Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach
Stadje,M.A.
Stadje,M.A.
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 -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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Date
2010
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Research Projects
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Citation
Stadje, M A 2010, 'Extending dynamic convex risk measures from discrete time to continuous time : A convergence approach', Insurance Mathematics & Economics, vol. 47, no. 3, pp. 391-404.
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info:eu-repo/semantics/restrictedAccess