## The Annals of Probability

### On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 34, Number 1 (2006), 284-320.

Dates
First available in Project Euclid: 17 February 2006

https://projecteuclid.org/euclid.aop/1140191539

Digital Object Identifier
doi:10.1214/009117905000000431

Mathematical Reviews number (MathSciNet)
MR2206349

Zentralblatt MATH identifier
1092.60030

#### Citation

Kaise, Hidehiro; Sheu, Shuenn-Jyi. On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control. Ann. Probab. 34 (2006), no. 1, 284--320. doi:10.1214/009117905000000431. https://projecteuclid.org/euclid.aop/1140191539

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