Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach
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.
Year of publication: |
2010
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Authors: | Stadje, Mitja |
Published in: |
Insurance: Mathematics and Economics. - Elsevier, ISSN 0167-6687. - Vol. 47.2010, 3, p. 391-404
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Publisher: |
Elsevier |
Keywords: | IM 10 IM 30 IE12 Dynamic convex risk measures Time-consistency g-expectation Discretization Convergence Special drivers |
Saved in:
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