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Generalized Gaussian bridges

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  • Sottinen, Tommi
  • Yazigi, Adil

Abstract

A generalized bridge is a stochastic process that is conditioned on N linear functionals of its path. We consider two types of representations: orthogonal and canonical. The orthogonal representation is constructed from the entire path of the process. Thus, the future knowledge of the path is needed. In the canonical representation the filtrations of the bridge and the underlying process coincide. The canonical representation is provided for prediction-invertible Gaussian processes. All martingales are trivially prediction-invertible. A typical non-semimartingale example of a prediction-invertible Gaussian process is the fractional Brownian motion. We apply the canonical bridges to insider trading.

Suggested Citation

  • Sottinen, Tommi & Yazigi, Adil, 2014. "Generalized Gaussian bridges," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3084-3105.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:9:p:3084-3105
    DOI: 10.1016/j.spa.2014.04.002
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    References listed on IDEAS

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    1. Alili, Larbi & Wu, Ching-Tang, 2009. "Further results on some singular linear stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1386-1399, April.
    2. Peter Imkeller, 2003. "Malliavin's Calculus in Insider Models: Additional Utility and Free Lunches," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 153-169, January.
    3. Hoyle, Edward & Hughston, Lane P. & Macrina, Andrea, 2011. "Lévy random bridges and the modelling of financial information," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 856-884, April.
    4. Campi, Luciano & Çetin, Umut & Danilova, Albina, 2011. "Dynamic Markov bridges motivated by models of insider trading," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 534-567, March.
    5. Baudoin, Fabrice, 0. "Conditioned stochastic differential equations: theory, examples and application to finance," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 109-145, July.
    6. Campi, Luciano & Cetin, Umut & Danilova, Albina, 2011. "Dynamic Markov bridges motivated by models of insider trading," LSE Research Online Documents on Economics 31538, London School of Economics and Political Science, LSE Library.
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    Cited by:

    1. Tommi Sottinen & Lauri Viitasaari, 2018. "Parameter estimation for the Langevin equation with stationary-increment Gaussian noise," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 569-601, October.
    2. Mengütürk, Levent Ali, 2018. "Gaussian random bridges and a geometric model for information equilibrium," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 465-483.
    3. Abel Azze & Bernardo D'Auria & Eduardo Garc'ia-Portugu'es, 2022. "Optimal stopping of Gauss-Markov bridges," Papers 2211.05835, arXiv.org, revised Jul 2024.
    4. Tommi Sottinen & Lauri Viitasaari, 2019. "Prediction Law of Mixed Gaussian Volterra Processes," Papers 1904.09799, arXiv.org.
    5. Levent Ali Mengütürk, 2023. "From Irrevocably Modulated Filtrations to Dynamical Equations Over Random Networks," Journal of Theoretical Probability, Springer, vol. 36(2), pages 845-875, June.
    6. Maleki Almani, Hamidreza & Shokrollahi, Foad & Sottinen, Tommi, 2024. "Prediction of Gaussian Volterra processes with compound Poisson jumps," Statistics & Probability Letters, Elsevier, vol. 208(C).
    7. Sottinen, Tommi & Viitasaari, Lauri, 2017. "Prediction law of fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 155-166.

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