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Teor. Veroyatnost. i Primenen., 2001, Volume 46, Issue 3, Pages 579–585 (Mi tvp3906)  

This article is cited in 26 scientific papers (total in 26 papers)

Short Communications

Time Change Representation of Stochastic Integrals

J. Kallsena, A. N. Shiryaevb

a Albert Ludwigs University of Freiburg
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: By the Dambis–Dubins–Schwarz theorem, any stochastic integral $M:=\int_0^\cdot H_sdW_s$ of Brownian motion can be written as a time-changed Brownian motion, i.e., $M=({\widehat{W}}_{\widehat{T_t}})_{t\in\mathbf{R}_+}$ for some Brownian motion $({\widehat{W}}_\theta)_{\theta\in\mathbf{R}_+}$ and some time change $({\widehat{T_t}})_{t\in\mathbf{R}_+}$. In [J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin–Heidelberg, 1987] and [O. Kallenberg, Stochastic Process. Appl., 40 (1992), pp. 199–223] it is shown that in this statement Brownian motion can be replaced with (symmetric) $\alpha$-stable Levy motion. Using the cumulant process of a semimartingale, we give new short proofs. Moreover, we show that the statement cannot be extended to any other Levy processes.

Keywords: stable Levy motions, cumulant process, stochastic integral, time change.

DOI: https://doi.org/10.4213/tvp3906

Full text: PDF file (829 kB)

English version:
Theory of Probability and its Applications, 2002, 46:3, 522–528

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Received: 04.05.2000

Citation: J. Kallsen, A. N. Shiryaev, “Time Change Representation of Stochastic Integrals”, Teor. Veroyatnost. i Primenen., 46:3 (2001), 579–585; Theory Probab. Appl., 46:3 (2002), 522–528

Citation in format AMSBIB
\by J.~Kallsen, A.~N.~Shiryaev
\paper Time Change Representation of Stochastic Integrals
\jour Teor. Veroyatnost. i Primenen.
\yr 2001
\vol 46
\issue 3
\pages 579--585
\jour Theory Probab. Appl.
\yr 2002
\vol 46
\issue 3
\pages 522--528

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    17. Kobayashi K., “Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations”, J. Theor. Probab., 24:3 (2011), 789–820  crossref  mathscinet  zmath  isi  scopus
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    20. Vakeroudis S., “on the Windings of Complex-Valued Ornstein–Uhlenbeck Processes Driven By a Brownian Motion and By a Stable Process”, Stochastics, 87:5 (2015), 766–793  crossref  mathscinet  zmath  isi  scopus
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