Criticality of viscous Hamilton–Jacobi equations and stochastic ergodic control

Ichihara, Naoyuki

doi:10.1016/j.matpur.2013.01.005

Cited by 20 publications

(33 citation statements)

References 14 publications

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“…In the rest of this section we show how the previous development can be used to obtain results analogous to those reported in [16], without imposing any smoothness assumptions on the coefficients. Withψ = − log Ψ * = −ψ * , we have − a ij ∂ ijψ − b i ∂ iψ + ψ , aψ + f = λ * (f ) .…”

Section: )mentioning

confidence: 99%

“…It is easy to see that (1.5) is related to an ergodic control problem with controlled drift b + 2au and running cost u, au − f (x). The parameter λ * (f ) can be thought of as the optimal ergodic value; see Ichihara [16]. Note then that the twisted process defined above corresponds to the optimally controlled diffusion.…”

Section: Introductionmentioning

confidence: 99%

“…Note then that the twisted process defined above corresponds to the optimally controlled diffusion. We refer to Ichihara [16,17] and Kaise and Sheu [9] for some important results in this direction. For a potential f that vanishes at infinity, Ichihara [16,18] considers the ergodic control problem in (1.5), with a more general Hamiltonian and under scaling of the potential.…”

Section: Introductionmentioning

confidence: 99%

“…Optimality for the ergodic problem is shown in [16,18] via the study of the optimal finite horizon problem (Cauchy parabolic problem). Inevitably, in doing so, optimality is shown in a certain class of controls.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Strict monotonicity of principal eigenvalues of elliptic operators in Rd and risk-sensitive control

Arapostathis

Biswas

Saha

2019

Journal de Mathématiques Pures et Appliquées

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This paper studies the eigenvalue problem on R d for a class of second order, elliptic operators of the form L f = a ij ∂ xi ∂ xj + b i ∂ xi + f , associated with non-degenerate diffusions. We show that strict monotonicity of the principal eigenvalue of the operator with respect to the potential function f fully characterizes the ergodic properties of the associated ground state diffusion, and the unicity of the ground state, and we present a comprehensive study of the eigenvalue problem from this point of view. This allows us to extend or strengthen various results in the literature for a class of viscous Hamilton-Jacobi equations of ergodic type with smooth coefficients to equations with measurable drift and potential. In addition, we establish the strong duality for the equivalent infinite dimensional linear programming formulation of these ergodic control problems. We also apply these results to the study of the infinite horizon risk-sensitive control problem for diffusions, and establish existence of optimal Markov controls, verification of optimality results, and the continuity of the controlled principal eigenvalue with respect to stationary Markov controls.

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Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Strict monotonicity of principal eigenvalues of elliptic operators in Rd and risk-sensitive control

Arapostathis

Biswas

Saha

2019

Journal de Mathématiques Pures et Appliquées

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“…Applying Itô's formula to (4.23) and following an argument similar to Lemma 3.3 we haveV (x) ≤ E x e τ 0 [c(Xs,v * 1 ,u * 2 )−Λ] ds V (Xτ) , for x ∈ B c ,(4 24). …”

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confidence: 98%

Zero-Sum Stochastic Differential Games with Risk-Sensitive Cost

Biswas

Saha

2018

Appl Math Optim

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Zero-sum games with risk-sensitive cost criterion are considered with underlying dynamics being given by controlled stochastic differential equations. Under the assumption of geometric stability on the dynamics, we completely characterize all possible saddle point strategies in the class of stationary Markov controls. In addition, we also establish existence-uniqueness result for the value function of the Hamilton-Jacobi-Isaacs equation.

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Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton–Jacobi equations

Chasseigne

Ichihara

2016

Nonlinear Differ. Equ. Appl.

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This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m > 2. We prove that the generalized principal eigenvalue of the equation converges to a constant as m → ∞, and that the limit coincides with the generalized principal eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized principal eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 < m < ∞, and the limiting case m = ∞.

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Criticality of viscous Hamilton–Jacobi equations and stochastic ergodic control

Cited by 20 publications

References 14 publications

Strict monotonicity of principal eigenvalues of elliptic operators in Rd and risk-sensitive control

Strict monotonicity of principal eigenvalues of elliptic operators in Rd and risk-sensitive control

Zero-Sum Stochastic Differential Games with Risk-Sensitive Cost

Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton–Jacobi equations

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