The C<sup>1,1</sup>—character of solutions of second order elliptic equations with gradient constraint

Wiegner, Michael

doi:10.1080/03605308108820181

Cited by 33 publications

(20 citation statements)

References 2 publications

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“…There are also recent applications in mathematical finance, where transactions costs are involved, see [28] and the references therein. When h is smooth, one can show that solutions are C 1,1 [29]. However, without the smoothness of h, the problem becomes much more delicate.…”

Section: (A) Gradient Constraintsmentioning

confidence: 99%

An overview of unconstrained free boundary problems

Figalli

Shahgholian

2015

Phil. Trans. R. Soc. A.

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In this paper, we present a survey concerning unconstrained free boundary problems of typewhere B 1 is the unit ball, Ω is an unknown open set, F 1 and F 2 are elliptic operators (admitting regular solutions), and S is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.

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Section: (A) Gradient Constraintsmentioning

confidence: 99%

An overview of unconstrained free boundary problems

Figalli

Shahgholian

2015

Phil. Trans. R. Soc. A.

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“…So, our main purpose is to investigate the behaviors of the free boundaries. 3 In addition, we are interested in the regularity of the solution to problem A. PDE problems related to problem A (variational inequalities with gradient constraints) have been studied by many researchers, including Evans (1979), Wiegner (1981), Ishii and Koike (1983), Hu (1986), Soner and Shreve (1991) and Zhu (1992). It can be shown that the solution to this type of problem belongs to…”

Section: Gradient Constraintsmentioning

confidence: 99%

Finite Horizon Optimal Investment and Consumption with Transaction Costs

Dai¹,

Jiang²,

Li³

et al. 2009

SIAM J. Control Optim.

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This paper concerns continuous-time optimal investment and consumption decision of a CRRA investor who faces proportional transaction costs and finite time horizon. In the no consumption case, it has been studied by Liu and Loewenstein (2002) and Dai and Yi (2006). Mathematically, it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraints. The problem gives rise to two free boundaries which stand for the optimal buying and selling strategies, respectively. We present an analytical approach to analyze the behaviors of free boundaries. The regularity of the value function is studied as well. Our approach is essentially based on the connection between singular control and optimal stopping, which is first revealed to the present problem.

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“…The previous lemma will also aid us in obtaining a pointwise bound on the second derivatives of u ǫ and thereby establish part (ii) of Propostion 3.1. To this end, we adapt the approach by M. Wiegner [12].…”

Section: Proof 1 It Suffices To Boundmentioning

confidence: 99%

“…We also use a penalization technique similar to the one introduced by L.C. Evans in [4] and refined by M. Wiegner [12] and H. Ishii et. al.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control

Hynd

2012

ESAIM: COCV

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We study the partial differential equationwhere u is the unknown function, L is a second-order elliptic operator, f is a given smooth function and H is a convex function. This is a model equation for HamiltonJacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Hölder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution is Lipschitz continuous.

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The C^1,1—character of solutions of second order elliptic equations with gradient constraint

Cited by 33 publications

References 2 publications

An overview of unconstrained free boundary problems

An overview of unconstrained free boundary problems

Finite Horizon Optimal Investment and Consumption with Transaction Costs

Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control

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The C1,1—character of solutions of second order elliptic equations with gradient constraint

Cited by 33 publications

References 2 publications

An overview of unconstrained free boundary problems

An overview of unconstrained free boundary problems

Finite Horizon Optimal Investment and Consumption with Transaction Costs

Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control

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The C^1,1—character of solutions of second order elliptic equations with gradient constraint