Viscosity solutions of nonlinear integro-differential equations

Alvarez, Olivier; Tourin, Agnès

doi:10.1016/s0294-1449(16)30106-8

Cited by 126 publications

(202 citation statements)

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“…Now we turn to the existence of viscosity solutions of the system of IPDEs (1.1); we will use Perron's method as developed by Ishii [25] and its adaptation to the scalar nonlocal equations by Alavarez & Tourin [1]. Different from [1], we face a system of equations and an unbounded Lévy measure ν.…”

Section: By (A1) and (A4) It Follows Thatmentioning

confidence: 99%

“…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%

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

2009

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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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Section: By (A1) and (A4) It Follows Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2009

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Fractal First-Order Partial Differential Equations

Droniou

Imbert

2006

Arch Rational Mech Anal

150

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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“…When the Lévy measure is bounded, general existence and comparison/uniqueness results for semicontinuous unbounded viscosity solutions of second order degenerate parabolic integro-partial differential equations are given in [1,2,3]. When the Lévy measure is unbounded near the origin, the existence and uniqueness of unbounded viscosity solutions of (systems of) semilinear degenerate parabolic integro-partial differential equations in R n is proved in [4].…”

Section: Introductionmentioning

confidence: 99%

Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

Amadori

Karlsen

Chioma

2004

Stochastics and Stochastic Reports

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Abstract. We prove a comparison principle for unbounded semicontinuous viscosity sub-and supersolutions of nonlinear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in which geometric Lévy processes act as the underlying stochastic processes for the assets dynamics. As a consequence of the "geometric form" of these processes, the comparison principle holds without assigning spatial boundary data. We present applications of our result to (i) backward stochastic differential equations and (ii) pricing of European and American derivatives via backward stochastic differential equations. Regarding (i), we extend previous results on backward stochastic differential equations in a Lévy setting and the connection to semilinear integro-partial differential equations.

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Viscosity solutions of nonlinear integro-differential equations

Cited by 126 publications

References 10 publications

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

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

Fractal First-Order Partial Differential Equations

Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

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