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Spectral Representation of Gaussian Semimartingales

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  • Andreas Basse

    (University of Aarhus)

Abstract

The aim of the present paper is to characterize the spectral representation of Gaussian semimartingales. That is, we provide necessary and sufficient conditions on the kernel K for X t =∫ K t (s) dN s to be a semimartingale. Here, N denotes an independently scattered Gaussian random measure on a general space S. We study the semimartingale property of X in three different filtrations. First, the ℱ X -semimartingale property is considered, and afterwards the ℱ X,∞-semimartingale property is treated in the case where X is a moving average process and ℱ t X,∞ =σ(X s :s∈(−∞,t]). Finally, we study a generalization of Gaussian Volterra processes. In particular, we provide necessary and sufficient conditions on K for the Gaussian Volterra process ∫ −∞ t K t (s) dW s to be an ℱ W,∞-semimartingale (W denotes a Wiener process). Hereby we generalize a result of Knight (Foundations of the Prediction Process, 1992) to the nonstationary case.

Suggested Citation

  • Andreas Basse, 2009. "Spectral Representation of Gaussian Semimartingales," Journal of Theoretical Probability, Springer, vol. 22(4), pages 811-826, December.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:4:d:10.1007_s10959-009-0246-2
    DOI: 10.1007/s10959-009-0246-2
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    References listed on IDEAS

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    1. Cheridito, Patrick, 2004. "Gaussian moving averages, semimartingales and option pricing," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 47-68, January.
    2. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    3. Basse, Andreas & Pedersen, Jan, 2009. "Lévy driven moving averages and semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2970-2991, September.
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    Cited by:

    1. Tomoyuki Ichiba & Guodong Pang & Murad S. Taqqu, 2022. "Path Properties of a Generalized Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 35(1), pages 550-574, March.

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