### Abstract

Original language | English |
---|---|

Pages (from-to) | 391-404 |

Journal | Insurance: Mathematics & Economics |

Volume | 47 |

Issue number | 3 |

Publication status | Published - 2010 |

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### Cite this

*Insurance: Mathematics & Economics*,

*47*(3), 391-404.

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*Insurance: Mathematics & Economics*, vol. 47, no. 3, pp. 391-404.

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

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Extending dynamic convex risk measures from discrete time to continuous time

T2 - A convergence approach

AU - Stadje, M.A.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

M3 - Article

VL - 47

SP - 391

EP - 404

JO - Insurance: Mathematics & Economics

JF - Insurance: Mathematics & Economics

SN - 0167-6687

IS - 3

ER -