Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data

Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o (1+| x | p ) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like O (1+| x | p ) at infinity. This latter case encompasses some equations related to backward stochastic differential equations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data

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Publisher
Springer-Verlag
Copyright
Copyright © 2011 by Springer Science+Business Media, LLC
Subject
Mathematics; Mathematical Methods in Physics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Numerical and Computational Physics; Theoretical, Mathematical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-010-9122-9
Publisher site
See Article on Publisher Site

Abstract

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o (1+| x | p ) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like O (1+| x | p ) at infinity. This latter case encompasses some equations related to backward stochastic differential equations.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2011

References

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