Singular stochastic control and its relations to Dynkin game and entry-exit problems
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Zusammenfassung
We consider a bounded variation singular stochastic control problem
with value V, the associated Dynkin game with value u and an
associated entry-exit or optimal switching problem. We establish the relation
dV/dx=u known from control of Bronwian motion for a general
situation with control of a diffusion and a nonlinear cost functional
defined as solution to a BSDE. A saddle point for the Dynkin game is
given by the pair of first action times of an optimal control.
Through an impulse control approximation scheme we construct a
solution to the control problem from solutions to the entry-exit
problem, and obtain an integral representation for the value V. As
a special case we deduce equivalence of monotone control and optimal
stopping.
In a Markovian setting we characterize the value of the control
problem in n dimensions as the largest viscosity solution to a
quasilinear Hamilton-Jacobi-Bellman PDE with gradient constraints. Due
to the gradient constraints, the latter has no unique solution in
general.
The methods are from stochastic analysis and include a priori estimates, pathwise
construction,
comparison theorems for FSDE and BSDE, Ito formula for convex
functions and nonlinear Feynman-Kac
formulae. Using this approach we can drop the condition of a
``proper'' operator in the HJB PDE
and alter the standard path for comparison towards a global argument.
Zusammenfassung in einer weiteren Sprache
We consider a bounded variation singular stochastic control problem
with value V, the associated Dynkin game with value u and an
associated entry-exit or optimal switching problem. We establish the relation
dV/dx=u known from control of Bronwian motion for a general
situation with control of a diffusion and a nonlinear cost functional
defined as solution to a BSDE. A saddle point for the Dynkin game is
given by the pair of first action times of an optimal control.
Through an impulse control approximation scheme we construct a
solution to the control problem from solutions to the entry-exit
problem, and obtain an integral representation for the value V. As
a special case we deduce equivalence of monotone control and optimal
stopping.
In a Markovian setting we characterize the value of the control
problem in n dimensions as the largest viscosity solution to a
quasilinear Hamilton-Jacobi-Bellman PDE with gradient constraints. Due
to the gradient constraints, the latter has no unique solution in
general.
The methods are from stochastic analysis and include a priori estimates, pathwise
construction,
comparison theorems for FSDE and BSDE, Ito formula for convex
functions and nonlinear Feynman-Kac
formulae. Using this approach we can drop the condition of a
``proper'' operator in the HJB PDE
and alter the standard path for comparison towards a global argument.
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BOETIUS, Frederik, 2001. Singular stochastic control and its relations to Dynkin game and entry-exit problems [Dissertation]. Konstanz: University of KonstanzBibTex
@phdthesis{Boetius2001Singu-584,
year={2001},
title={Singular stochastic control and its relations to Dynkin game and entry-exit problems},
author={Boetius, Frederik},
address={Konstanz},
school={Universität Konstanz}
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<dcterms:abstract xml:lang="deu">We consider a bounded variation singular stochastic control problem<br />with value V, the associated Dynkin game with value u and an<br />associated entry-exit or optimal switching problem. We establish the relation<br />dV/dx=u known from control of Bronwian motion for a general<br />situation with control of a diffusion and a nonlinear cost functional<br />defined as solution to a BSDE. A saddle point for the Dynkin game is<br />given by the pair of first action times of an optimal control.<br />Through an impulse control approximation scheme we construct a<br />solution to the control problem from solutions to the entry-exit<br />problem, and obtain an integral representation for the value V. As<br />a special case we deduce equivalence of monotone control and optimal<br />stopping.<br /><br />In a Markovian setting we characterize the value of the control<br />problem in n dimensions as the largest viscosity solution to a<br />quasilinear Hamilton-Jacobi-Bellman PDE with gradient constraints. Due<br />to the gradient constraints, the latter has no unique solution in<br />general.<br /><br />The methods are from stochastic analysis and include a priori estimates, pathwise<br />construction,<br />comparison theorems for FSDE and BSDE, Ito formula for convex<br />functions and nonlinear Feynman-Kac<br />formulae. Using this approach we can drop the condition of a<br />``proper'' operator in the HJB PDE<br />and alter the standard path for comparison towards a global argument.</dcterms:abstract>
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