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Special Semester on Stochastics with Emphasis on Finance
Linz, September 2008 - December 2008
Presentation: Stochastic Target Problems with Controlled Loss. (joint work with N. Touzi and R. Elie).

Workshop: Optimization and Optimal Control

Time: Tue, October 21, 2008, 16:20-17:10

Speaker: Bruno Bouchard

Abstract

We consider the problem of finding the minimal initial
data of a controlled process which guarantees to reach a controlled target
with a given probability of success or, more generally, with a given level
of expected loss. By suitably increasing the state space and the controls,
we show that this problem can be converted into a stochastic target
problem, i.e. find the minimal initial data of a controlled process which
guarantees to reach a controlled target with probability one. Unlike the
existing literature on stochastic target problems, our increased controls
are valued in an unbounded set. In this paper, we provide a new
derivation of the dynamic programming equation for general stochastic
target problems with unbounded controls, together with the appropriate
boundary conditions. These results are applied to the problem of quantile
hedging in financial mathematics, and are shown to recover the explicit
solution of Follmer and Leukert.

Presentation slides (pdf, 177 KB)

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