A worst-case optimization approach to impulse perturbed stochastic control with application to financial risk management
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Date
2012-08-07
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Abstract
This work presents the main ideas, methods and results of the theory of impulse perturbed
stochastic control as an extension of the classic stochastic control theory. Apart from the introduction
and the motivation of the basic concept, two stochastic optimization problems are
the focus of the investigations. On the one hand we consider a differential game as analogue
of the expected utility maximization problem in the situation with impulse perturbation,
and on the other hand we study an appropriate version of a target problem. By dynamic
optimization principles we characterize the associated value functions by systems of partial
differential equations (PDEs). More precisely, we deal with variational inequalities whose
single inequalities comprise constrained optimization problems, where the corresponding admissibility
sets again are given by the seeked value functions. Using the concept of viscosity
solutions as weak solutions of PDEs, we avoid strong regularity assumptions on the value
functions. To use this concept as sufficient verification method, we additionally have to prove
the uniqueness of the solutions of the PDEs.
As a second major part of this work we apply the presented theory of impulse perturbed
stochastic control in the field of financial risk management where extreme events have to be
taken into account in order to control risks in a reasonable way. Such extreme scenarios are
modelled by impulse controls and the financial decisions are made with respect to the worstcase
scenario. In a first example we discuss portfolio problems as well as pricing problems on
a capital market with crash risk. In particular, we consider the possibility of trading options
and study their in
uence on the investor's performance measured by the expected utility of
terminal wealth. This brings up the question of crash-adjusted option prices and leads to
the introduction of crash insurance. The second application concerns an insurance company
which faces potentially large losses from extreme damages. We propose a dynamic model
where the insurance company controls its risk process by reinsurance in form of proportional
reinsurance and catastrophe reinsurance. Optimal reinsurance strategies are obtained by
maximizing expected utility of the terminal surplus value and by minimizing the required
capital reserves associated to the risk process.
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Keywords
Crash hedging, Impulse perturbed stochastic control, Optimal reinsurance, Option pricing under crash risk, Worst-case optimization