Singular stochastic control and its relations to Dynkin game and entry-exit problems ; Singuläre stochastische Kontrolle und ihre Beziehungen zu Dynkin-Spiel- und -Eintritt-Austritt-Problemen
Titel:
Singular stochastic control and its relations to Dynkin game and entry-exit problems ; Singuläre stochastische Kontrolle und ihre Beziehungen zu Dynkin-Spiel- und -Eintritt-Austritt-Problemen
Beteiligte:
Boetius, Frederik
[VerfasserIn]
Erschienen:
KOPS - The Institutional Repository of the University of Konstanz, 2001
Anmerkungen:
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Beschreibung:
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. ; published