Time Change Representation of Stochastic Integrals

By Jan Kallsen, A. Shiraev. 2002

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Esscher transform and the duality principle for multidimensional semimartingales (2008, methodology): “(3.8) Now, we can apply the stochastic logarithm on both sides of (3.8) since K(ei ) > −1 [cf. Kallsen and Shiryaev (2002), page 405] which leads to the required result.”
TIME‐CHANGED ORNSTEIN–UHLENBECK PROCESSES AND THEIR APPLICATIONS IN COMMODITY DERIVATIVE MODELS (2012, background): “tion G t:= F Tt . It is clear that Tand Zare adapted to (G t) t0. Dene a new process Y by Y t:= X Tt , with Y 0 = y 0 = x 0. From Jacod (1979) Corollary 10.12, Y is a (G t)-semimartingale, and from Kallsen and Shiryaev (2002) Lemma 5 it admits the following local characteristics (B;C;): B t(!) = Z t”
Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations (2009, background): “In the literature, Jacod [6], Kallsen and Shiryaev [8] use the expression (Tt)-adapted in describing a process being in synchronization with a time-change (Tt).”
A Didactic Note on Affine Stochastic Volatility Models (2006, methodology): “The process X̃ is called the exponential transform of X in Kallsen & Shiryaev (2002a).”
Power Variation and Time Change (2006, unknown): “For a proof and the history of these result, see Kallsen and Shiryaev (2002). These authors also show that, in essence, the results cannot be extended to more general Lévy processes.”