Workshop: Optimization and Optimal Control
Time: Wed, October 22, 2008, 09:30-10:20
Speaker: Huyên Pham
Abstract
The theory of impulse and switching control provides a suitable mathematical framework
for various contexts in finance. Indeed, an essential feature of these control problems
is the choice of a sequence of intervention times associated to a sequence of actions,
which result in costs. Thus, impulse control problems arise naturally in portfolio
selection under transaction costs where trading times do not accumulate continuously
but are discrete. Another important financial application of switching control concerns
real options problem for the valuation of firm's investment (energy station, oil tanker,
etc ...) under uncertainty, or the pricing of swing options.
A classical approach for studying impulse/switching control problems is the dynamic
programming method, which relates these control problems to a system of quasivariational
inequalities (QVIs). The main theoretical and numerical difficulty for solving
these QVIs comes from the nonlocal term in the obstacle part of the variational
inequality, and a standard approach is to approximate the impulse control problem by
a sequence of iterated optimal stopping problems.
In this talk, we shall present a new approach for dealing with general QVIs, which
includes in particular impulse control problems. We introduce backward stochastic differential
equations (BSDEs) driven by Brownian motion and Poisson random measure,
and subject to constraints on the jump component. Existence and uniqueness of a
minimal solution is proved by a penalization approach. We show under mild conditions
that the minimal solution to these constrained BSDEs is characterized as the unique
viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic
representation for solutions to QVIs. Such a representation in particular gives a new
stochastic formula for value functions of impulse control problems. As a consequence,
we obtain a numerical scheme for QVIs by simulation of the penalized BSDEs.
This is based on joint work with I. Kharroubi, J. Ma and J. Zhang.
Presentation slides (pdf, 575 KB)
URL: www.ricam.oeaw.ac.at/specsem/sef/events/program/presentation.php
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