A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations

Jakobsen, Espen R.; Karlsen, Kenneth H.

doi:10.1007/s00030-005-0031-6

Cited by 103 publications

(158 citation statements)

References 26 publications

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“…We can also write an integral equation satisfied by this second derivative, provided that we begin at an initial time t 0 > 0 instead of 0; this equation is of the kind (60). An induction process, using Proposition 5 on the successive equations satisfied by the spatial derivatives of u, then proves that (20) holds for spatial derivatives (all the regularities and bounds we obtain are local in time, but since the time span on which they hold is controlled, we also obtain global bounds).…”

Section: Sketch Of the Proof Of Propositionmentioning

confidence: 74%

“…(16), then there exists at most one function defined on ]0, T [×R N which satisfies (20), (21) and (22).…”

Section: ∞ Estimates and Uniquenessmentioning

confidence: 99%

“…Indeed, take u a weak solution to (15) on [0, T ] given by Theorem 4. By (17), F satisfies (25) with h = Λ T ; since u satisfies (20) and (21), Proposition 2 shows that, for 0 < t (22) holds for u, and we can therefore let t ′ → 0 to deduce that (23) is valid. We have (24) by Proposition 3.…”

Section: Proof (Of Proposition 3)mentioning

confidence: 99%

“…See also [25,23,1] and more recently [6][7][8]. A general theory for non-linear integro-partial differential equations is developed by Jakobsen and Karlsen [19,20]. Some of the ideas of [15] are adapted in [18] to prove that (2) has regular solutions if λ > 1.…”

Section: Introductionmentioning

confidence: 99%

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Fractal First-Order Partial Differential Equations

Droniou

Imbert

2006

Arch Rational Mech Anal

148

222

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The present paper is concerned with semilinear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton-Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one side, the key a priori estimates for the scalar conservation law and the Hamilton-Jacobi equation and, on the other side, the smoothing effect of the operator. As far as Hamilton-Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law. Mathematical subject classifications: 35B45, 35B65, 35A35, 35S30Key words. Lévy operator -fractal conservation laws -maximum principle -non-local Hamilton-Jacobi equations -smoothing effect -a priori estimates -global-in-time existence -rate of convergence.

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Section: Sketch Of the Proof Of Propositionmentioning

confidence: 74%

“…(16), then there exists at most one function defined on ]0, T [×R N which satisfies (20), (21) and (22).…”

Section: ∞ Estimates and Uniquenessmentioning

confidence: 99%

Section: Proof (Of Proposition 3)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fractal First-Order Partial Differential Equations

Droniou

Imbert

2006

Arch Rational Mech Anal

148

222

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“…On the other hand, the viscosity solution approach to nonlocal equations is still under development and is currently an active research area, cf. for example [1,2,6,7,17,29,30,31,38,39]. Contrary to its pure PDE counterpart, the available literature applying viscosity solutions to systems of integro-PDEs is very limited, but see [5] (switching systems are not covered).…”

Section: Introductionmentioning

confidence: 99%

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

2009

Self Cite

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Abstract. We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.

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Partial Integro‐Differential Equations (PIDES)

Voltchkova

2010

Encyclopedia of Quantitative Finance

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Option pricing in models with jumps leads to the solution of partial integro‐differential equations (PIDEs). We present the specific features of the PIDEs compared with the partial differential equations. In particular, we highlight the difficulties that arise for the numerical solution of these equations. We show that the use of standard methods, such as finite differences or finite elements, is not straightforward. We then present a simple explicit–implicit finite difference scheme for pricing European vanilla and barrier options in exponential Lévy models and survey the other existing numerical techniques. We also discuss American option pricing, which leads to variational inequalities with the same integro‐differential operator.

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A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations

Cited by 103 publications

References 26 publications

Fractal First-Order Partial Differential Equations

Fractal First-Order Partial Differential Equations

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

Partial Integro‐Differential Equations (PIDES)

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