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An S-related DCV generated by a convex function in a jump market

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2010

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Xiong, Dewen

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Stochastic analysis and applications. 2010, 28(2), pp. 202-225. Available under: doi: 10.1080/07362990903546389

Zusammenfassung

We consider an incomplete market with general jumps, in which the discounted price process S of a risky asset is a given bounded semimartingale. We continue our work on the S-related dynamic convex valuation (DCV) initiated in Xiong and Kohlmann [23] by considering here an S-related DCV Cĝ whose dynamic penalty functional αĝ is generated by a convex function ĝ. So the penalty Junctional takes the following form is the density process of an equivalent martingale measure (EMM) Q for S with respect to the fundamental EMM Q0. For a given ∈ L∞ (FT), we prove that (Cĝ(ξ) is the first component of the minimal bounded solution of a backward semimartingale equation (BSE) generated by a convex, possibly non-Lipschitz g. If this BSE has a bounded solution (Y, θ1, θ2, L) such that θ2 is also bounded and 〈L〉T ∈ L∞ (FT), we prove that Cĝt(ξ) = Yt, Q0-a.s., for all t ∈ [0, T]. Finally, we introduce the concept of a bounded Cĝ-(super-)martingale and derive a decomposition for a Cĝ-supermartingale.

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510 Mathematik

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Backward semimartingale equation (BSE), Dynamic convex risk measure, Dynamic convex valuation (DCV), Time-consistent property

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ISO 690XIONG, Dewen, Michael KOHLMANN, 2010. An S-related DCV generated by a convex function in a jump market. In: Stochastic analysis and applications. 2010, 28(2), pp. 202-225. Available under: doi: 10.1080/07362990903546389
BibTex
@article{Xiong2010Srela-835,
  year={2010},
  doi={10.1080/07362990903546389},
  title={An S-related DCV generated by a convex function in a jump market},
  number={2},
  volume={28},
  journal={Stochastic analysis and applications},
  pages={202--225},
  author={Xiong, Dewen and Kohlmann, Michael}
}
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