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Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach

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  • Stadje, Mitja

Abstract

We present an approach for the transition from convex risk measures in a certain discrete time setting to their counterparts in continuous time. The aim of this paper is to show that a large class of convex risk measures in continuous time can be obtained as limits of discrete time-consistent convex risk measures. The discrete time risk measures are constructed from properly rescaled ([`]tilted') one-period convex risk measures, using a d-dimensional random walk converging to a Brownian motion. Under suitable conditions (covering many standard one-period risk measures) we obtain convergence of the discrete risk measures to the solution of a BSDE, defining a convex risk measure in continuous time, whose driver can then be viewed as the continuous time analogue of the discrete [`]driver' characterizing the one-period risk. We derive the limiting drivers for the semi-deviation risk measure, Value at Risk, Average Value at Risk, and the Gini risk measure in closed form.

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Bibliographic Info

Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

Volume (Year): 47 (2010)
Issue (Month): 3 (December)
Pages: 391-404

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Handle: RePEc:eee:insuma:v:47:y:2010:i:3:p:391-404

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Web page: http://www.elsevier.com/locate/inca/505554

Related research

Keywords: IM 10 IM 30 IE12 Dynamic convex risk measures Time-consistency g-expectation Discretization Convergence Special drivers;

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Citations

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Cited by:
  1. Pelsser, A. & Stadje, M.A., 2012. "Time-Consistent and Market-Consistent Evaluations (Revised version of CentER DP 2011-063)," Discussion Paper 2012-086, Tilburg University, Center for Economic Research.
  2. Stadje, M.A. & Pelsser, A., 2014. "Time-Consistent and Market-Consistent Evaluations (Revised version of 2012-086)," Discussion Paper 2014-002, Tilburg University, Center for Economic Research.
  3. Tianxiao Wang, 2012. "Risk minimizing of derivatives via dynamic g-expectation and related topics," Papers 1208.2068, arXiv.org.
  4. Ronnie Sircar & Stephan Sturm, 2011. "From Smile Asymptotics to Market Risk Measures," Papers 1107.4632, arXiv.org, revised Jul 2012.
  5. Dilip Madan & Martijn Pistorius & Mitja Stadje, 2013. "On consistent valuations based on distorted expectations: from multinomial random walks to L\'{e}vy processes," Papers 1301.3531, arXiv.org.
  6. Babacar Seck & Robert J. Elliott & Jean-Pierre Gueyie, 2013. "Computational Dynamic Market Risk Measures in Discrete Time Setting," Papers 1306.5705, arXiv.org.

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