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A conditional dichotomy theorem for stochastic processes with independent increments

Author

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  • Brockett, Patrick L.
  • Tucker, Howard G.

Abstract

Let X(t) and Y(t) be two stochastically continuous processes with independent increments over [0, T] and Lévy spectral measures Mt and Nt, respectively, and let the "time-jump" measures M and N be defined over [0, T] - [-45 degree rule]{0} by M((t1, t2] - A) = Mt2(A) - Mt1(A) and N((T1, t2] - A) = Nt2(A) - Nt1(A). Under the assumption that M is equivalent to N, it is shown that the measures induced on function space by X(t) and Y(t) are either equivalent or orthogonal, and necessary and sufficient conditions for equivalence are given. As a corollary a complete characterization of the set of admissible translates of such processes is obtained: a function f is an admissible translate for X(t) if and only if it is an admissible translate for the Gaussian component of X(t). In particular, if X(t) has no Gaussian component, then every nontrivial translate of X(t) is orthogonal to it.

Suggested Citation

  • Brockett, Patrick L. & Tucker, Howard G., 1977. "A conditional dichotomy theorem for stochastic processes with independent increments," Journal of Multivariate Analysis, Elsevier, vol. 7(1), pages 13-27, March.
    • Handle: RePEc:eee:jmvana:v:7:y:1977:i:1:p:13-27
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    Cited by:

    1. Fred Espen Benth & Claudia Kluppelberg & Gernot Muller & Linda Vos, 2012. "Futures pricing in electricity markets based on stable CARMA spot models," Papers 1201.1151, arXiv.org.
    2. Rosinski, Jan, 2007. "Tempering stable processes," Stochastic Processes and their Applications, Elsevier, vol. 117(6), pages 677-707, June.

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