A Pair of Explicitly Solvable Singular Stochastic Control Problems

A Pair of Explicitly Solvable Singular Stochastic Control Problems We consider a general model of singular stochastic control with infinite time horizon and we prove a ``verification theorem'' under the assumption that the Hamilton—Jacobi—Bellman (HJB) equation has a C 2 solution. In the one-dimensional case, under the assumption that the HJB equation has a solution in W loc 2,p(R) with $p \geq 1$ , we prove a very general ``verification theorem'' by employing the generalized Meyer—Ito change of variables formula with local times. In what follows, we consider two special cases which we explicitly solve. These are the formal equivalent of the one-dimensional infinite time horizon LQG problem and a simple example with radial symmetry in an arbitrary Euclidean space. The value function of either of these problems is C 2 and is expressed in terms of special functions, and, in particular, the confluent hypergeometric function and the modified Bessel function of the first kind, respectively. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

A Pair of Explicitly Solvable Singular Stochastic Control Problems

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Publisher
Springer-Verlag
Copyright
Copyright © Inc. by 1998 Springer-Verlag New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459900094
Publisher site
See Article on Publisher Site

Abstract

We consider a general model of singular stochastic control with infinite time horizon and we prove a ``verification theorem'' under the assumption that the Hamilton—Jacobi—Bellman (HJB) equation has a C 2 solution. In the one-dimensional case, under the assumption that the HJB equation has a solution in W loc 2,p(R) with $p \geq 1$ , we prove a very general ``verification theorem'' by employing the generalized Meyer—Ito change of variables formula with local times. In what follows, we consider two special cases which we explicitly solve. These are the formal equivalent of the one-dimensional infinite time horizon LQG problem and a simple example with radial symmetry in an arbitrary Euclidean space. The value function of either of these problems is C 2 and is expressed in terms of special functions, and, in particular, the confluent hypergeometric function and the modified Bessel function of the first kind, respectively.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Aug 1, 2090

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