On the Structure of Solutions of Ergodic Type Bellman Equation Related to Risk-Sensitive Control

Bellman equations of ergodic type related to risk-sensitive control are considered. We treat the case that the nonlinear term is positive quadratic form on first-order partial derivatives of solution, which includes linear exponential quadratic Gaussian control problem. In this paper we prove that t...

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Bibliographic Details
Published in:The Annals of probability 2006-01, Vol.34 (1), p.284-320
Main Authors: Kaise, Hidehiro, Sheu, Shuenn-Jyi
Format: Article
Language:English
Subjects:
60G35
60H30
93E20
Academia
classification of solutions
Coefficients
Control theory
Ergodic theory
Ergodic type Bellman equations
Exact sciences and technology
General topics
Global behavior
Investment risk
Limit theorems
Markov processes
Mathematical models
Mathematical theorems
Mathematics
Optimal control
Probabilities
Probability
Probability and statistics
Probability theory and stochastic processes
risk-senstive control
Sciences and techniques of general use
Studies
transience and ergodicity
variational representation
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Summary:Bellman equations of ergodic type related to risk-sensitive control are considered. We treat the case that the nonlinear term is positive quadratic form on first-order partial derivatives of solution, which includes linear exponential quadratic Gaussian control problem. In this paper we prove that the equation in general has multiple solutions. We shall specify the set of all the classical solutions and classify the solutions by a global behavior of the diffusion process associated with the given solution. The solution associated with ergodic diffusion process plays particular role. We shall also prove the uniqueness of such solution. Furthermore, the solution which gives us ergodicity is stable under perturbation of coefficients. Finally, we have a representation result for the solution corresponding to the ergodic diffusion.
ISSN:0091-1798
2168-894X
DOI:10.1214/009117905000000431