016/j.spa.2012.01.002
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    References listed on IDEAS

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    1. Behme, Anita & Lindner, Alexander & Maller, Ross, 2011. "Stationary solutions of the stochastic differential equation with Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 91-108, January.
    2. Lindner, Alexander & Maller, Ross, 2005. "Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1701-1722, October.
    3. Sato, Ken-iti & Yamazato, Makoto, 1984. "Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type," Stochastic Processes and their Applications, Elsevier, vol. 17(1), pages 73-100, May.
    4. de Haan, L. & Karandikar, R. L., 1989. "Embedding a stochastic difference equation into a continuous-time process," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 225-235, August.
    5. Paulsen, Jostein, 1993. "Risk theory in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 327-361, June.
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    Cited by:

    1. Behme, Anita, 2012. "Moments of MGOU processes and positive semidefinite matrix processes," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 183-197.
    2. Marko Voutilainen & Lauri Viitasaari & Pauliina Ilmonen & Soledad Torres & Ciprian Tudor, 2022. "Vector‐valued generalized Ornstein–Uhlenbeck processes: Properties and parameter estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 992-1022, September.

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