Large time asymptotic problems for optimal stochastic control with superlinear cost

Ichihara, Naoyuki

doi:10.1016/j.spa.2011.12.005

Cited by 36 publications

(77 citation statements)

References 11 publications

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“…with equality if and only if ξ = D p H(x, p) or p = D ξ L(x, ξ). The model above is slightly more general than the model in [2,3]. A more restrictive assumption on H is used in [2], while H does not depend on x in [3].…”

Section: )mentioning

confidence: 99%

“…The model above is slightly more general than the model in [2,3]. A more restrictive assumption on H is used in [2], while H does not depend on x in [3]. For the properties mentioned above see As mentioned earlier, [2] imposes the assumptions in (1.2) for f , for both the subquadratic and superquadratic cases.…”

Section: )mentioning

confidence: 99%

“…(1.1) This equation which has a long history in the literature, has been studied in [2,3] for somewhat more general Hamiltonians, and was recently revisited by Barles in [1]. What is of interest here, is to characterize the solutions of (EP) which are bounded from below, that is, without loss of generality, the positive solutions.…”

Section: Introductionmentioning

confidence: 99%

“…In the superquadratic case (γ ≥ 2), [1,Theorem 2.6] shows that (EP) has a unique positive solution for any coercive f , and in addition, for this solution, λ = λ * . In the subquadratic case (γ ∈ (1, 2)), [1] adopts assumption (H2) in [2], which states that f satisfies a bound of the form c −1 |x| β − c ≤ f (x) ≤ c 1 + |x| β , and Df (x) ≤ c −1 1 + |x| (β−1) + (1.2) for some positive constants β and c for all x ∈ R d . Without enforcing this assumption, to our knowledge, there are no results on uniqueness in the literature, and therefore also no verification of optimality results.…”

Section: Introductionmentioning

confidence: 99%

“…There is substantial literature on viscous HJB equations, other than [1][2][3] mentioned above. It is not our intent to review this literature, since it does not address the problem studied in this paper, but we should at least mention [4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

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On uniqueness of solutions to viscous HJB equations with a subquadratic nonlinearity in the gradient

Arapostathis

Biswas

Caffarelli

2019

Communications in Partial Differential Equations

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Uniqueness of positive solutions to viscous Hamilton-Jacobi-Bellman (HJB) equations of the form −∆u(x) + 1 γ |Du(x)| γ = f (x) − λ, with f a coercive function and λ a constant, in the subquadratic case, that is, γ ∈ (1, 2), appears to be an open problem. Barles and Meireles [Comm. Partial Differential Equations 41 (2016)] show uniqueness in the case that f (x) ≈ |x| β and |Df (x)| |x| (β−1) + for some β > 0, essentially matching earlier results of Ichihara, who considered more general Hamiltonians but with better regularity for f . Without enforcing this assumption, to our knowledge, there are no results on uniqueness in the literature. In this short article, we show that the equation has a unique positive solution for any locally Lipschitz continuous, coercive f which satisfies Df (x) ≤ κ 1 + |f (x)| 2−1/γ for some positive constant κ. Since 2 − 1 γ > 1, this assumption imposes very mild restrictions on the growth of the potential f . We also show that this solution fully characterizes optimality for the associated ergodic problem. Our method involves the study of an infinite dimensional linear program for elliptic equations for measures, and is very different from earlier approaches. It also applies to the larger class of Hamiltonians studied by Ichihara, and we show that it is well suited to provide verification of optimality results for the associated ergodic control problems, even in a pathwise sense, and without resorting to the parabolic problem.

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