Systems of Bellman Equations to Stochastic Differential Games with Discount Control

Alain Bensoussan; Jens Frehse

Bollettino dell'Unione Matematica Italiana (2008)

  • Volume: 1, Issue: 3, page 663-681
  • ISSN: 0392-4041

Abstract

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We consider two dimensional diagonal elliptic systems Δ ⁢ u + a ⁢ u = H ⁢ ( x , u , ∇ ⁡ u ) which arise from stochastic differential games with discount control. The Hamiltonians H have quadratic growth in ∇ ⁡ u and a special structure which has notyet been covered by regularity theory. Without smallness condition on H , the existence of a regular solution is established.

How to cite

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Bensoussan, Alain, and Frehse, Jens. "Systems of Bellman Equations to Stochastic Differential Games with Discount Control." Bollettino dell'Unione Matematica Italiana 1.3 (2008): 663-681. <http://eudml.org/doc/290484>.

@article{Bensoussan2008,
abstract = {We consider two dimensional diagonal elliptic systems $\Delta u + au = H(x, u, \nabla u)$ which arise from stochastic differential games with discount control. The Hamiltonians $H$ have quadratic growth in $\nabla u$ and a special structure which has notyet been covered by regularity theory. Without smallness condition on $H$, the existence of a regular solution is established.},
author = {Bensoussan, Alain, Frehse, Jens},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {663-681},
publisher = {Unione Matematica Italiana},
title = {Systems of Bellman Equations to Stochastic Differential Games with Discount Control},
url = {http://eudml.org/doc/290484},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Bensoussan, Alain
AU - Frehse, Jens
TI - Systems of Bellman Equations to Stochastic Differential Games with Discount Control
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/10//
PB - Unione Matematica Italiana
VL - 1
IS - 3
SP - 663
EP - 681
AB - We consider two dimensional diagonal elliptic systems $\Delta u + au = H(x, u, \nabla u)$ which arise from stochastic differential games with discount control. The Hamiltonians $H$ have quadratic growth in $\nabla u$ and a special structure which has notyet been covered by regularity theory. Without smallness condition on $H$, the existence of a regular solution is established.
LA - eng
UR - http://eudml.org/doc/290484
ER -

References

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  2. BENSOUSSAN, ALAIN and FREHSE, JENS, Ergodic Bellman systems for stochastic games. In Differential equations, dynamical systems, and control science, volume 152 of Lecture Notes in Pure and Appl. Math., pages 411-421. Dekker, New York, 1994. Zbl0830.90142MR1243215
  3. BENSOUSSAN, A. - FREHSE, J., Ergodic Bellman systems for stochastic games in arbitrary dimension. Proc. Roy. Soc. London Ser. A, 449 (1935), 65-77, 1995. Zbl0833.90141MR1328140DOI10.1098/rspa.1995.0032
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  9. HILDEBRANDT, STEFAN, Nonlinear elliptic systems and harmonic mappings. In Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980), pages 481-615, Beijing, 1982. Science Press. MR714341
  10. JOST, JÜRGEN, Riemannian geometry and geometric analysis. Universitext. Springer-Verlag, Berlin, third edition, 2002. MR1871261DOI10.1007/978-3-662-04672-2
  11. KINDERLEHRER, DAVID - STAMPACCHIA, GUIDO, An introduction to variational inequalities and their applications, volume 88 of Pure and Applied Mathematics. Academic Press Inc.[Harcourt Brace Jovanovich Publishers], New York, 1980. Zbl0457.35001MR567696
  12. LADYZHENSKAYA, A. OLGA - NINA URAL'TSEVA, N., Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York, 1968. MR244627
  13. CHARLES MORREY, B-, Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer-VerlagNew York, Inc., New York, 1966. MR202511
  14. STAMPACCHIA, GUIDO, Équations elliptiques du second ordre á coefficients discontinus. Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965). Les Presses de l'Université de Montréal, Montreal, Que., 1966. Zbl0151.15401MR192177
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