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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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  • Stadje, Mitja, 2010. "Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 391-404, December.
    • Handle: RePEc:eee:insuma:v:47:y:2010:i:3:p:391-404
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    5. Wayne King Ming Chan, 2015. "RAROC-Based Contingent Claim Valuation," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 3-2015.
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