Extending dynamic convex risk measures from discrete time to continuous time

A convergence approach

M.A. Stadje

Research output: Contribution to journalArticleScientificpeer-review

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.
Original languageEnglish
Pages (from-to)391-404
JournalInsurance: Mathematics & Economics
Volume47
Issue number3
Publication statusPublished - 2010

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Convex Risk Measures
Risk Measures
Continuous Time
Discrete-time
Driver
Value at Risk
Brownian motion
Random walk
Closed-form
Covering
Deviation
Limiting
Risk measures
Convex risk measures
Continuous time
Analogue

Cite this

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Extending dynamic convex risk measures from discrete time to continuous time : A convergence approach. / Stadje, M.A.

In: Insurance: Mathematics & Economics, Vol. 47, No. 3, 2010, p. 391-404.

Research output: Contribution to journalArticleScientificpeer-review

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