Dynamic Risk Measures And Stochastic Calculus Mitja Stadje ...
Dynamic Risk Measures and Stochastic Calculus
Mitja Stadje
Abstract:
The main aim of this talk is to present an approach for the transition from
risk measures in discrete time to their counterparts in continuous time. Af-
ter a general introduction to risk assessment in mathematical finance it is
shown that a large class of risk measures in continuous time can be ob-
tained very naturally as limits of time-consistent risk measures in a dis-
crete setting. The discrete-time risk measures are constructed from properly
rescaled (‘tilted’) one-period risk measures, using a d-dimensional random
walk converging to a Brownian Motion. Under suitable conditions (covering
the classical one-period risk measures) we obtain convergence of the discrete
risk measures to the solution of a backward stochastic differential equation,
defining a 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, Average Value at Risk, and the Gini risk measure in closed form.
This is joint work with my PhD advisor Patrick Cheridito.
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