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Cox Point Processes Driven by Ornstein–Uhlenbeck Type Processes

Author

Listed:
  • R. Lechnerová

    (Private College of Economic Studies, Ltd.)

  • K. Helisová

    (Charles University in Prague)

  • V. Beneš

    (Charles University in Prague)

Abstract

The paper is devoted to the development of Cox point processes driven by nonnegative processes of Ornstein–Uhlenbeck (OU) type. Starting with multivariate temporal processes we develop formula for the cross pair correlation function. Further filtering problem is studied by means of two different approaches, either with discretization in time or through the point process densities with respect to the Poisson process. The first approach is described mainly analytically while in the second case we obtain numerical solution by means of MCMC. The Metropolis–Hastings birth–death chain for filtering can be also used when estimating the parameters of the model. In the second part we try to develop spatial and spatio-temporal Cox point processes driven by a stationary OU process. The generating functional of the point process is derived which enables evaluation of basic characteristics. Finally a simulation algorithm is given and applied.

Suggested Citation

  • R. Lechnerová & K. Helisová & V. Beneš, 2008. "Cox Point Processes Driven by Ornstein–Uhlenbeck Type Processes," Methodology and Computing in Applied Probability, Springer, vol. 10(3), pages 315-335, September.
    • Handle: RePEc:spr:metcap:v:10:y:2008:i:3:d:10.1007_s11009-007-9055-1
    DOI: 10.1007/s11009-007-9055-1
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    References listed on IDEAS

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    1. Anders Brix & Peter J. Diggle, 2001. "Spatiotemporal prediction for log‐Gaussian Cox processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(4), pages 823-841.
    2. Karr, Alan F., 1983. "State estimation for cox processes on general spaces," Stochastic Processes and their Applications, Elsevier, vol. 14(3), pages 209-232, March.
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