Convexity Preserving for Fully Nonlinear Parabolic Integro-Differential Equations

Bian, Baojun; Guan, Pengfei

doi:10.4310/maa.2008.v15.n1.a5

Cited by 10 publications

(19 citation statements)

References 16 publications

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“…(2.3) has continuous flow in the following sense. 7 Is is then easy to show, by Hölder's inequality, that estimate (2.32) holds with…”

Section: Transformed Into a Normed Space Via A Given Norm ·mentioning

confidence: 99%

“…Related to our discussion here is also [7] where convexity preservation results (in space variable) for HJB PIDEs, and their significance to financial applications are addressed, however under important restrictions: equations are linear in the integro-differential part, and the second-order fully nonlinear local part of the equation is assumed to be strictly elliptic. Finally, for reader's benefit let us mention the following references regarding semiconcavity results in space variable: for semiconcavity results (even in the generalized sense of Definition 1.1) in deterministic optimal control (or first-order Hamilton-Jacobi PDEs) see [16], for classical semiconcavity estimates in optimal control of diffusions (or second-order HJB PDEs) see [17] or [27]; semiconcavity estimates can also be proved via comparison principles as in [20,22].…”

Section: Introductionmentioning

confidence: 99%

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Generalized semiconcavity of the value function of a jump diffusion optimal control problem

Feleqi

2014

Nonlinear Differ. Equ. Appl.

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Generalized semiconcavity results for the value function of a jump diffusion optimal control problem are established, in the state variable, uniformly in time. Moreover, the semiconcavity modulus of the value function is expressed rather explicitly in terms of the semiconcavity or regularity moduli of the data (Lagrangian, terminal cost, and terms comprising the controlled SDE), at least under appropriate restrictions either on the class of the moduli, or on the SDEs. In particular, if the moduli of the data are of power type, then the semiconcavity modulus of the value function is also of power type. An immediate corollary are analogous regularity properties for (viscosity) solutions of certain integro-differential Hamilton-Jacobi-Bellman equations, which may be represented as value functions of appropriate optimal control problems for jump diffusion processes.Mathematics Subject Classification. 35D10, 35E10, 60H30, 93E20.

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“…(2.3) has continuous flow in the following sense. 7 Is is then easy to show, by Hölder's inequality, that estimate (2.32) holds with…”

Section: Transformed Into a Normed Space Via A Given Norm ·mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized semiconcavity of the value function of a jump diffusion optimal control problem

Feleqi

2014

Nonlinear Differ. Equ. Appl.

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“…The regularity is well known in the PDE theory. The key idea to improve convexity is connected to the convexity preserving and constant rank principle for solutions of PDEs, see [1,2,8,11]. The techniques used here are likely to be useful in solving other problems involving nonlinear equations.…”

Section: Smooth Solutions To Hjb Equationmentioning

confidence: 99%

Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems

Bian¹,

Miao²,

Zheng³

2011

SIAM J. Finan. Math.

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Abstract. In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is continuous on the closure of its domain. The key idea is to work on the dual control problem and the dual HJB equation. We construct a smooth, strictly convex solution to the dual HJB equation and show that its conjugate function is a smooth, strictly concave solution to the primal HJB equation satisfying the terminal and boundary conditions.

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“…Remark A.2. We say V satisfies (A.1) if V is twice differentiable in (the components of) x and once differentiable in t on Ω × (t n , t n+1 ), 4 continuous on Ω × (t n , t n+1 ], 5 and satisfies (A.1) pointwise.…”

Section: A Preservation Of Convexity and Monotonicitymentioning

confidence: 99%

The Existence of Optimal Bang-Bang Controls for GMxB Contracts

Azimzadeh¹

2015

SIAM J. Finan. Math.

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A large collection of financial contracts offering guaranteed minimum benefits are often posed as control problems, in which at any point in the solution domain, a control is able to take any one of an uncountable number of values from the admissible set. Often, such contracts specify that the holder exert control at a finite number of deterministic times. The existence of an optimal bang-bang control, an optimal control taking on only a finite subset of values from the admissible set, is a common assumption in the literature. In this case, the numerical complexity of searching for an optimal control is considerably reduced. However, no rigorous treatment as to when an optimal bang-bang control exists is present in the literature. We provide the reader with a bang-bang principle from which the existence of such a control can be established for contracts satisfying some simple conditions. The bang-bang principle relies on the convexity and monotonicity of the solution and is developed using basic results in convex analysis and parabolic partial differential equations. We show that a guaranteed lifelong withdrawal benefit (GLWB) contract admits an optimal bang-bang control. In particular, we find that the holder of a GLWB can maximize a writer's losses by only ever performing nonwithdrawal, withdrawal at exactly the contract rate, or full surrender. We demonstrate that the related guaranteed minimum withdrawal benefit contract is not convexity preserving, and hence does not satisfy the bang-bang principle other than in certain degenerate cases.

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Convexity Preserving for Fully Nonlinear Parabolic Integro-Differential Equations

Cited by 10 publications

References 16 publications

Generalized semiconcavity of the value function of a jump diffusion optimal control problem

Generalized semiconcavity of the value function of a jump diffusion optimal control problem

Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems

The Existence of Optimal Bang-Bang Controls for GMxB Contracts

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